From be1c7c6fc18326d40bf4cab905d55c755b6804d0 Mon Sep 17 00:00:00 2001
From: EeethB <59264453+EeethB@users.noreply.github.com>
Date: Mon, 11 Sep 2023 22:04:17 +0000
Subject: [PATCH] =?UTF-8?q?Deploying=20to=20gh-pages=20from=20@=20Gilead-B?=
 =?UTF-8?q?ioStats/graphicalMCP@669d503c9448631cf1a5f173cffd8159354ef358?=
 =?UTF-8?q?=20=F0=9F=9A=80?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 404.html                                      |   2 +-
 LICENSE-text.html                             |   2 +-
 LICENSE.html                                  |   2 +-
 articles/generate-closure.html                | 532 +++++++++---------
 .../figure-html/base-graph-1.png              | Bin 61456 -> 52109 bytes
 articles/graph-examples.html                  | 211 +++++--
 .../figure-html/bonferroni-1.png              | Bin 0 -> 42516 bytes
 .../figure-html/bonferroni-holm-1.png         | Bin 190863 -> 184828 bytes
 .../figure-html/fixed-sequence-1.png          | Bin 24996 -> 23258 bytes
 .../figure-html/huque-alosh-bhore-1.png       | Bin 51418 -> 48568 bytes
 .../improved-parallel-gatekeeping-1.png       | Bin 71855 -> 64222 bytes
 ...improved-parallel-gatekeeping-update-1.png | Bin 0 -> 80204 bytes
 ...improved-parallel-gatekeeping-update-2.png | Bin 0 -> 62656 bytes
 ...improved-parallel-gatekeeping-update-3.png | Bin 0 -> 47729 bytes
 .../figure-html/parallel-gatekeeping-1.png    | Bin 50148 -> 46840 bytes
 .../parallel-gatekeeping-update-1.png         | Bin 0 -> 30841 bytes
 .../figure-html/simple-successive-1.png       | Bin 40298 -> 37265 bytes
 .../figure-html/simple-successive-var-1.png   | Bin 84520 -> 54332 bytes
 .../figure-html/simple-successive-var-2.png   | Bin 83616 -> 52836 bytes
 .../figure-html/wiens-dmitrienko-1.png        | Bin 39061 -> 38734 bytes
 articles/index.html                           |   4 +-
 articles/shortuct-testing.html                |   2 +-
 authors.html                                  |   2 +-
 index.html                                    |   2 +-
 pkgdown.yml                                   |   2 +-
 reference/adjust_p.html                       |   2 +-
 reference/adjusted-weights.html               |   2 +-
 reference/as-graph.html                       |   2 +-
 reference/calc-test_values.html               |   2 +-
 reference/example-graphs.html                 |   2 +-
 reference/graph_calculate_power.html          |   2 +-
 reference/graph_create.html                   |   2 +-
 reference/graph_generate_weights.html         |   2 +-
 reference/graph_rejection_orderings.html      |   2 +-
 reference/graph_update.html                   |   2 +-
 reference/graphicalMCP-package.html           |   2 +-
 reference/index.html                          |   2 +-
 reference/plot.initial_graph-1.png            | Bin 86766 -> 82119 bytes
 reference/plot.initial_graph.html             |   2 +-
 reference/plot.updated_graph-1.png            | Bin 41308 -> 31591 bytes
 reference/plot.updated_graph.html             |   2 +-
 reference/print.graph_report.html             |   2 +-
 reference/print.initial_graph.html            |   2 +-
 reference/print.power_report.html             |   2 +-
 reference/print.updated_graph.html            |   2 +-
 reference/testing-fast.html                   |   2 +-
 reference/testing.html                        |   2 +-
 search.json                                   |   2 +-
 48 files changed, 455 insertions(+), 348 deletions(-)
 create mode 100644 articles/graph-examples_files/figure-html/bonferroni-1.png
 create mode 100644 articles/graph-examples_files/figure-html/improved-parallel-gatekeeping-update-1.png
 create mode 100644 articles/graph-examples_files/figure-html/improved-parallel-gatekeeping-update-2.png
 create mode 100644 articles/graph-examples_files/figure-html/improved-parallel-gatekeeping-update-3.png
 create mode 100644 articles/graph-examples_files/figure-html/parallel-gatekeeping-update-1.png

diff --git a/404.html b/404.html
index 4837619d..4685d57f 100644
--- a/404.html
+++ b/404.html
@@ -47,7 +47,7 @@
   <a href="https://urban-sniffle-p11zlpj.pages.github.io/#" class="nav-link dropdown-toggle" data-bs-toggle="dropdown" role="button" aria-expanded="false" aria-haspopup="true" id="dropdown-articles">Articles</a>
   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
diff --git a/LICENSE-text.html b/LICENSE-text.html
index e4747518..c3d4bd76 100644
--- a/LICENSE-text.html
+++ b/LICENSE-text.html
@@ -25,7 +25,7 @@
   <a href="#" class="nav-link dropdown-toggle" data-bs-toggle="dropdown" role="button" aria-expanded="false" aria-haspopup="true" id="dropdown-articles">Articles</a>
   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
diff --git a/LICENSE.html b/LICENSE.html
index eee1987a..7e43fec4 100644
--- a/LICENSE.html
+++ b/LICENSE.html
@@ -25,7 +25,7 @@
   <a href="#" class="nav-link dropdown-toggle" data-bs-toggle="dropdown" role="button" aria-expanded="false" aria-haspopup="true" id="dropdown-articles">Articles</a>
   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
diff --git a/articles/generate-closure.html b/articles/generate-closure.html
index 0f216a80..1cef0abf 100644
--- a/articles/generate-closure.html
+++ b/articles/generate-closure.html
@@ -49,7 +49,7 @@
   <a href="#" class="nav-link dropdown-toggle" data-bs-toggle="dropdown" role="button" aria-expanded="false" aria-haspopup="true" id="dropdown-articles">Articles</a>
   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
@@ -116,23 +116,23 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
 parts: An indicator matrix showing which hypotheses are contained in
 each sub-graph (The so-called powerset of a set), and a weights matrix
 containing the induced weights of each sub-graph.</p>
-<div id="nsezewpgah" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
-<style>#nsezewpgah table {
+<div id="cpszpmwcaj" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
+<style>#cpszpmwcaj table {
   font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
   -webkit-font-smoothing: antialiased;
   -moz-osx-font-smoothing: grayscale;
 }
 
-#nsezewpgah thead, #nsezewpgah tbody, #nsezewpgah tfoot, #nsezewpgah tr, #nsezewpgah td, #nsezewpgah th {
+#cpszpmwcaj thead, #cpszpmwcaj tbody, #cpszpmwcaj tfoot, #cpszpmwcaj tr, #cpszpmwcaj td, #cpszpmwcaj th {
   border-style: none;
 }
 
-#nsezewpgah p {
+#cpszpmwcaj p {
   margin: 0;
   padding: 0;
 }
 
-#nsezewpgah .gt_table {
+#cpszpmwcaj .gt_table {
   display: table;
   border-collapse: collapse;
   line-height: normal;
@@ -158,12 +158,12 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   border-left-color: #D3D3D3;
 }
 
-#nsezewpgah .gt_caption {
+#cpszpmwcaj .gt_caption {
   padding-top: 4px;
   padding-bottom: 4px;
 }
 
-#nsezewpgah .gt_title {
+#cpszpmwcaj .gt_title {
   color: #333333;
   font-size: 125%;
   font-weight: initial;
@@ -175,7 +175,7 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   border-bottom-width: 0;
 }
 
-#nsezewpgah .gt_subtitle {
+#cpszpmwcaj .gt_subtitle {
   color: #333333;
   font-size: 85%;
   font-weight: initial;
@@ -187,7 +187,7 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   border-top-width: 0;
 }
 
-#nsezewpgah .gt_heading {
+#cpszpmwcaj .gt_heading {
   background-color: #FFFFFF;
   text-align: center;
   border-bottom-color: #FFFFFF;
@@ -199,13 +199,13 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   border-right-color: #D3D3D3;
 }
 
-#nsezewpgah .gt_bottom_border {
+#cpszpmwcaj .gt_bottom_border {
   border-bottom-style: solid;
   border-bottom-width: 2px;
   border-bottom-color: #D3D3D3;
 }
 
-#nsezewpgah .gt_col_headings {
+#cpszpmwcaj .gt_col_headings {
   border-top-style: solid;
   border-top-width: 2px;
   border-top-color: #D3D3D3;
@@ -220,7 +220,7 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   border-right-color: #D3D3D3;
 }
 
-#nsezewpgah .gt_col_heading {
+#cpszpmwcaj .gt_col_heading {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -240,7 +240,7 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   overflow-x: hidden;
 }
 
-#nsezewpgah .gt_column_spanner_outer {
+#cpszpmwcaj .gt_column_spanner_outer {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -252,15 +252,15 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   padding-right: 4px;
 }
 
-#nsezewpgah .gt_column_spanner_outer:first-child {
+#cpszpmwcaj .gt_column_spanner_outer:first-child {
   padding-left: 0;
 }
 
-#nsezewpgah .gt_column_spanner_outer:last-child {
+#cpszpmwcaj .gt_column_spanner_outer:last-child {
   padding-right: 0;
 }
 
-#nsezewpgah .gt_column_spanner {
+#cpszpmwcaj .gt_column_spanner {
   border-bottom-style: solid;
   border-bottom-width: 2px;
   border-bottom-color: #D3D3D3;
@@ -272,11 +272,11 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   width: 100%;
 }
 
-#nsezewpgah .gt_spanner_row {
+#cpszpmwcaj .gt_spanner_row {
   border-bottom-style: hidden;
 }
 
-#nsezewpgah .gt_group_heading {
+#cpszpmwcaj .gt_group_heading {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -302,7 +302,7 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   text-align: left;
 }
 
-#nsezewpgah .gt_empty_group_heading {
+#cpszpmwcaj .gt_empty_group_heading {
   padding: 0.5px;
   color: #333333;
   background-color: #FFFFFF;
@@ -317,15 +317,15 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   vertical-align: middle;
 }
 
-#nsezewpgah .gt_from_md > :first-child {
+#cpszpmwcaj .gt_from_md > :first-child {
   margin-top: 0;
 }
 
-#nsezewpgah .gt_from_md > :last-child {
+#cpszpmwcaj .gt_from_md > :last-child {
   margin-bottom: 0;
 }
 
-#nsezewpgah .gt_row {
+#cpszpmwcaj .gt_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -344,7 +344,7 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   overflow-x: hidden;
 }
 
-#nsezewpgah .gt_stub {
+#cpszpmwcaj .gt_stub {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -357,7 +357,7 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   padding-right: 5px;
 }
 
-#nsezewpgah .gt_stub_row_group {
+#cpszpmwcaj .gt_stub_row_group {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -371,15 +371,15 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   vertical-align: top;
 }
 
-#nsezewpgah .gt_row_group_first td {
+#cpszpmwcaj .gt_row_group_first td {
   border-top-width: 2px;
 }
 
-#nsezewpgah .gt_row_group_first th {
+#cpszpmwcaj .gt_row_group_first th {
   border-top-width: 2px;
 }
 
-#nsezewpgah .gt_summary_row {
+#cpszpmwcaj .gt_summary_row {
   color: #333333;
   background-color: #FFFFFF;
   text-transform: inherit;
@@ -389,16 +389,16 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   padding-right: 5px;
 }
 
-#nsezewpgah .gt_first_summary_row {
+#cpszpmwcaj .gt_first_summary_row {
   border-top-style: solid;
   border-top-color: #D3D3D3;
 }
 
-#nsezewpgah .gt_first_summary_row.thick {
+#cpszpmwcaj .gt_first_summary_row.thick {
   border-top-width: 2px;
 }
 
-#nsezewpgah .gt_last_summary_row {
+#cpszpmwcaj .gt_last_summary_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -408,7 +408,7 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   border-bottom-color: #D3D3D3;
 }
 
-#nsezewpgah .gt_grand_summary_row {
+#cpszpmwcaj .gt_grand_summary_row {
   color: #333333;
   background-color: #FFFFFF;
   text-transform: inherit;
@@ -418,7 +418,7 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   padding-right: 5px;
 }
 
-#nsezewpgah .gt_first_grand_summary_row {
+#cpszpmwcaj .gt_first_grand_summary_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -428,7 +428,7 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   border-top-color: #D3D3D3;
 }
 
-#nsezewpgah .gt_last_grand_summary_row_top {
+#cpszpmwcaj .gt_last_grand_summary_row_top {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -438,11 +438,11 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   border-bottom-color: #D3D3D3;
 }
 
-#nsezewpgah .gt_striped {
+#cpszpmwcaj .gt_striped {
   background-color: rgba(128, 128, 128, 0.05);
 }
 
-#nsezewpgah .gt_table_body {
+#cpszpmwcaj .gt_table_body {
   border-top-style: solid;
   border-top-width: 2px;
   border-top-color: #D3D3D3;
@@ -451,7 +451,7 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   border-bottom-color: #D3D3D3;
 }
 
-#nsezewpgah .gt_footnotes {
+#cpszpmwcaj .gt_footnotes {
   color: #333333;
   background-color: #FFFFFF;
   border-bottom-style: none;
@@ -465,7 +465,7 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   border-right-color: #D3D3D3;
 }
 
-#nsezewpgah .gt_footnote {
+#cpszpmwcaj .gt_footnote {
   margin: 0px;
   font-size: 90%;
   padding-top: 4px;
@@ -474,7 +474,7 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   padding-right: 5px;
 }
 
-#nsezewpgah .gt_sourcenotes {
+#cpszpmwcaj .gt_sourcenotes {
   color: #333333;
   background-color: #FFFFFF;
   border-bottom-style: none;
@@ -488,7 +488,7 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   border-right-color: #D3D3D3;
 }
 
-#nsezewpgah .gt_sourcenote {
+#cpszpmwcaj .gt_sourcenote {
   font-size: 90%;
   padding-top: 4px;
   padding-bottom: 4px;
@@ -496,63 +496,63 @@ <h3 id="components-of-the-closure">Components of the closure<a class="anchor" ar
   padding-right: 5px;
 }
 
-#nsezewpgah .gt_left {
+#cpszpmwcaj .gt_left {
   text-align: left;
 }
 
-#nsezewpgah .gt_center {
+#cpszpmwcaj .gt_center {
   text-align: center;
 }
 
-#nsezewpgah .gt_right {
+#cpszpmwcaj .gt_right {
   text-align: right;
   font-variant-numeric: tabular-nums;
 }
 
-#nsezewpgah .gt_font_normal {
+#cpszpmwcaj .gt_font_normal {
   font-weight: normal;
 }
 
-#nsezewpgah .gt_font_bold {
+#cpszpmwcaj .gt_font_bold {
   font-weight: bold;
 }
 
-#nsezewpgah .gt_font_italic {
+#cpszpmwcaj .gt_font_italic {
   font-style: italic;
 }
 
-#nsezewpgah .gt_super {
+#cpszpmwcaj .gt_super {
   font-size: 65%;
 }
 
-#nsezewpgah .gt_footnote_marks {
+#cpszpmwcaj .gt_footnote_marks {
   font-size: 75%;
   vertical-align: 0.4em;
   position: initial;
 }
 
-#nsezewpgah .gt_asterisk {
+#cpszpmwcaj .gt_asterisk {
   font-size: 100%;
   vertical-align: 0;
 }
 
-#nsezewpgah .gt_indent_1 {
+#cpszpmwcaj .gt_indent_1 {
   text-indent: 5px;
 }
 
-#nsezewpgah .gt_indent_2 {
+#cpszpmwcaj .gt_indent_2 {
   text-indent: 10px;
 }
 
-#nsezewpgah .gt_indent_3 {
+#cpszpmwcaj .gt_indent_3 {
   text-indent: 15px;
 }
 
-#nsezewpgah .gt_indent_4 {
+#cpszpmwcaj .gt_indent_4 {
   text-indent: 20px;
 }
 
-#nsezewpgah .gt_indent_5 {
+#cpszpmwcaj .gt_indent_5 {
   text-indent: 25px;
 }
 </style>
@@ -770,23 +770,23 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
 This reveals a pattern between the bottom half and top half, where each
 row’s parent in the bottom half is the corresponding row in the top
 half, eight rows up.</p>
-<div id="yhxbofofru" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
-<style>#yhxbofofru table {
+<div id="kgleaynter" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
+<style>#kgleaynter table {
   font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
   -webkit-font-smoothing: antialiased;
   -moz-osx-font-smoothing: grayscale;
 }
 
-#yhxbofofru thead, #yhxbofofru tbody, #yhxbofofru tfoot, #yhxbofofru tr, #yhxbofofru td, #yhxbofofru th {
+#kgleaynter thead, #kgleaynter tbody, #kgleaynter tfoot, #kgleaynter tr, #kgleaynter td, #kgleaynter th {
   border-style: none;
 }
 
-#yhxbofofru p {
+#kgleaynter p {
   margin: 0;
   padding: 0;
 }
 
-#yhxbofofru .gt_table {
+#kgleaynter .gt_table {
   display: table;
   border-collapse: collapse;
   line-height: normal;
@@ -812,12 +812,12 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-left-color: #D3D3D3;
 }
 
-#yhxbofofru .gt_caption {
+#kgleaynter .gt_caption {
   padding-top: 4px;
   padding-bottom: 4px;
 }
 
-#yhxbofofru .gt_title {
+#kgleaynter .gt_title {
   color: #333333;
   font-size: 125%;
   font-weight: initial;
@@ -829,7 +829,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-bottom-width: 0;
 }
 
-#yhxbofofru .gt_subtitle {
+#kgleaynter .gt_subtitle {
   color: #333333;
   font-size: 85%;
   font-weight: initial;
@@ -841,7 +841,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-top-width: 0;
 }
 
-#yhxbofofru .gt_heading {
+#kgleaynter .gt_heading {
   background-color: #FFFFFF;
   text-align: center;
   border-bottom-color: #FFFFFF;
@@ -853,13 +853,13 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-right-color: #D3D3D3;
 }
 
-#yhxbofofru .gt_bottom_border {
+#kgleaynter .gt_bottom_border {
   border-bottom-style: solid;
   border-bottom-width: 2px;
   border-bottom-color: #D3D3D3;
 }
 
-#yhxbofofru .gt_col_headings {
+#kgleaynter .gt_col_headings {
   border-top-style: solid;
   border-top-width: 2px;
   border-top-color: #D3D3D3;
@@ -874,7 +874,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-right-color: #D3D3D3;
 }
 
-#yhxbofofru .gt_col_heading {
+#kgleaynter .gt_col_heading {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -894,7 +894,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   overflow-x: hidden;
 }
 
-#yhxbofofru .gt_column_spanner_outer {
+#kgleaynter .gt_column_spanner_outer {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -906,15 +906,15 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   padding-right: 4px;
 }
 
-#yhxbofofru .gt_column_spanner_outer:first-child {
+#kgleaynter .gt_column_spanner_outer:first-child {
   padding-left: 0;
 }
 
-#yhxbofofru .gt_column_spanner_outer:last-child {
+#kgleaynter .gt_column_spanner_outer:last-child {
   padding-right: 0;
 }
 
-#yhxbofofru .gt_column_spanner {
+#kgleaynter .gt_column_spanner {
   border-bottom-style: solid;
   border-bottom-width: 2px;
   border-bottom-color: #D3D3D3;
@@ -926,11 +926,11 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   width: 100%;
 }
 
-#yhxbofofru .gt_spanner_row {
+#kgleaynter .gt_spanner_row {
   border-bottom-style: hidden;
 }
 
-#yhxbofofru .gt_group_heading {
+#kgleaynter .gt_group_heading {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -956,7 +956,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   text-align: left;
 }
 
-#yhxbofofru .gt_empty_group_heading {
+#kgleaynter .gt_empty_group_heading {
   padding: 0.5px;
   color: #333333;
   background-color: #FFFFFF;
@@ -971,15 +971,15 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   vertical-align: middle;
 }
 
-#yhxbofofru .gt_from_md > :first-child {
+#kgleaynter .gt_from_md > :first-child {
   margin-top: 0;
 }
 
-#yhxbofofru .gt_from_md > :last-child {
+#kgleaynter .gt_from_md > :last-child {
   margin-bottom: 0;
 }
 
-#yhxbofofru .gt_row {
+#kgleaynter .gt_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -998,7 +998,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   overflow-x: hidden;
 }
 
-#yhxbofofru .gt_stub {
+#kgleaynter .gt_stub {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -1011,7 +1011,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   padding-right: 5px;
 }
 
-#yhxbofofru .gt_stub_row_group {
+#kgleaynter .gt_stub_row_group {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -1025,15 +1025,15 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   vertical-align: top;
 }
 
-#yhxbofofru .gt_row_group_first td {
+#kgleaynter .gt_row_group_first td {
   border-top-width: 2px;
 }
 
-#yhxbofofru .gt_row_group_first th {
+#kgleaynter .gt_row_group_first th {
   border-top-width: 2px;
 }
 
-#yhxbofofru .gt_summary_row {
+#kgleaynter .gt_summary_row {
   color: #333333;
   background-color: #FFFFFF;
   text-transform: inherit;
@@ -1043,16 +1043,16 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   padding-right: 5px;
 }
 
-#yhxbofofru .gt_first_summary_row {
+#kgleaynter .gt_first_summary_row {
   border-top-style: solid;
   border-top-color: #D3D3D3;
 }
 
-#yhxbofofru .gt_first_summary_row.thick {
+#kgleaynter .gt_first_summary_row.thick {
   border-top-width: 2px;
 }
 
-#yhxbofofru .gt_last_summary_row {
+#kgleaynter .gt_last_summary_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -1062,7 +1062,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-bottom-color: #D3D3D3;
 }
 
-#yhxbofofru .gt_grand_summary_row {
+#kgleaynter .gt_grand_summary_row {
   color: #333333;
   background-color: #FFFFFF;
   text-transform: inherit;
@@ -1072,7 +1072,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   padding-right: 5px;
 }
 
-#yhxbofofru .gt_first_grand_summary_row {
+#kgleaynter .gt_first_grand_summary_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -1082,7 +1082,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-top-color: #D3D3D3;
 }
 
-#yhxbofofru .gt_last_grand_summary_row_top {
+#kgleaynter .gt_last_grand_summary_row_top {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -1092,11 +1092,11 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-bottom-color: #D3D3D3;
 }
 
-#yhxbofofru .gt_striped {
+#kgleaynter .gt_striped {
   background-color: rgba(128, 128, 128, 0.05);
 }
 
-#yhxbofofru .gt_table_body {
+#kgleaynter .gt_table_body {
   border-top-style: solid;
   border-top-width: 2px;
   border-top-color: #D3D3D3;
@@ -1105,7 +1105,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-bottom-color: #D3D3D3;
 }
 
-#yhxbofofru .gt_footnotes {
+#kgleaynter .gt_footnotes {
   color: #333333;
   background-color: #FFFFFF;
   border-bottom-style: none;
@@ -1119,7 +1119,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-right-color: #D3D3D3;
 }
 
-#yhxbofofru .gt_footnote {
+#kgleaynter .gt_footnote {
   margin: 0px;
   font-size: 90%;
   padding-top: 4px;
@@ -1128,7 +1128,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   padding-right: 5px;
 }
 
-#yhxbofofru .gt_sourcenotes {
+#kgleaynter .gt_sourcenotes {
   color: #333333;
   background-color: #FFFFFF;
   border-bottom-style: none;
@@ -1142,7 +1142,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-right-color: #D3D3D3;
 }
 
-#yhxbofofru .gt_sourcenote {
+#kgleaynter .gt_sourcenote {
   font-size: 90%;
   padding-top: 4px;
   padding-bottom: 4px;
@@ -1150,63 +1150,63 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   padding-right: 5px;
 }
 
-#yhxbofofru .gt_left {
+#kgleaynter .gt_left {
   text-align: left;
 }
 
-#yhxbofofru .gt_center {
+#kgleaynter .gt_center {
   text-align: center;
 }
 
-#yhxbofofru .gt_right {
+#kgleaynter .gt_right {
   text-align: right;
   font-variant-numeric: tabular-nums;
 }
 
-#yhxbofofru .gt_font_normal {
+#kgleaynter .gt_font_normal {
   font-weight: normal;
 }
 
-#yhxbofofru .gt_font_bold {
+#kgleaynter .gt_font_bold {
   font-weight: bold;
 }
 
-#yhxbofofru .gt_font_italic {
+#kgleaynter .gt_font_italic {
   font-style: italic;
 }
 
-#yhxbofofru .gt_super {
+#kgleaynter .gt_super {
   font-size: 65%;
 }
 
-#yhxbofofru .gt_footnote_marks {
+#kgleaynter .gt_footnote_marks {
   font-size: 75%;
   vertical-align: 0.4em;
   position: initial;
 }
 
-#yhxbofofru .gt_asterisk {
+#kgleaynter .gt_asterisk {
   font-size: 100%;
   vertical-align: 0;
 }
 
-#yhxbofofru .gt_indent_1 {
+#kgleaynter .gt_indent_1 {
   text-indent: 5px;
 }
 
-#yhxbofofru .gt_indent_2 {
+#kgleaynter .gt_indent_2 {
   text-indent: 10px;
 }
 
-#yhxbofofru .gt_indent_3 {
+#kgleaynter .gt_indent_3 {
   text-indent: 15px;
 }
 
-#yhxbofofru .gt_indent_4 {
+#kgleaynter .gt_indent_4 {
   text-indent: 20px;
 }
 
-#yhxbofofru .gt_indent_5 {
+#kgleaynter .gt_indent_5 {
   text-indent: 25px;
 }
 </style>
@@ -1335,23 +1335,23 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
 <p>The pattern then repeats within each box recursively, with the top
 half of each box matching the bottom half, if the first missing
 hypothesis is flipped from 0 to 1.</p>
-<div id="jxxwgzkyai" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
-<style>#jxxwgzkyai table {
+<div id="coeqwqwaso" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
+<style>#coeqwqwaso table {
   font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
   -webkit-font-smoothing: antialiased;
   -moz-osx-font-smoothing: grayscale;
 }
 
-#jxxwgzkyai thead, #jxxwgzkyai tbody, #jxxwgzkyai tfoot, #jxxwgzkyai tr, #jxxwgzkyai td, #jxxwgzkyai th {
+#coeqwqwaso thead, #coeqwqwaso tbody, #coeqwqwaso tfoot, #coeqwqwaso tr, #coeqwqwaso td, #coeqwqwaso th {
   border-style: none;
 }
 
-#jxxwgzkyai p {
+#coeqwqwaso p {
   margin: 0;
   padding: 0;
 }
 
-#jxxwgzkyai .gt_table {
+#coeqwqwaso .gt_table {
   display: table;
   border-collapse: collapse;
   line-height: normal;
@@ -1377,12 +1377,12 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-left-color: #D3D3D3;
 }
 
-#jxxwgzkyai .gt_caption {
+#coeqwqwaso .gt_caption {
   padding-top: 4px;
   padding-bottom: 4px;
 }
 
-#jxxwgzkyai .gt_title {
+#coeqwqwaso .gt_title {
   color: #333333;
   font-size: 125%;
   font-weight: initial;
@@ -1394,7 +1394,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-bottom-width: 0;
 }
 
-#jxxwgzkyai .gt_subtitle {
+#coeqwqwaso .gt_subtitle {
   color: #333333;
   font-size: 85%;
   font-weight: initial;
@@ -1406,7 +1406,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-top-width: 0;
 }
 
-#jxxwgzkyai .gt_heading {
+#coeqwqwaso .gt_heading {
   background-color: #FFFFFF;
   text-align: center;
   border-bottom-color: #FFFFFF;
@@ -1418,13 +1418,13 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-right-color: #D3D3D3;
 }
 
-#jxxwgzkyai .gt_bottom_border {
+#coeqwqwaso .gt_bottom_border {
   border-bottom-style: solid;
   border-bottom-width: 2px;
   border-bottom-color: #D3D3D3;
 }
 
-#jxxwgzkyai .gt_col_headings {
+#coeqwqwaso .gt_col_headings {
   border-top-style: solid;
   border-top-width: 2px;
   border-top-color: #D3D3D3;
@@ -1439,7 +1439,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-right-color: #D3D3D3;
 }
 
-#jxxwgzkyai .gt_col_heading {
+#coeqwqwaso .gt_col_heading {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -1459,7 +1459,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   overflow-x: hidden;
 }
 
-#jxxwgzkyai .gt_column_spanner_outer {
+#coeqwqwaso .gt_column_spanner_outer {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -1471,15 +1471,15 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   padding-right: 4px;
 }
 
-#jxxwgzkyai .gt_column_spanner_outer:first-child {
+#coeqwqwaso .gt_column_spanner_outer:first-child {
   padding-left: 0;
 }
 
-#jxxwgzkyai .gt_column_spanner_outer:last-child {
+#coeqwqwaso .gt_column_spanner_outer:last-child {
   padding-right: 0;
 }
 
-#jxxwgzkyai .gt_column_spanner {
+#coeqwqwaso .gt_column_spanner {
   border-bottom-style: solid;
   border-bottom-width: 2px;
   border-bottom-color: #D3D3D3;
@@ -1491,11 +1491,11 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   width: 100%;
 }
 
-#jxxwgzkyai .gt_spanner_row {
+#coeqwqwaso .gt_spanner_row {
   border-bottom-style: hidden;
 }
 
-#jxxwgzkyai .gt_group_heading {
+#coeqwqwaso .gt_group_heading {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -1521,7 +1521,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   text-align: left;
 }
 
-#jxxwgzkyai .gt_empty_group_heading {
+#coeqwqwaso .gt_empty_group_heading {
   padding: 0.5px;
   color: #333333;
   background-color: #FFFFFF;
@@ -1536,15 +1536,15 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   vertical-align: middle;
 }
 
-#jxxwgzkyai .gt_from_md > :first-child {
+#coeqwqwaso .gt_from_md > :first-child {
   margin-top: 0;
 }
 
-#jxxwgzkyai .gt_from_md > :last-child {
+#coeqwqwaso .gt_from_md > :last-child {
   margin-bottom: 0;
 }
 
-#jxxwgzkyai .gt_row {
+#coeqwqwaso .gt_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -1563,7 +1563,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   overflow-x: hidden;
 }
 
-#jxxwgzkyai .gt_stub {
+#coeqwqwaso .gt_stub {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -1576,7 +1576,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   padding-right: 5px;
 }
 
-#jxxwgzkyai .gt_stub_row_group {
+#coeqwqwaso .gt_stub_row_group {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -1590,15 +1590,15 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   vertical-align: top;
 }
 
-#jxxwgzkyai .gt_row_group_first td {
+#coeqwqwaso .gt_row_group_first td {
   border-top-width: 2px;
 }
 
-#jxxwgzkyai .gt_row_group_first th {
+#coeqwqwaso .gt_row_group_first th {
   border-top-width: 2px;
 }
 
-#jxxwgzkyai .gt_summary_row {
+#coeqwqwaso .gt_summary_row {
   color: #333333;
   background-color: #FFFFFF;
   text-transform: inherit;
@@ -1608,16 +1608,16 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   padding-right: 5px;
 }
 
-#jxxwgzkyai .gt_first_summary_row {
+#coeqwqwaso .gt_first_summary_row {
   border-top-style: solid;
   border-top-color: #D3D3D3;
 }
 
-#jxxwgzkyai .gt_first_summary_row.thick {
+#coeqwqwaso .gt_first_summary_row.thick {
   border-top-width: 2px;
 }
 
-#jxxwgzkyai .gt_last_summary_row {
+#coeqwqwaso .gt_last_summary_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -1627,7 +1627,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-bottom-color: #D3D3D3;
 }
 
-#jxxwgzkyai .gt_grand_summary_row {
+#coeqwqwaso .gt_grand_summary_row {
   color: #333333;
   background-color: #FFFFFF;
   text-transform: inherit;
@@ -1637,7 +1637,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   padding-right: 5px;
 }
 
-#jxxwgzkyai .gt_first_grand_summary_row {
+#coeqwqwaso .gt_first_grand_summary_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -1647,7 +1647,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-top-color: #D3D3D3;
 }
 
-#jxxwgzkyai .gt_last_grand_summary_row_top {
+#coeqwqwaso .gt_last_grand_summary_row_top {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -1657,11 +1657,11 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-bottom-color: #D3D3D3;
 }
 
-#jxxwgzkyai .gt_striped {
+#coeqwqwaso .gt_striped {
   background-color: rgba(128, 128, 128, 0.05);
 }
 
-#jxxwgzkyai .gt_table_body {
+#coeqwqwaso .gt_table_body {
   border-top-style: solid;
   border-top-width: 2px;
   border-top-color: #D3D3D3;
@@ -1670,7 +1670,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-bottom-color: #D3D3D3;
 }
 
-#jxxwgzkyai .gt_footnotes {
+#coeqwqwaso .gt_footnotes {
   color: #333333;
   background-color: #FFFFFF;
   border-bottom-style: none;
@@ -1684,7 +1684,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-right-color: #D3D3D3;
 }
 
-#jxxwgzkyai .gt_footnote {
+#coeqwqwaso .gt_footnote {
   margin: 0px;
   font-size: 90%;
   padding-top: 4px;
@@ -1693,7 +1693,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   padding-right: 5px;
 }
 
-#jxxwgzkyai .gt_sourcenotes {
+#coeqwqwaso .gt_sourcenotes {
   color: #333333;
   background-color: #FFFFFF;
   border-bottom-style: none;
@@ -1707,7 +1707,7 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   border-right-color: #D3D3D3;
 }
 
-#jxxwgzkyai .gt_sourcenote {
+#coeqwqwaso .gt_sourcenote {
   font-size: 90%;
   padding-top: 4px;
   padding-bottom: 4px;
@@ -1715,63 +1715,63 @@ <h4 id="repeating-recursive-blocks">Repeating recursive blocks<a class="anchor"
   padding-right: 5px;
 }
 
-#jxxwgzkyai .gt_left {
+#coeqwqwaso .gt_left {
   text-align: left;
 }
 
-#jxxwgzkyai .gt_center {
+#coeqwqwaso .gt_center {
   text-align: center;
 }
 
-#jxxwgzkyai .gt_right {
+#coeqwqwaso .gt_right {
   text-align: right;
   font-variant-numeric: tabular-nums;
 }
 
-#jxxwgzkyai .gt_font_normal {
+#coeqwqwaso .gt_font_normal {
   font-weight: normal;
 }
 
-#jxxwgzkyai .gt_font_bold {
+#coeqwqwaso .gt_font_bold {
   font-weight: bold;
 }
 
-#jxxwgzkyai .gt_font_italic {
+#coeqwqwaso .gt_font_italic {
   font-style: italic;
 }
 
-#jxxwgzkyai .gt_super {
+#coeqwqwaso .gt_super {
   font-size: 65%;
 }
 
-#jxxwgzkyai .gt_footnote_marks {
+#coeqwqwaso .gt_footnote_marks {
   font-size: 75%;
   vertical-align: 0.4em;
   position: initial;
 }
 
-#jxxwgzkyai .gt_asterisk {
+#coeqwqwaso .gt_asterisk {
   font-size: 100%;
   vertical-align: 0;
 }
 
-#jxxwgzkyai .gt_indent_1 {
+#coeqwqwaso .gt_indent_1 {
   text-indent: 5px;
 }
 
-#jxxwgzkyai .gt_indent_2 {
+#coeqwqwaso .gt_indent_2 {
   text-indent: 10px;
 }
 
-#jxxwgzkyai .gt_indent_3 {
+#coeqwqwaso .gt_indent_3 {
   text-indent: 15px;
 }
 
-#jxxwgzkyai .gt_indent_4 {
+#coeqwqwaso .gt_indent_4 {
   text-indent: 20px;
 }
 
-#jxxwgzkyai .gt_indent_5 {
+#coeqwqwaso .gt_indent_5 {
   text-indent: 25px;
 }
 </style>
@@ -1912,23 +1912,23 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
 <code>1010</code>. When interpreted as binary, this is
 <code>1 * 8 + 0 * 4 + 1 * 2 + 0 * 1 = 10</code> in base 10, and
 <code>6 == 15 - 10 + 1</code>.</p>
-<div id="qlfavycsif" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
-<style>#qlfavycsif table {
+<div id="gihwgxbnxi" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
+<style>#gihwgxbnxi table {
   font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
   -webkit-font-smoothing: antialiased;
   -moz-osx-font-smoothing: grayscale;
 }
 
-#qlfavycsif thead, #qlfavycsif tbody, #qlfavycsif tfoot, #qlfavycsif tr, #qlfavycsif td, #qlfavycsif th {
+#gihwgxbnxi thead, #gihwgxbnxi tbody, #gihwgxbnxi tfoot, #gihwgxbnxi tr, #gihwgxbnxi td, #gihwgxbnxi th {
   border-style: none;
 }
 
-#qlfavycsif p {
+#gihwgxbnxi p {
   margin: 0;
   padding: 0;
 }
 
-#qlfavycsif .gt_table {
+#gihwgxbnxi .gt_table {
   display: table;
   border-collapse: collapse;
   line-height: normal;
@@ -1954,12 +1954,12 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   border-left-color: #D3D3D3;
 }
 
-#qlfavycsif .gt_caption {
+#gihwgxbnxi .gt_caption {
   padding-top: 4px;
   padding-bottom: 4px;
 }
 
-#qlfavycsif .gt_title {
+#gihwgxbnxi .gt_title {
   color: #333333;
   font-size: 125%;
   font-weight: initial;
@@ -1971,7 +1971,7 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   border-bottom-width: 0;
 }
 
-#qlfavycsif .gt_subtitle {
+#gihwgxbnxi .gt_subtitle {
   color: #333333;
   font-size: 85%;
   font-weight: initial;
@@ -1983,7 +1983,7 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   border-top-width: 0;
 }
 
-#qlfavycsif .gt_heading {
+#gihwgxbnxi .gt_heading {
   background-color: #FFFFFF;
   text-align: center;
   border-bottom-color: #FFFFFF;
@@ -1995,13 +1995,13 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   border-right-color: #D3D3D3;
 }
 
-#qlfavycsif .gt_bottom_border {
+#gihwgxbnxi .gt_bottom_border {
   border-bottom-style: solid;
   border-bottom-width: 2px;
   border-bottom-color: #D3D3D3;
 }
 
-#qlfavycsif .gt_col_headings {
+#gihwgxbnxi .gt_col_headings {
   border-top-style: solid;
   border-top-width: 2px;
   border-top-color: #D3D3D3;
@@ -2016,7 +2016,7 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   border-right-color: #D3D3D3;
 }
 
-#qlfavycsif .gt_col_heading {
+#gihwgxbnxi .gt_col_heading {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -2036,7 +2036,7 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   overflow-x: hidden;
 }
 
-#qlfavycsif .gt_column_spanner_outer {
+#gihwgxbnxi .gt_column_spanner_outer {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -2048,15 +2048,15 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   padding-right: 4px;
 }
 
-#qlfavycsif .gt_column_spanner_outer:first-child {
+#gihwgxbnxi .gt_column_spanner_outer:first-child {
   padding-left: 0;
 }
 
-#qlfavycsif .gt_column_spanner_outer:last-child {
+#gihwgxbnxi .gt_column_spanner_outer:last-child {
   padding-right: 0;
 }
 
-#qlfavycsif .gt_column_spanner {
+#gihwgxbnxi .gt_column_spanner {
   border-bottom-style: solid;
   border-bottom-width: 2px;
   border-bottom-color: #D3D3D3;
@@ -2068,11 +2068,11 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   width: 100%;
 }
 
-#qlfavycsif .gt_spanner_row {
+#gihwgxbnxi .gt_spanner_row {
   border-bottom-style: hidden;
 }
 
-#qlfavycsif .gt_group_heading {
+#gihwgxbnxi .gt_group_heading {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -2098,7 +2098,7 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   text-align: left;
 }
 
-#qlfavycsif .gt_empty_group_heading {
+#gihwgxbnxi .gt_empty_group_heading {
   padding: 0.5px;
   color: #333333;
   background-color: #FFFFFF;
@@ -2113,15 +2113,15 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   vertical-align: middle;
 }
 
-#qlfavycsif .gt_from_md > :first-child {
+#gihwgxbnxi .gt_from_md > :first-child {
   margin-top: 0;
 }
 
-#qlfavycsif .gt_from_md > :last-child {
+#gihwgxbnxi .gt_from_md > :last-child {
   margin-bottom: 0;
 }
 
-#qlfavycsif .gt_row {
+#gihwgxbnxi .gt_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -2140,7 +2140,7 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   overflow-x: hidden;
 }
 
-#qlfavycsif .gt_stub {
+#gihwgxbnxi .gt_stub {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -2153,7 +2153,7 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   padding-right: 5px;
 }
 
-#qlfavycsif .gt_stub_row_group {
+#gihwgxbnxi .gt_stub_row_group {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -2167,15 +2167,15 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   vertical-align: top;
 }
 
-#qlfavycsif .gt_row_group_first td {
+#gihwgxbnxi .gt_row_group_first td {
   border-top-width: 2px;
 }
 
-#qlfavycsif .gt_row_group_first th {
+#gihwgxbnxi .gt_row_group_first th {
   border-top-width: 2px;
 }
 
-#qlfavycsif .gt_summary_row {
+#gihwgxbnxi .gt_summary_row {
   color: #333333;
   background-color: #FFFFFF;
   text-transform: inherit;
@@ -2185,16 +2185,16 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   padding-right: 5px;
 }
 
-#qlfavycsif .gt_first_summary_row {
+#gihwgxbnxi .gt_first_summary_row {
   border-top-style: solid;
   border-top-color: #D3D3D3;
 }
 
-#qlfavycsif .gt_first_summary_row.thick {
+#gihwgxbnxi .gt_first_summary_row.thick {
   border-top-width: 2px;
 }
 
-#qlfavycsif .gt_last_summary_row {
+#gihwgxbnxi .gt_last_summary_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -2204,7 +2204,7 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   border-bottom-color: #D3D3D3;
 }
 
-#qlfavycsif .gt_grand_summary_row {
+#gihwgxbnxi .gt_grand_summary_row {
   color: #333333;
   background-color: #FFFFFF;
   text-transform: inherit;
@@ -2214,7 +2214,7 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   padding-right: 5px;
 }
 
-#qlfavycsif .gt_first_grand_summary_row {
+#gihwgxbnxi .gt_first_grand_summary_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -2224,7 +2224,7 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   border-top-color: #D3D3D3;
 }
 
-#qlfavycsif .gt_last_grand_summary_row_top {
+#gihwgxbnxi .gt_last_grand_summary_row_top {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -2234,11 +2234,11 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   border-bottom-color: #D3D3D3;
 }
 
-#qlfavycsif .gt_striped {
+#gihwgxbnxi .gt_striped {
   background-color: rgba(128, 128, 128, 0.05);
 }
 
-#qlfavycsif .gt_table_body {
+#gihwgxbnxi .gt_table_body {
   border-top-style: solid;
   border-top-width: 2px;
   border-top-color: #D3D3D3;
@@ -2247,7 +2247,7 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   border-bottom-color: #D3D3D3;
 }
 
-#qlfavycsif .gt_footnotes {
+#gihwgxbnxi .gt_footnotes {
   color: #333333;
   background-color: #FFFFFF;
   border-bottom-style: none;
@@ -2261,7 +2261,7 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   border-right-color: #D3D3D3;
 }
 
-#qlfavycsif .gt_footnote {
+#gihwgxbnxi .gt_footnote {
   margin: 0px;
   font-size: 90%;
   padding-top: 4px;
@@ -2270,7 +2270,7 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   padding-right: 5px;
 }
 
-#qlfavycsif .gt_sourcenotes {
+#gihwgxbnxi .gt_sourcenotes {
   color: #333333;
   background-color: #FFFFFF;
   border-bottom-style: none;
@@ -2284,7 +2284,7 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   border-right-color: #D3D3D3;
 }
 
-#qlfavycsif .gt_sourcenote {
+#gihwgxbnxi .gt_sourcenote {
   font-size: 90%;
   padding-top: 4px;
   padding-bottom: 4px;
@@ -2292,63 +2292,63 @@ <h4 id="binary-counting">Binary counting<a class="anchor" aria-label="anchor" hr
   padding-right: 5px;
 }
 
-#qlfavycsif .gt_left {
+#gihwgxbnxi .gt_left {
   text-align: left;
 }
 
-#qlfavycsif .gt_center {
+#gihwgxbnxi .gt_center {
   text-align: center;
 }
 
-#qlfavycsif .gt_right {
+#gihwgxbnxi .gt_right {
   text-align: right;
   font-variant-numeric: tabular-nums;
 }
 
-#qlfavycsif .gt_font_normal {
+#gihwgxbnxi .gt_font_normal {
   font-weight: normal;
 }
 
-#qlfavycsif .gt_font_bold {
+#gihwgxbnxi .gt_font_bold {
   font-weight: bold;
 }
 
-#qlfavycsif .gt_font_italic {
+#gihwgxbnxi .gt_font_italic {
   font-style: italic;
 }
 
-#qlfavycsif .gt_super {
+#gihwgxbnxi .gt_super {
   font-size: 65%;
 }
 
-#qlfavycsif .gt_footnote_marks {
+#gihwgxbnxi .gt_footnote_marks {
   font-size: 75%;
   vertical-align: 0.4em;
   position: initial;
 }
 
-#qlfavycsif .gt_asterisk {
+#gihwgxbnxi .gt_asterisk {
   font-size: 100%;
   vertical-align: 0;
 }
 
-#qlfavycsif .gt_indent_1 {
+#gihwgxbnxi .gt_indent_1 {
   text-indent: 5px;
 }
 
-#qlfavycsif .gt_indent_2 {
+#gihwgxbnxi .gt_indent_2 {
   text-indent: 10px;
 }
 
-#qlfavycsif .gt_indent_3 {
+#gihwgxbnxi .gt_indent_3 {
   text-indent: 15px;
 }
 
-#qlfavycsif .gt_indent_4 {
+#gihwgxbnxi .gt_indent_4 {
   text-indent: 20px;
 }
 
-#qlfavycsif .gt_indent_5 {
+#gihwgxbnxi .gt_indent_5 {
   text-indent: 25px;
 }
 </style>
@@ -2655,23 +2655,23 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
 formula calculates which hypothesis to delete:
 <code>rep(rev(seq_len(num_hyps)), 2^(seq_len(num_hyps) - 1))</code>.
 This also applies to rows 2 through 16.</p>
-<div id="wowtdxzaub" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
-<style>#wowtdxzaub table {
+<div id="yrxusaexyd" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
+<style>#yrxusaexyd table {
   font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
   -webkit-font-smoothing: antialiased;
   -moz-osx-font-smoothing: grayscale;
 }
 
-#wowtdxzaub thead, #wowtdxzaub tbody, #wowtdxzaub tfoot, #wowtdxzaub tr, #wowtdxzaub td, #wowtdxzaub th {
+#yrxusaexyd thead, #yrxusaexyd tbody, #yrxusaexyd tfoot, #yrxusaexyd tr, #yrxusaexyd td, #yrxusaexyd th {
   border-style: none;
 }
 
-#wowtdxzaub p {
+#yrxusaexyd p {
   margin: 0;
   padding: 0;
 }
 
-#wowtdxzaub .gt_table {
+#yrxusaexyd .gt_table {
   display: table;
   border-collapse: collapse;
   line-height: normal;
@@ -2697,12 +2697,12 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   border-left-color: #D3D3D3;
 }
 
-#wowtdxzaub .gt_caption {
+#yrxusaexyd .gt_caption {
   padding-top: 4px;
   padding-bottom: 4px;
 }
 
-#wowtdxzaub .gt_title {
+#yrxusaexyd .gt_title {
   color: #333333;
   font-size: 125%;
   font-weight: initial;
@@ -2714,7 +2714,7 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   border-bottom-width: 0;
 }
 
-#wowtdxzaub .gt_subtitle {
+#yrxusaexyd .gt_subtitle {
   color: #333333;
   font-size: 85%;
   font-weight: initial;
@@ -2726,7 +2726,7 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   border-top-width: 0;
 }
 
-#wowtdxzaub .gt_heading {
+#yrxusaexyd .gt_heading {
   background-color: #FFFFFF;
   text-align: center;
   border-bottom-color: #FFFFFF;
@@ -2738,13 +2738,13 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   border-right-color: #D3D3D3;
 }
 
-#wowtdxzaub .gt_bottom_border {
+#yrxusaexyd .gt_bottom_border {
   border-bottom-style: solid;
   border-bottom-width: 2px;
   border-bottom-color: #D3D3D3;
 }
 
-#wowtdxzaub .gt_col_headings {
+#yrxusaexyd .gt_col_headings {
   border-top-style: solid;
   border-top-width: 2px;
   border-top-color: #D3D3D3;
@@ -2759,7 +2759,7 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   border-right-color: #D3D3D3;
 }
 
-#wowtdxzaub .gt_col_heading {
+#yrxusaexyd .gt_col_heading {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -2779,7 +2779,7 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   overflow-x: hidden;
 }
 
-#wowtdxzaub .gt_column_spanner_outer {
+#yrxusaexyd .gt_column_spanner_outer {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -2791,15 +2791,15 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   padding-right: 4px;
 }
 
-#wowtdxzaub .gt_column_spanner_outer:first-child {
+#yrxusaexyd .gt_column_spanner_outer:first-child {
   padding-left: 0;
 }
 
-#wowtdxzaub .gt_column_spanner_outer:last-child {
+#yrxusaexyd .gt_column_spanner_outer:last-child {
   padding-right: 0;
 }
 
-#wowtdxzaub .gt_column_spanner {
+#yrxusaexyd .gt_column_spanner {
   border-bottom-style: solid;
   border-bottom-width: 2px;
   border-bottom-color: #D3D3D3;
@@ -2811,11 +2811,11 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   width: 100%;
 }
 
-#wowtdxzaub .gt_spanner_row {
+#yrxusaexyd .gt_spanner_row {
   border-bottom-style: hidden;
 }
 
-#wowtdxzaub .gt_group_heading {
+#yrxusaexyd .gt_group_heading {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -2841,7 +2841,7 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   text-align: left;
 }
 
-#wowtdxzaub .gt_empty_group_heading {
+#yrxusaexyd .gt_empty_group_heading {
   padding: 0.5px;
   color: #333333;
   background-color: #FFFFFF;
@@ -2856,15 +2856,15 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   vertical-align: middle;
 }
 
-#wowtdxzaub .gt_from_md > :first-child {
+#yrxusaexyd .gt_from_md > :first-child {
   margin-top: 0;
 }
 
-#wowtdxzaub .gt_from_md > :last-child {
+#yrxusaexyd .gt_from_md > :last-child {
   margin-bottom: 0;
 }
 
-#wowtdxzaub .gt_row {
+#yrxusaexyd .gt_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -2883,7 +2883,7 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   overflow-x: hidden;
 }
 
-#wowtdxzaub .gt_stub {
+#yrxusaexyd .gt_stub {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -2896,7 +2896,7 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   padding-right: 5px;
 }
 
-#wowtdxzaub .gt_stub_row_group {
+#yrxusaexyd .gt_stub_row_group {
   color: #333333;
   background-color: #FFFFFF;
   font-size: 100%;
@@ -2910,15 +2910,15 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   vertical-align: top;
 }
 
-#wowtdxzaub .gt_row_group_first td {
+#yrxusaexyd .gt_row_group_first td {
   border-top-width: 2px;
 }
 
-#wowtdxzaub .gt_row_group_first th {
+#yrxusaexyd .gt_row_group_first th {
   border-top-width: 2px;
 }
 
-#wowtdxzaub .gt_summary_row {
+#yrxusaexyd .gt_summary_row {
   color: #333333;
   background-color: #FFFFFF;
   text-transform: inherit;
@@ -2928,16 +2928,16 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   padding-right: 5px;
 }
 
-#wowtdxzaub .gt_first_summary_row {
+#yrxusaexyd .gt_first_summary_row {
   border-top-style: solid;
   border-top-color: #D3D3D3;
 }
 
-#wowtdxzaub .gt_first_summary_row.thick {
+#yrxusaexyd .gt_first_summary_row.thick {
   border-top-width: 2px;
 }
 
-#wowtdxzaub .gt_last_summary_row {
+#yrxusaexyd .gt_last_summary_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -2947,7 +2947,7 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   border-bottom-color: #D3D3D3;
 }
 
-#wowtdxzaub .gt_grand_summary_row {
+#yrxusaexyd .gt_grand_summary_row {
   color: #333333;
   background-color: #FFFFFF;
   text-transform: inherit;
@@ -2957,7 +2957,7 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   padding-right: 5px;
 }
 
-#wowtdxzaub .gt_first_grand_summary_row {
+#yrxusaexyd .gt_first_grand_summary_row {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -2967,7 +2967,7 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   border-top-color: #D3D3D3;
 }
 
-#wowtdxzaub .gt_last_grand_summary_row_top {
+#yrxusaexyd .gt_last_grand_summary_row_top {
   padding-top: 8px;
   padding-bottom: 8px;
   padding-left: 5px;
@@ -2977,11 +2977,11 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   border-bottom-color: #D3D3D3;
 }
 
-#wowtdxzaub .gt_striped {
+#yrxusaexyd .gt_striped {
   background-color: rgba(128, 128, 128, 0.05);
 }
 
-#wowtdxzaub .gt_table_body {
+#yrxusaexyd .gt_table_body {
   border-top-style: solid;
   border-top-width: 2px;
   border-top-color: #D3D3D3;
@@ -2990,7 +2990,7 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   border-bottom-color: #D3D3D3;
 }
 
-#wowtdxzaub .gt_footnotes {
+#yrxusaexyd .gt_footnotes {
   color: #333333;
   background-color: #FFFFFF;
   border-bottom-style: none;
@@ -3004,7 +3004,7 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   border-right-color: #D3D3D3;
 }
 
-#wowtdxzaub .gt_footnote {
+#yrxusaexyd .gt_footnote {
   margin: 0px;
   font-size: 90%;
   padding-top: 4px;
@@ -3013,7 +3013,7 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   padding-right: 5px;
 }
 
-#wowtdxzaub .gt_sourcenotes {
+#yrxusaexyd .gt_sourcenotes {
   color: #333333;
   background-color: #FFFFFF;
   border-bottom-style: none;
@@ -3027,7 +3027,7 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   border-right-color: #D3D3D3;
 }
 
-#wowtdxzaub .gt_sourcenote {
+#yrxusaexyd .gt_sourcenote {
   font-size: 90%;
   padding-top: 4px;
   padding-bottom: 4px;
@@ -3035,63 +3035,63 @@ <h3 id="formula-shortcut">Formula shortcut<a class="anchor" aria-label="anchor"
   padding-right: 5px;
 }
 
-#wowtdxzaub .gt_left {
+#yrxusaexyd .gt_left {
   text-align: left;
 }
 
-#wowtdxzaub .gt_center {
+#yrxusaexyd .gt_center {
   text-align: center;
 }
 
-#wowtdxzaub .gt_right {
+#yrxusaexyd .gt_right {
   text-align: right;
   font-variant-numeric: tabular-nums;
 }
 
-#wowtdxzaub .gt_font_normal {
+#yrxusaexyd .gt_font_normal {
   font-weight: normal;
 }
 
-#wowtdxzaub .gt_font_bold {
+#yrxusaexyd .gt_font_bold {
   font-weight: bold;
 }
 
-#wowtdxzaub .gt_font_italic {
+#yrxusaexyd .gt_font_italic {
   font-style: italic;
 }
 
-#wowtdxzaub .gt_super {
+#yrxusaexyd .gt_super {
   font-size: 65%;
 }
 
-#wowtdxzaub .gt_footnote_marks {
+#yrxusaexyd .gt_footnote_marks {
   font-size: 75%;
   vertical-align: 0.4em;
   position: initial;
 }
 
-#wowtdxzaub .gt_asterisk {
+#yrxusaexyd .gt_asterisk {
   font-size: 100%;
   vertical-align: 0;
 }
 
-#wowtdxzaub .gt_indent_1 {
+#yrxusaexyd .gt_indent_1 {
   text-indent: 5px;
 }
 
-#wowtdxzaub .gt_indent_2 {
+#yrxusaexyd .gt_indent_2 {
   text-indent: 10px;
 }
 
-#wowtdxzaub .gt_indent_3 {
+#yrxusaexyd .gt_indent_3 {
   text-indent: 15px;
 }
 
-#wowtdxzaub .gt_indent_4 {
+#yrxusaexyd .gt_indent_4 {
   text-indent: 20px;
 }
 
-#wowtdxzaub .gt_indent_5 {
+#yrxusaexyd .gt_indent_5 {
   text-indent: 25px;
 }
 </style>
diff --git a/articles/generate-closure_files/figure-html/base-graph-1.png b/articles/generate-closure_files/figure-html/base-graph-1.png
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diff --git a/articles/graph-examples.html b/articles/graph-examples.html
index f4dabbac..8532bfcc 100644
--- a/articles/graph-examples.html
+++ b/articles/graph-examples.html
@@ -6,7 +6,7 @@
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 <meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no">
 <meta name="description" content="graphicalMCP">
-<title>Commonly-used graph examples • graphicalMCP</title>
+<title>Common procedures • graphicalMCP</title>
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+<!-- bootstrap-toc --><script src="https://cdn.jsdelivr.net/gh/afeld/bootstrap-toc@v1.0.1/dist/bootstrap-toc.min.js" integrity="sha256-4veVQbu7//Lk5TSmc7YV48MxtMy98e26cf5MrgZYnwo=" crossorigin="anonymous"></script><!-- headroom.js --><script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.11.0/headroom.min.js" integrity="sha256-AsUX4SJE1+yuDu5+mAVzJbuYNPHj/WroHuZ8Ir/CkE0=" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.11.0/jQuery.headroom.min.js" integrity="sha256-ZX/yNShbjqsohH1k95liqY9Gd8uOiE1S4vZc+9KQ1K4=" crossorigin="anonymous"></script><!-- clipboard.js --><script src="https://cdnjs.cloudflare.com/ajax/libs/clipboard.js/2.0.6/clipboard.min.js" integrity="sha256-inc5kl9MA1hkeYUt+EC3BhlIgyp/2jDIyBLS6k3UxPI=" crossorigin="anonymous"></script><!-- search --><script src="https://cdnjs.cloudflare.com/ajax/libs/fuse.js/6.4.6/fuse.js" integrity="sha512-zv6Ywkjyktsohkbp9bb45V6tEMoWhzFzXis+LrMehmJZZSys19Yxf1dopHx7WzIKxr5tK2dVcYmaCk2uqdjF4A==" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/autocomplete.js/0.38.0/autocomplete.jquery.min.js" integrity="sha512-GU9ayf+66Xx2TmpxqJpliWbT5PiGYxpaG8rfnBEk1LL8l1KGkRShhngwdXK1UgqhAzWpZHSiYPc09/NwDQIGyg==" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/mark.js/8.11.1/mark.min.js" integrity="sha512-5CYOlHXGh6QpOFA/TeTylKLWfB3ftPsde7AnmhuitiTX4K5SqCLBeKro6sPS8ilsz1Q4NRx3v8Ko2IBiszzdww==" crossorigin="anonymous"></script><!-- pkgdown --><script src="../pkgdown.js"></script><meta property="og:title" content="Common procedures">
 <meta property="og:description" content="graphicalMCP">
 <meta property="og:image" content="https://urban-sniffle-p11zlpj.pages.github.io/logo.png">
 <!-- mathjax --><script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js" integrity="sha256-nvJJv9wWKEm88qvoQl9ekL2J+k/RWIsaSScxxlsrv8k=" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/config/TeX-AMS-MML_HTMLorMML.js" integrity="sha256-84DKXVJXs0/F8OTMzX4UR909+jtl4G7SPypPavF+GfA=" crossorigin="anonymous"></script><!--[if lt IE 9]>
@@ -49,7 +49,7 @@
   <a href="#" class="nav-link dropdown-toggle" data-bs-toggle="dropdown" role="button" aria-expanded="false" aria-haspopup="true" id="dropdown-articles">Articles</a>
   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
@@ -77,7 +77,7 @@
 
 <div class="row">
   <main id="main" class="col-md-9"><div class="page-header">
-      <img src="../logo.png" class="logo" alt=""><h1>Commonly-used graph examples</h1>
+      <img src="../logo.png" class="logo" alt=""><h1>Common procedures</h1>
             
       
       <small class="dont-index">Source: <a href="https://github.com/Gilead-BioStats/graphicalMCP/blob/HEAD/vignettes/graph-examples.Rmd" class="external-link"><code>vignettes/graph-examples.Rmd</code></a></small>
@@ -91,31 +91,57 @@
 <div class="section level2">
 <h2 id="introduction">Introduction<a class="anchor" aria-label="anchor" href="#introduction"></a>
 </h2>
-<div class="section level3">
-<h3 id="multiple-comparison-procedures">Multiple comparison procedures<a class="anchor" aria-label="anchor" href="#multiple-comparison-procedures"></a>
-</h3>
-<p>In confirmatory clinical trials with multiple endpoints,</p>
-<p>Any valid graph can be made with <code><a href="../reference/graph_create.html">graph_create()</a></code>, but
-some patterns have been established over time. We demonstrate how to run
-some of these patterns with graphicalMCP.</p>
-</div>
+<p>In confirmatory clinical trials with multiple endpoints, regulations
+mandate the strong control of the family-wise error rate to reduce the
+risk of false positives. Multiple comparison procedures (MCPs) are used
+to spread the total significance out over multiple hypotheses. In this
+article we’ll demonstrate how to go from a few standard procedures to
+more advanced procedures using the graphical approach.</p>
 </div>
 <div class="section level2">
-<h2 id="bonferroni-holm">Bonferroni-Holm<a class="anchor" aria-label="anchor" href="#bonferroni-holm"></a>
+<h2 id="simple-procedures">Simple procedures<a class="anchor" aria-label="anchor" href="#simple-procedures"></a>
 </h2>
+<div class="section level3">
+<h3 id="bonferroni">Bonferroni<a class="anchor" aria-label="anchor" href="#bonferroni"></a>
+</h3>
+<p>Probably the most common MCP is the Bonferroni correction. It simply
+divides the alpha level evenly between all hypotheses. No significance
+adjustment is made if a hypothesis is rejected. The Bonferroni
+correction can be performed easily without a graphical approach, but
+here is a graphical example of it.</p>
 <div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="va">transitions</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/matrix.html" class="external-link">matrix</a></span><span class="op">(</span><span class="fl">0</span>, <span class="fl">5</span>, <span class="fl">5</span><span class="op">)</span></span>
+<span></span>
+<span><span class="va">bonferroni_graph</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/graph_create.html">graph_create</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/rep.html" class="external-link">rep</a></span><span class="op">(</span><span class="fl">1</span> <span class="op">/</span> <span class="fl">5</span>, <span class="fl">5</span><span class="op">)</span>, <span class="va">transitions</span><span class="op">)</span></span>
+<span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">bonferroni_graph</span>, layout <span class="op">=</span> <span class="fu">igraph</span><span class="fu">::</span><span class="va"><a href="https://r.igraph.org/reference/layout_in_circle.html" class="external-link">layout_in_circle</a></span>, vertex.size <span class="op">=</span> <span class="fl">30</span><span class="op">)</span></span></code></pre></div>
+<p><img src="graph-examples_files/figure-html/bonferroni-1.png" width="480" style="display: block; margin: auto;"></p>
+</div>
+<div class="section level3">
+<h3 id="holm">Holm<a class="anchor" aria-label="anchor" href="#holm"></a>
+</h3>
+<p>The Holm procedure starts with identical weights to Bonferroni, but
+weights are updated uniformly when a hypothesis is rejected. Because of
+this updating, it is always at least as powerful as the Bonferroni
+procedure.</p>
+<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r">
 <code class="sourceCode R"><span><span class="va">transitions</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/matrix.html" class="external-link">matrix</a></span><span class="op">(</span><span class="fl">1</span> <span class="op">/</span> <span class="fl">4</span>, <span class="fl">5</span>, <span class="fl">5</span><span class="op">)</span></span>
 <span><span class="fu"><a href="https://rdrr.io/r/base/diag.html" class="external-link">diag</a></span><span class="op">(</span><span class="va">transitions</span><span class="op">)</span> <span class="op">&lt;-</span> <span class="fl">0</span></span>
 <span></span>
-<span><span class="va">bonferroni_holm_graph</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/graph_create.html">graph_create</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/rep.html" class="external-link">rep</a></span><span class="op">(</span><span class="fl">1</span> <span class="op">/</span> <span class="fl">5</span>, <span class="fl">5</span><span class="op">)</span>, <span class="va">transitions</span><span class="op">)</span></span>
+<span><span class="va">holm_graph</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/graph_create.html">graph_create</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/rep.html" class="external-link">rep</a></span><span class="op">(</span><span class="fl">1</span> <span class="op">/</span> <span class="fl">5</span>, <span class="fl">5</span><span class="op">)</span>, <span class="va">transitions</span><span class="op">)</span></span>
 <span></span>
-<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">bonferroni_holm_graph</span><span class="op">)</span></span></code></pre></div>
-<p><img src="graph-examples_files/figure-html/bonferroni-holm-1.png" width="576" style="display: block; margin: auto;"></p>
+<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">holm_graph</span>, layout <span class="op">=</span> <span class="fu">igraph</span><span class="fu">::</span><span class="va"><a href="https://r.igraph.org/reference/layout_in_circle.html" class="external-link">layout_in_circle</a></span>, vertex.size <span class="op">=</span> <span class="fl">30</span><span class="op">)</span></span></code></pre></div>
+<p><img src="graph-examples_files/figure-html/bonferroni-holm-1.png" width="480" style="display: block; margin: auto;"></p>
 </div>
-<div class="section level2">
-<h2 id="fixed-sequence">Fixed sequence<a class="anchor" aria-label="anchor" href="#fixed-sequence"></a>
-</h2>
-<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r">
+<div class="section level3">
+<h3 id="fixed-sequence">Fixed sequence<a class="anchor" aria-label="anchor" href="#fixed-sequence"></a>
+</h3>
+<p>Another common procedure because of its simplicity is the fixed
+sequence procedure. It places more emphasis on rejecting hypotheses when
+most hypotheses are expected to have p-values only slightly below alpha,
+where the reduced weights of the Holm procedure would cause most or all
+hypotheses to not be rejected from the start.</p>
+<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
 <code class="sourceCode R"><span><span class="va">fixed_sequence_graph</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/graph_create.html">graph_create</a></span><span class="op">(</span></span>
 <span>  <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">1</span>, <span class="fl">0</span>, <span class="fl">0</span>, <span class="fl">0</span><span class="op">)</span>,</span>
 <span>  <span class="fu"><a href="https://rdrr.io/r/base/cbind.html" class="external-link">rbind</a></span><span class="op">(</span></span>
@@ -126,13 +152,31 @@ <h2 id="fixed-sequence">Fixed sequence<a class="anchor" aria-label="anchor" href
 <span>  <span class="op">)</span></span>
 <span><span class="op">)</span></span>
 <span></span>
-<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">fixed_sequence_graph</span>, layout <span class="op">=</span> <span class="st">"grid"</span>, nrow <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
+<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">fixed_sequence_graph</span>, nrow <span class="op">=</span> <span class="fl">1</span>, asp <span class="op">=</span> <span class="fl">.1</span><span class="op">)</span></span></code></pre></div>
 <p><img src="graph-examples_files/figure-html/fixed-sequence-1.png" width="576" style="display: block; margin: auto;"></p>
+<!-- This paragraph needs work -->
+<p>These procedures are popular because they control the FWER while
+being simple, allowing them to be easily understood and communicated
+about. However, they’re not always the procedures which fit the study
+best, and using more complex procedures can make communication
+difficult. The graphical approach can enable more powerful procedures to
+be used while enabling clinical teams to understand and communicate the
+procedures.</p>
+</div>
 </div>
 <div class="section level2">
-<h2 id="simple-successive">Simple successive<a class="anchor" aria-label="anchor" href="#simple-successive"></a>
+<h2 id="complex-procedures">Complex procedures<a class="anchor" aria-label="anchor" href="#complex-procedures"></a>
 </h2>
-<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
+<div class="section level3">
+<h3 id="simple-successive">Simple successive<a class="anchor" aria-label="anchor" href="#simple-successive"></a>
+</h3>
+<p>The simple successive procedure was introduced by [names] in their
+[year] paper, [<em>title</em>]. It is designed for experiments involving
+four hypotheses: Two pairs of hypotheses, each containing a primary and
+a related secondary hypothesis. Initial weight is split equally between
+pairs and is passed from one pair to the other only when both hypotheses
+in a pair are rejected.</p>
+<div class="sourceCode" id="cb5"><pre class="downlit sourceCode r">
 <code class="sourceCode R"><span><span class="va">simple_successive_graph</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/graph_create.html">graph_create</a></span><span class="op">(</span></span>
 <span>  <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">.5</span>, <span class="fl">.5</span>, <span class="fl">0</span>, <span class="fl">0</span><span class="op">)</span>,</span>
 <span>  <span class="fu"><a href="https://rdrr.io/r/base/cbind.html" class="external-link">rbind</a></span><span class="op">(</span></span>
@@ -143,13 +187,13 @@ <h2 id="simple-successive">Simple successive<a class="anchor" aria-label="anchor
 <span>  <span class="op">)</span></span>
 <span><span class="op">)</span></span>
 <span></span>
-<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">simple_successive_graph</span>, layout <span class="op">=</span> <span class="st">"grid"</span>, nrow <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
+<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">simple_successive_graph</span>, layout <span class="op">=</span> <span class="st">"grid"</span>, nrow <span class="op">=</span> <span class="fl">2</span>, vertex.size <span class="op">=</span> <span class="fl">30</span><span class="op">)</span></span></code></pre></div>
 <p><img src="graph-examples_files/figure-html/simple-successive-1.png" width="480" style="display: block; margin: auto;"></p>
 <p>General successive graphs are a good example where multiple
 variations may be useful with slight differences in starting edge
 weights. Variable edge weights are not currently supported, but they can
 be done for a particular graph with a light wrapper.</p>
-<div class="sourceCode" id="cb5"><pre class="downlit sourceCode r">
+<div class="sourceCode" id="cb6"><pre class="downlit sourceCode r">
 <code class="sourceCode R"><span><span class="va">simple_successive_var</span> <span class="op">&lt;-</span> <span class="kw">function</span><span class="op">(</span><span class="va">gamma</span><span class="op">)</span> <span class="op">{</span></span>
 <span>  <span class="fu"><a href="../reference/graph_create.html">graph_create</a></span><span class="op">(</span></span>
 <span>    <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">.5</span>, <span class="fl">.5</span>, <span class="fl">0</span>, <span class="fl">0</span><span class="op">)</span>,</span>
@@ -161,29 +205,35 @@ <h2 id="simple-successive">Simple successive<a class="anchor" aria-label="anchor
 <span>    <span class="op">)</span></span>
 <span>  <span class="op">)</span></span>
 <span><span class="op">}</span></span></code></pre></div>
-<p>Then multiple variations can be created and compared easily.</p>
-<div class="sourceCode" id="cb6"><pre class="downlit sourceCode r">
+<p>Then multiple variations can be created and compared easily. A
+similar approach could be used for deviating from the equal split of
+hypothesis weights as well.</p>
+<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
 <code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span></span>
 <span>  <span class="fu">simple_successive_var</span><span class="op">(</span><span class="fl">.75</span><span class="op">)</span>,</span>
 <span>  layout <span class="op">=</span> <span class="st">"grid"</span>,</span>
 <span>  nrow <span class="op">=</span> <span class="fl">2</span>,</span>
-<span>  vertex.label.cex <span class="op">=</span> <span class="fl">.7</span>,</span>
-<span>  edge.label.cex <span class="op">=</span> <span class="fl">.7</span></span>
+<span>  vertex.label.cex <span class="op">=</span> <span class="fl">1.4</span>,</span>
+<span>  edge.label.cex <span class="op">=</span> <span class="fl">1.4</span>,</span>
+<span>  vertex.size <span class="op">=</span> <span class="fl">30</span></span>
 <span><span class="op">)</span></span>
 <span></span>
 <span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span></span>
 <span>  <span class="fu">simple_successive_var</span><span class="op">(</span><span class="fl">.9</span><span class="op">)</span>,</span>
 <span>  layout <span class="op">=</span> <span class="st">"grid"</span>,</span>
 <span>  nrow <span class="op">=</span> <span class="fl">2</span>,</span>
-<span>  vertex.label.cex <span class="op">=</span> <span class="fl">.7</span>,</span>
-<span>  edge.label.cex <span class="op">=</span> <span class="fl">.7</span></span>
+<span>  vertex.label.cex <span class="op">=</span> <span class="fl">1.4</span>,</span>
+<span>  edge.label.cex <span class="op">=</span> <span class="fl">1.4</span>,</span>
+<span>  vertex.size <span class="op">=</span> <span class="fl">30</span></span>
 <span><span class="op">)</span></span></code></pre></div>
 <p><img src="graph-examples_files/figure-html/simple-successive-var-1.png" width="47%"><img src="graph-examples_files/figure-html/simple-successive-var-2.png" width="47%"></p>
 </div>
-<div class="section level2">
-<h2 id="huque-alosh-bhore-2011">Huque-Alosh-Bhore (2011)<a class="anchor" aria-label="anchor" href="#huque-alosh-bhore-2011"></a>
-</h2>
-<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
+<div class="section level3">
+<h3 id="huque-alosh-bhore-2011">Huque-Alosh-Bhore (2011)<a class="anchor" aria-label="anchor" href="#huque-alosh-bhore-2011"></a>
+</h3>
+<p>In 2011, Huque, Alosh, and Bhore proposed another size 4 procedure.
+What’s this one particularly meant for?</p>
+<div class="sourceCode" id="cb8"><pre class="downlit sourceCode r">
 <code class="sourceCode R"><span><span class="va">hab_2011_graph</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/graph_create.html">graph_create</a></span><span class="op">(</span></span>
 <span>  <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">1</span>, <span class="fl">0</span>, <span class="fl">0</span>, <span class="fl">0</span><span class="op">)</span>,</span>
 <span>  <span class="fu"><a href="https://rdrr.io/r/base/cbind.html" class="external-link">rbind</a></span><span class="op">(</span></span>
@@ -194,13 +244,13 @@ <h2 id="huque-alosh-bhore-2011">Huque-Alosh-Bhore (2011)<a class="anchor" aria-l
 <span>  <span class="op">)</span></span>
 <span><span class="op">)</span></span>
 <span></span>
-<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">hab_2011_graph</span>, layout <span class="op">=</span> <span class="st">"grid"</span><span class="op">)</span></span></code></pre></div>
+<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">hab_2011_graph</span>, layout <span class="op">=</span> <span class="st">"grid"</span>, vertex.size <span class="op">=</span> <span class="fl">30</span><span class="op">)</span></span></code></pre></div>
 <p><img src="graph-examples_files/figure-html/huque-alosh-bhore-1.png" width="480" style="display: block; margin: auto;"></p>
 </div>
-<div class="section level2">
-<h2 id="wiens-dmitrienko-2005">Wiens-Dmitrienko (2005)<a class="anchor" aria-label="anchor" href="#wiens-dmitrienko-2005"></a>
-</h2>
-<div class="sourceCode" id="cb8"><pre class="downlit sourceCode r">
+<div class="section level3">
+<h3 id="wiens-dmitrienko-2005">Wiens-Dmitrienko (2005)<a class="anchor" aria-label="anchor" href="#wiens-dmitrienko-2005"></a>
+</h3>
+<div class="sourceCode" id="cb9"><pre class="downlit sourceCode r">
 <code class="sourceCode R"><span><span class="va">wd_2005_graph</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/graph_create.html">graph_create</a></span><span class="op">(</span></span>
 <span>  <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">1</span> <span class="op">/</span> <span class="fl">3</span>, <span class="fl">1</span> <span class="op">/</span> <span class="fl">3</span>, <span class="fl">1</span> <span class="op">/</span> <span class="fl">3</span><span class="op">)</span>,</span>
 <span>  <span class="fu"><a href="https://rdrr.io/r/base/cbind.html" class="external-link">rbind</a></span><span class="op">(</span></span>
@@ -214,14 +264,18 @@ <h2 id="wiens-dmitrienko-2005">Wiens-Dmitrienko (2005)<a class="anchor" aria-lab
 <span>  <span class="va">wd_2005_graph</span>,</span>
 <span>  layout <span class="op">=</span> <span class="st">"grid"</span>,</span>
 <span>  nrow <span class="op">=</span> <span class="fl">1</span>,</span>
-<span>  edge_curves <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>pairs <span class="op">=</span> <span class="op">-</span><span class="fl">6</span>, <span class="st">"H3|H1"</span> <span class="op">=</span> <span class="op">-</span><span class="fl">6</span><span class="op">)</span></span>
+<span>  edge_curves <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>pairs <span class="op">=</span> <span class="op">-</span><span class="fl">4</span>, <span class="st">"H3|H1"</span> <span class="op">=</span> <span class="op">-</span><span class="fl">4</span><span class="op">)</span>,</span>
+<span>  vertex.size <span class="op">=</span> <span class="fl">30</span>,</span>
+<span>  asp <span class="op">=</span> <span class="fl">.1</span></span>
 <span><span class="op">)</span></span></code></pre></div>
 <p><img src="graph-examples_files/figure-html/wiens-dmitrienko-1.png" width="576" style="display: block; margin: auto;"></p>
 </div>
-<div class="section level2">
-<h2 id="parallel-gate-keeping">Parallel gate-keeping<a class="anchor" aria-label="anchor" href="#parallel-gate-keeping"></a>
-</h2>
-<div class="sourceCode" id="cb9"><pre class="downlit sourceCode r">
+<div class="section level3">
+<h3 id="parallel-gate-keeping">Parallel gate-keeping<a class="anchor" aria-label="anchor" href="#parallel-gate-keeping"></a>
+</h3>
+<p>In [year], [names] introduced the parallel gate-keeping procedure. It
+is useful when…?</p>
+<div class="sourceCode" id="cb10"><pre class="downlit sourceCode r">
 <code class="sourceCode R"><span><span class="va">par_gate_graph</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/graph_create.html">graph_create</a></span><span class="op">(</span></span>
 <span>  <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">.5</span>, <span class="fl">.5</span>, <span class="fl">0</span>, <span class="fl">0</span><span class="op">)</span>,</span>
 <span>  <span class="fu"><a href="https://rdrr.io/r/base/cbind.html" class="external-link">rbind</a></span><span class="op">(</span></span>
@@ -233,13 +287,26 @@ <h2 id="parallel-gate-keeping">Parallel gate-keeping<a class="anchor" aria-label
 <span><span class="op">)</span></span>
 <span></span>
 <span></span>
-<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">par_gate_graph</span>, layout <span class="op">=</span> <span class="st">"grid"</span>, nrow <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
+<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">par_gate_graph</span>, layout <span class="op">=</span> <span class="st">"grid"</span>, nrow <span class="op">=</span> <span class="fl">2</span>, vertex.size <span class="op">=</span> <span class="fl">30</span><span class="op">)</span></span></code></pre></div>
 <p><img src="graph-examples_files/figure-html/parallel-gatekeeping-1.png" width="480" style="display: block; margin: auto;"></p>
 </div>
-<div class="section level2">
-<h2 id="improved-parallel-gate-keeping">Improved parallel gate-keeping<a class="anchor" aria-label="anchor" href="#improved-parallel-gate-keeping"></a>
-</h2>
-<div class="sourceCode" id="cb10"><pre class="downlit sourceCode r">
+<div class="section level3">
+<h3 id="improved-parallel-gate-keeping">Improved parallel gate-keeping<a class="anchor" aria-label="anchor" href="#improved-parallel-gate-keeping"></a>
+</h3>
+<p>One shortcoming of the parallel gate-keeping approach is in the case
+where e.g. H2, H3, and H4 are found to be significant. Once these three
+have been deleted, H1 is left with only 0.5 weight. This makes the graph
+sub-optimal.</p>
+<div class="sourceCode" id="cb11"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="fu"><a href="../reference/graph_update.html">graph_update</a></span><span class="op">(</span><span class="va">par_gate_graph</span>, <span class="fl">2</span><span class="op">:</span><span class="fl">4</span><span class="op">)</span>, vertex.size <span class="op">=</span> <span class="fl">30</span><span class="op">)</span></span></code></pre></div>
+<p><img src="graph-examples_files/figure-html/parallel-gatekeeping-update-1.png" width="480" style="display: block; margin: auto;"></p>
+<p>The graph can be made optimal by adding epsilon edges going back up
+from the secondary to the primary hypotheses. Epsilon edges have some
+infinitesimal weight which creates a connection between hypotheses that
+effectively only propagates weight when it’s the only edge left exiting
+a hypothesis. This yields the improved parallel gate-keeping
+procedure.</p>
+<div class="sourceCode" id="cb12"><pre class="downlit sourceCode r">
 <code class="sourceCode R"><span><span class="va">imp_par_gate_graph</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/graph_create.html">graph_create</a></span><span class="op">(</span></span>
 <span>  <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">.5</span>, <span class="fl">.5</span>, <span class="fl">0</span>, <span class="fl">0</span><span class="op">)</span>,</span>
 <span>  <span class="fu"><a href="https://rdrr.io/r/base/cbind.html" class="external-link">rbind</a></span><span class="op">(</span></span>
@@ -250,9 +317,49 @@ <h2 id="improved-parallel-gate-keeping">Improved parallel gate-keeping<a class="
 <span>  <span class="op">)</span></span>
 <span><span class="op">)</span></span>
 <span></span>
-<span></span>
-<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">imp_par_gate_graph</span>, layout <span class="op">=</span> <span class="st">"grid"</span>, nrow <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
+<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span></span>
+<span>  <span class="va">imp_par_gate_graph</span>,</span>
+<span>  layout <span class="op">=</span> <span class="st">"grid"</span>,</span>
+<span>  nrow <span class="op">=</span> <span class="fl">2</span>,</span>
+<span>  eps <span class="op">=</span> <span class="fl">.001</span>,</span>
+<span>  vertex.size <span class="op">=</span> <span class="fl">30</span></span>
+<span><span class="op">)</span></span></code></pre></div>
 <p><img src="graph-examples_files/figure-html/improved-parallel-gatekeeping-1.png" width="480" style="display: block; margin: auto;"></p>
+<div class="sourceCode" id="cb13"><pre class="downlit sourceCode r">
+<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span></span>
+<span>  <span class="fu"><a href="../reference/graph_update.html">graph_update</a></span><span class="op">(</span><span class="va">imp_par_gate_graph</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">2</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>  vertex.size <span class="op">=</span> <span class="fl">60</span>,</span>
+<span>  vertex.label.cex <span class="op">=</span> <span class="fl">2.5</span>,</span>
+<span>  edge.label.cex <span class="op">=</span> <span class="fl">2.5</span>,</span>
+<span>  eps <span class="op">=</span> <span class="fl">1e-3</span></span>
+<span><span class="op">)</span></span>
+<span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span></span>
+<span>  <span class="fu"><a href="../reference/graph_update.html">graph_update</a></span><span class="op">(</span><span class="va">imp_par_gate_graph</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">2</span>, <span class="fl">4</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>  vertex.size <span class="op">=</span> <span class="fl">60</span>,</span>
+<span>  vertex.label.cex <span class="op">=</span> <span class="fl">2.5</span>,</span>
+<span>  edge.label.cex <span class="op">=</span> <span class="fl">2.5</span></span>
+<span><span class="op">)</span></span>
+<span></span>
+<span><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span></span>
+<span>  <span class="fu"><a href="../reference/graph_update.html">graph_update</a></span><span class="op">(</span><span class="va">imp_par_gate_graph</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">2</span>, <span class="fl">4</span>, <span class="fl">3</span><span class="op">)</span><span class="op">)</span>,</span>
+<span>  vertex.size <span class="op">=</span> <span class="fl">60</span>,</span>
+<span>  vertex.label.cex <span class="op">=</span> <span class="fl">2.5</span>,</span>
+<span>  edge.label.cex <span class="op">=</span> <span class="fl">2.5</span></span>
+<span><span class="op">)</span></span></code></pre></div>
+<p><img src="graph-examples_files/figure-html/improved-parallel-gatekeeping-update-1.png" width="30%"><img src="graph-examples_files/figure-html/improved-parallel-gatekeeping-update-2.png" width="30%"><img src="graph-examples_files/figure-html/improved-parallel-gatekeeping-update-3.png" width="30%"></p>
+</div>
+</div>
+<div class="section level2">
+<h2 id="conclusion">Conclusion<a class="anchor" aria-label="anchor" href="#conclusion"></a>
+</h2>
+<p>These examples are just a few of the unlimited possibilities enabled
+by the graphical approach to MCPs. Graphs can be customized based on
+priorities in trial design, marginal power of the endpoints, and other
+assumptions, such as expected or known correlation between endpoints.
+The graphical approach allows these complex procedures to maximize the
+chances of having a successful trial while enabling better communication
+and insight about the procedures.</p>
 </div>
   </main><aside class="col-md-3"><nav id="toc"><h2>On this page</h2>
     </nav></aside>
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diff --git a/articles/graph-examples_files/figure-html/bonferroni-holm-1.png b/articles/graph-examples_files/figure-html/bonferroni-holm-1.png
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diff --git a/articles/index.html b/articles/index.html
index ea576f17..1934ae48 100644
--- a/articles/index.html
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@@ -25,7 +25,7 @@
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   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
@@ -54,7 +54,7 @@ <h3>All vignettes</h3>
 
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         <dd>
-        </dd><dt><a href="graph-examples.html">Commonly-used graph examples</a></dt>
+        </dd><dt><a href="graph-examples.html">Common procedures</a></dt>
         <dd>
         </dd><dt><a href="shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a></dt>
         <dd>
diff --git a/articles/shortuct-testing.html b/articles/shortuct-testing.html
index fdbbdc4d..74c8a059 100644
--- a/articles/shortuct-testing.html
+++ b/articles/shortuct-testing.html
@@ -49,7 +49,7 @@
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   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
diff --git a/authors.html b/authors.html
index 94d13d67..7472f172 100644
--- a/authors.html
+++ b/authors.html
@@ -25,7 +25,7 @@
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   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
diff --git a/index.html b/index.html
index dc0910cc..fa737bc6 100644
--- a/index.html
+++ b/index.html
@@ -65,7 +65,7 @@
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   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
diff --git a/pkgdown.yml b/pkgdown.yml
index bb4c0600..53e0da18 100644
--- a/pkgdown.yml
+++ b/pkgdown.yml
@@ -5,7 +5,7 @@ articles:
   generate-closure: generate-closure.html
   graph-examples: graph-examples.html
   shortuct-testing: shortuct-testing.html
-last_built: 2023-09-11T13:54Z
+last_built: 2023-09-11T22:03Z
 urls:
   reference: https://urban-sniffle-p11zlpj.pages.github.io/reference
   article: https://urban-sniffle-p11zlpj.pages.github.io/articles
diff --git a/reference/adjust_p.html b/reference/adjust_p.html
index 196f05c7..414e4bcf 100644
--- a/reference/adjust_p.html
+++ b/reference/adjust_p.html
@@ -25,7 +25,7 @@
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   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
diff --git a/reference/adjusted-weights.html b/reference/adjusted-weights.html
index 66fba23b..e9206323 100644
--- a/reference/adjusted-weights.html
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   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
diff --git a/reference/as-graph.html b/reference/as-graph.html
index 567d6d1b..6517ce57 100644
--- a/reference/as-graph.html
+++ b/reference/as-graph.html
@@ -25,7 +25,7 @@
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   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
diff --git a/reference/calc-test_values.html b/reference/calc-test_values.html
index 29b1769f..4b574708 100644
--- a/reference/calc-test_values.html
+++ b/reference/calc-test_values.html
@@ -25,7 +25,7 @@
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   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
diff --git a/reference/example-graphs.html b/reference/example-graphs.html
index 8b03f126..99730c12 100644
--- a/reference/example-graphs.html
+++ b/reference/example-graphs.html
@@ -25,7 +25,7 @@
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   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
diff --git a/reference/graph_calculate_power.html b/reference/graph_calculate_power.html
index 3ea8f1f8..487d51d0 100644
--- a/reference/graph_calculate_power.html
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     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
diff --git a/reference/graph_create.html b/reference/graph_create.html
index c2922a70..e03a3f8b 100644
--- a/reference/graph_create.html
+++ b/reference/graph_create.html
@@ -33,7 +33,7 @@
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   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
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diff --git a/reference/graph_generate_weights.html b/reference/graph_generate_weights.html
index fad6bad0..34aa5b57 100644
--- a/reference/graph_generate_weights.html
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     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
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diff --git a/reference/graph_rejection_orderings.html b/reference/graph_rejection_orderings.html
index 9014621c..a421e5b5 100644
--- a/reference/graph_rejection_orderings.html
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@@ -33,7 +33,7 @@
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   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
diff --git a/reference/graph_update.html b/reference/graph_update.html
index 7d3c9ecb..3de90c7c 100644
--- a/reference/graph_update.html
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@@ -31,7 +31,7 @@
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   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
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diff --git a/reference/graphicalMCP-package.html b/reference/graphicalMCP-package.html
index 6148a351..8040d7b3 100644
--- a/reference/graphicalMCP-package.html
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@@ -27,7 +27,7 @@
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-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
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index d1646c9e..44b7e0d2 100644
--- a/reference/index.html
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-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
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diff --git a/reference/plot.initial_graph.html b/reference/plot.initial_graph.html
index b68ffc1a..60555d85 100644
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   <div class="dropdown-menu" aria-labelledby="dropdown-articles">
     <a class="dropdown-item" href="../articles/generate-closure.html">Generating the closure of a graph</a>
-    <a class="dropdown-item" href="../articles/graph-examples.html">Commonly-used graph examples</a>
+    <a class="dropdown-item" href="../articles/graph-examples.html">Common procedures</a>
     <a class="dropdown-item" href="../articles/shortuct-testing.html">Sequentially rejective graphical approaches based on Bonferroni tests</a>
   </div>
 </li>
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GNU General Public License permit incorporating program proprietary programs. program subroutine library, may consider useful permit linking proprietary applications library. want , use GNU Lesser General Public License instead License. first, please read <http://www.gnu.org/philosophy/--lgpl.html>.","code":"<one line to give the program's name and a brief idea of what it does.> Copyright (C) <year>  <name of author>  This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.  This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more details.  You should have received a copy of the GNU General Public License along with this program.  If not, see <http://www.gnu.org/licenses/>. <program>  Copyright (C) <year>  <name of author> This program comes with ABSOLUTELY NO WARRANTY; for details type 'show w'. This is free software, and you are welcome to redistribute it under certain conditions; type 'show c' for details."},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"what-is-the-closure","dir":"Articles","previous_headings":"","what":"What is the closure?","title":"Generating the closure of a graph","text":"closure graph set sub-graphs, along weights calculated according algorithm 1 Bretz et al (2011). primarily used closed testing, sub-graphs tested significance, results aggregated determine null hypotheses significant globally. Throughout article common example used demonstrations - simple successive graph. two primary hypotheses, \\(H_1\\) \\(H_2\\), initial weight evenly split . secondary hypotheses, \\(H_3\\) \\(H_4\\), weight propagated \\(H_1\\) \\(H_2\\) deleted, respectively.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"components-of-the-closure","dir":"Articles","previous_headings":"What is the closure?","what":"Components of the closure","title":"Generating the closure of a graph","text":"graphicalMCP, closure represented matrix, row represents sub-graph (also called intersection hypothesis), column corresponds individual hypothesis. matrix can created graph_generate_weights(), two parts: indicator matrix showing hypotheses contained sub-graph (-called powerset set), weights matrix containing induced weights sub-graph.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"properties-of-the-closure","dir":"Articles","previous_headings":"What is the closure?","what":"Properties of the closure","title":"Generating the closure of a graph","text":"rows closure generated particular way order give useful properties.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"repeating-recursive-blocks","dir":"Articles","previous_headings":"What is the closure? > Properties of the closure","what":"Repeating recursive blocks","title":"Generating the closure of a graph","text":"First, notice row can obtained row higher matrix flipping single 1 0. example, go row 1 row 3 flipping \\(H_3\\), go row 10 row 14 flipping \\(H_2\\). upper row pairing like can thought “parent” sub-graph, lower row “child” sub-graph. Flipping 0 1 moving matrix called “finding sub-graph’s parent.” Now consider parent-finding strategy left-0 row flipped. reveals pattern bottom half top half, row’s parent bottom half corresponding row top half, eight rows . pattern repeats within box recursively, top half box matching bottom half, first missing hypothesis flipped 0 1.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"binary-counting","dir":"Articles","previous_headings":"What is the closure? > Properties of the closure","what":"Binary counting","title":"Generating the closure of a graph","text":"second useful property somewhat re-framing first, perhaps viewed reason first true. Starting bottom row, powerset particular order counts 1 binary, incrementing 1 per row. means row number can directly calculated vector showing hypotheses currently deleted graph: row_number == number_of_rows - incl_excl_vec_converted_to_base_10 + 1. example, intersection number 6 hypothesis vector 1010. interpreted binary, 1 * 8 + 0 * 4 + 1 * 2 + 0 * 1 = 10 base 10, 6 == 15 - 10 + 1.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"strategies","dir":"Articles","previous_headings":"","what":"Strategies","title":"Generating the closure of a graph","text":"size closure grows quickly graph size increases (n-graph 2^n - 1 sub-graphs), calculating full closure large graphs can computationally intensive. Optimizing process led three main strategies: simplest approach, uses full graph starting point every sub-graph, deletes appropriate hypotheses recursive method, traverses closure tree, deleting one hypothesis time step graphs formulaic shortcut using order graphs generated recursive method Note discussion methods focuses primarily weights side rather powerset side closure. fastest methods discovered generate powerset implicitly missing values weights side. methods simple approach, rely powerset order generate weights, powerset can created efficiently. num_hyps refers number hypotheses initial graph.","code":"powerset <- as.matrix(rev(expand.grid(rep(list(1:0), num_hyps))[-2^num_hyps, ]))"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"simple-approach","dir":"Articles","previous_headings":"Strategies","what":"Simple approach","title":"Generating the closure of a graph","text":"simplest approach generate weights closure apply graph_update() initial graph sub-graph. short sweet write, ’s inefficient hypothesis gets deleted graph multiple times. code simple method.","code":"ggw_simple <- function(graph) {   num_hyps <- length(graph$hypotheses)    matrix_intersections <-     as.matrix(rev(expand.grid(rep(list(1:0), num_hyps))[-2^num_hyps, ]))   colnames(matrix_intersections) <- names(graph$hypotheses)    matrix_weights <- apply(     matrix_intersections,     1,     function(h) graph_update(graph, !h)$updated_graph$hypotheses,     simplify = FALSE   )    cbind(matrix_intersections, do.call(rbind, matrix_weights)) }"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"recursive","dir":"Articles","previous_headings":"Strategies","what":"Recursive","title":"Generating the closure of a graph","text":"recursive method treats space sub-graphs tree, initial graph root, sub-graphs decreasing size going branches. essence recursive step delete hypothesis current graph. every hypothesis every sub-graph tree result taking multiple paths many graphs. key part recursive step memory - graph tree delete hypotheses come hypothesis just deleted reach current graph. base case also needs memory - reached graph one hypothesis left, last-deleted hypothesis number larger current hypotheses. memory recursion enables tree traversal reach unique graph state exactly . implementation recursion, followed wrapper processing sub-graph list standard matrix form.","code":"delete_nodes_recursive <- function(graph, last = 0) {   init_hypotheses <- hypotheses <- graph$hypotheses   init_transitions <- transitions <- graph$transitions    ### base case   int_hyp <- as.integer(names(hypotheses))    is_single_node <- length(hypotheses) == 1   last_is_bigger <- last > max(int_hyp)    if (is_single_node || last_is_bigger) {     return(list(graph))   }    ### recursive step   children <- list()    for (orig_hyp_num in int_hyp[int_hyp > last]) {     del_index <- match(orig_hyp_num, int_hyp)     hyp_nums <- seq_along(hypotheses)[seq_along(hypotheses) != del_index]      for (hyp_num in hyp_nums) {       hypotheses[[hyp_num]] <-         init_hypotheses[[hyp_num]] +         init_hypotheses[[del_index]] * init_transitions[[del_index, hyp_num]]        denominator <- 1 - init_transitions[[hyp_num, del_index]] *         init_transitions[[del_index, hyp_num]]        for (end_num in hyp_nums) {         if (hyp_num == end_num || denominator <= 0) {           transitions[[hyp_num, end_num]] <- 0         } else {           transitions[[hyp_num, end_num]] <- (             init_transitions[[hyp_num, end_num]] +               init_transitions[[hyp_num, del_index]] *                 init_transitions[[del_index, end_num]]           ) / denominator         }       }     }      smaller_graph <- structure(       list(         hypotheses = hypotheses[-del_index],         transitions = as.matrix(transitions[-del_index, -del_index])       ),       class = \"initial_graph\"     )      children[[del_index]] <- delete_nodes_recursive(       smaller_graph,       orig_hyp_num     )   }    c(     unlist(children, recursive = FALSE),     list(graph)   ) } ggw_recursive <- function(graph) {   # The recursion requires the hypotheses to be named sequentially as actual   # numbers for the memory property to work   hyp_names <- names(graph$hypotheses)   names(graph$hypotheses) <- seq_along(graph$hypotheses)   colnames(graph$transitions) <- names(graph$hypotheses)   rownames(graph$transitions) <- names(graph$hypotheses)    # Recursively generate a list of all sub-graphs   list_subgraphs <- delete_nodes_recursive(graph)    # Process the list of graphs into the normal matrix form   matrix_weights <- structure(     do.call(       rbind,       lapply(         list_subgraphs,         function(graph) graph$hypotheses[as.character(seq_along(hyp_names))]       )     ),     dimnames = list(1:(2^length(hyp_names) - 1), hyp_names)   )    matrix_intersections <- !is.na(matrix_weights)   matrix_weights[is.na(matrix_weights)] <- 0    cbind(matrix_intersections, matrix_weights) }"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"formula-shortcut","dir":"Articles","previous_headings":"Strategies","what":"Formula shortcut","title":"Generating the closure of a graph","text":"Finally, fastest method found far - formula shortcut. recursion can save lot time first method, still bit overhead get list sub-graphs matrix form standard. “repeating block” property closure mentioned earlier useful. Instead using recursion connect parent sub-graphs children, pair formulas can used. One formula generates parent graph obtained flipping left-0 1: .call(c, lapply(2^(seq_len(num_hyps) - 1), seq_len)). Note rows 2 (non-existent) row 16, empty graph. Row 1 parent graph. left-0 child graph easy find, parent graph’s perspective, formula calculates hypothesis delete: rep(rev(seq_len(num_hyps)), 2^(seq_len(num_hyps) - 1)). also applies rows 2 16. formula shortcut also results simpler code recursive solution.","code":""},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"generating-the-closure","dir":"Articles","previous_headings":"Performance gains","what":"Generating the closure","title":"Generating the closure of a graph","text":"formula method reduces step generating closure single deletion prior graph, almost additional overhead. ’s different methods fare, including version gMCP package reference. Also worthy note lrstat package, contains MCP-related functions, including generating weights closure. uses excellent C++ code perform even faster, since speed current formula-based method acceptable, adding Rcpp dependency considered worth additional time savings.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"power-simulations","dir":"Articles","previous_headings":"Performance gains","what":"Power simulations","title":"Generating the closure of a graph","text":"time savings closure nice, graphs savings make big difference compared longer run-times e.g. power simulations. However, paying attention closure important reasons , improving power algorithm.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"standard-algorithm","dir":"Articles","previous_headings":"Performance gains > Power simulations","what":"Standard algorithm","title":"Generating the closure of a graph","text":"typical method running Bonferroni shortcut procedure graph : Search graph single hypothesis can rejected Delete rejected hypothesis update weights Repeat significant hypotheses Running process either R low-level language like C fast single procedure, ’s run 100,000 times power simulation, R version becomes onerous. However, ’s lot duplication power simulation using method. many simulations, steps taken, means re-calculating set weights many times.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"closure-shortcut","dir":"Articles","previous_headings":"Performance gains > Power simulations","what":"Closure shortcut","title":"Generating the closure of a graph","text":"binary counting property closure admits shortcut can implemented R get scalable competitor typical algorithm: Generate closure single time get sub-graph weights efficiently simulation: 1. Search graph hypotheses can rejected - fast vectorization 1. Using binary counting property, index row closure corresponding hypotheses rejected far get updated weights - substantially faster updating graph re-calculating weights, especially larger graphs 1. Repeat using updated weights method fast gMCP’s C implementation, still runs substantial power simulations matter seconds. biggest drawback can scale differently different graph structures. instance fixed sequence procedure can take longer balanced graph, like Holm procedure, takes steps reject possible hypotheses.","code":""},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"multiple-comparison-procedures","dir":"Articles","previous_headings":"Introduction","what":"Multiple comparison procedures","title":"Commonly-used graph examples","text":"confirmatory clinical trials multiple endpoints, valid graph can made graph_create(), patterns established time. demonstrate run patterns graphicalMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"bonferroni-holm","dir":"Articles","previous_headings":"","what":"Bonferroni-Holm","title":"Commonly-used graph examples","text":"","code":"transitions <- matrix(1 / 4, 5, 5) diag(transitions) <- 0  bonferroni_holm_graph <- graph_create(rep(1 / 5, 5), transitions)  plot(bonferroni_holm_graph)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"fixed-sequence","dir":"Articles","previous_headings":"","what":"Fixed sequence","title":"Commonly-used graph examples","text":"","code":"fixed_sequence_graph <- graph_create(   c(1, 0, 0, 0),   rbind(     c(0, 1, 0, 0),     c(0, 0, 1, 0),     c(0, 0, 0, 1),     c(0, 0, 0, 0)   ) )  plot(fixed_sequence_graph, layout = \"grid\", nrow = 1)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"simple-successive","dir":"Articles","previous_headings":"","what":"Simple successive","title":"Commonly-used graph examples","text":"General successive graphs good example multiple variations may useful slight differences starting edge weights. Variable edge weights currently supported, can done particular graph light wrapper. multiple variations can created compared easily.","code":"simple_successive_graph <- graph_create(   c(.5, .5, 0, 0),   rbind(     c(0, 0, 1, 0),     c(0, 0, 0, 1),     c(0, 1, 0, 0),     c(1, 0, 0, 0)   ) )  plot(simple_successive_graph, layout = \"grid\", nrow = 2) simple_successive_var <- function(gamma) {   graph_create(     c(.5, .5, 0, 0),     rbind(       c(0, gamma, 1 - gamma, 0),       c(gamma, 0, 0, 1 - gamma),       c(0, 1, 0, 0),       c(1, 0, 0, 0)     )   ) } plot(   simple_successive_var(.75),   layout = \"grid\",   nrow = 2,   vertex.label.cex = .7,   edge.label.cex = .7 )  plot(   simple_successive_var(.9),   layout = \"grid\",   nrow = 2,   vertex.label.cex = .7,   edge.label.cex = .7 )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"huque-alosh-bhore-2011","dir":"Articles","previous_headings":"","what":"Huque-Alosh-Bhore (2011)","title":"Commonly-used graph examples","text":"","code":"hab_2011_graph <- graph_create(   c(1, 0, 0, 0),   rbind(     c(0, .5, .5, 0),     c(0, 0, 0, 1),     c(0, .5, 0, .5),     c(0, 1, 0, 0)   ) )  plot(hab_2011_graph, layout = \"grid\")"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"wiens-dmitrienko-2005","dir":"Articles","previous_headings":"","what":"Wiens-Dmitrienko (2005)","title":"Commonly-used graph examples","text":"","code":"wd_2005_graph <- graph_create(   c(1 / 3, 1 / 3, 1 / 3),   rbind(     c(0, 1, 0),     c(0, 0, 1),     c(.5, .5, 0)   ) )  plot(   wd_2005_graph,   layout = \"grid\",   nrow = 1,   edge_curves = c(pairs = -6, \"H3|H1\" = -6) )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"parallel-gate-keeping","dir":"Articles","previous_headings":"","what":"Parallel gate-keeping","title":"Commonly-used graph examples","text":"","code":"par_gate_graph <- graph_create(   c(.5, .5, 0, 0),   rbind(     c(0, 0, .5, .5),     c(0, 0, .5, .5),     c(0, 0, 0, 1),     c(0, 0, 1, 0)   ) )   plot(par_gate_graph, layout = \"grid\", nrow = 2)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"improved-parallel-gate-keeping","dir":"Articles","previous_headings":"","what":"Improved parallel gate-keeping","title":"Commonly-used graph examples","text":"","code":"imp_par_gate_graph <- graph_create(   c(.5, .5, 0, 0),   rbind(     c(0, 0, .5, .5),     c(0, 0, .5, .5),     c(.0001, 0, 0, .9999),     c(0, .0001, .9999, 0)   ) )   plot(imp_par_gate_graph, layout = \"grid\", nrow = 2)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"multiple-comparison-problem","dir":"Articles","previous_headings":"","what":"Multiple comparison problem","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"Consider confirmatory clinical trial comparing test treatment (treatment) control treatment (control) disease. two doses treatment: low dose high dose. two endpoints included multiplicity adjustment strategy, primary endpoint (PE) secondary endpoint (SE). total, four null hypotheses: \\(H_1\\) \\(H_3\\) primary secondary hypotheses respectively low dose versus control; \\(H_2\\) \\(H_4\\) primary secondary hypotheses respectively high dose versus control. discussing appropriate multiple comparison procedures, clinical considerations, adds structure multiple comparison procedures, can flexibly incorporated using graphical approaches. First, low high doses considered equally important, means rejecting primary hypothesis either dose versus control leads successful trial. Regarding secondary hypotheses, one tested corresponding primary hypothesis rejected. means \\(H_3\\) tested \\(H_1\\) rejected; \\(H_4\\) tested \\(H_2\\) rejected. addition, statistical considerations complete graph. primary hypotheses \\(H_1\\) \\(H_2\\) equal hypothesis weight 0.5. secondary hypotheses hypothesis weight 0. primary hypothesis rejected, weight propagate along two outgoing edges: one primary hypothesis one descendant secondary hypothesis. two edges equal transition weight 0.5. primary secondary hypotheses rejected dose-control comparison, hypothesis weights propagated primary hypothesis dose-control comparison. specifications, can create following graph.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"create-a-graph","dir":"Articles","previous_headings":"","what":"Create a graph","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"","code":"hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   c(0, 0.5, 0.5, 0),   c(0.5, 0, 0, 0.5),   c(0, 1, 0, 0),   c(1, 0, 0, 0) ) hyp_names <- c(\"H1\", \"H2\", \"H3\", \"H4\")  g <- graph_create(hypotheses, transitions, hyp_names)  plot(g, layout = \"grid\", vertex.size = 20)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"perform-the-graphical-multiple-comparison-procedure","dir":"Articles","previous_headings":"","what":"Perform the graphical multiple comparison procedure","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"Given set p-values \\(H_1, \\ldots, H_4\\), graphical multiple comparison procedure can performed control familywise error rate (FWER) significance level alpha. graph_test_shortcut function agnostic one-sided two-sided tests. one-sided p-values, alpha often set 0.025 (default); two-sided p-values, alpha often set 0.05. consider one-sided tests . hypothesis rejected adjusted p-value less equal alpha. running procedure, hypotheses \\(H_1\\), \\(H_2\\), \\(H_4\\) rejected adjusted p-value calculated.","code":"p_values <- c(.013, .012, .105, .005) test_results <- graph_test_shortcut(g, p = p_values, alpha = 0.025) test_results$outputs # Adjusted p-values and rejections #> $adjusted_p #>    H1    H2    H3    H4  #> 0.024 0.024 0.105 0.024  #>  #> $rejected #>    H1    H2    H3    H4  #>  TRUE  TRUE FALSE  TRUE  #>  #> $graph #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0 #> H2: 0 #> H3: 1 #> H4: 0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  0  0 #>  H2  0  0  0  0 #>  H3  0  0  0  0 #>  H4  0  0  0  0"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"obtain-final-and-intermediate-graphs-after-rejections","dir":"Articles","previous_headings":"Perform the graphical multiple comparison procedure","what":"Obtain final and intermediate graphs after rejections","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"addition, two graphs provided: one initial graph one final graph removing rejected hypotheses \\(H_1\\), \\(H_2\\), \\(H_4\\). also interested intermediate graphs - example, graph \\(H_1\\) \\(H_2\\) rejected - can specify verbose = TRUE. Note intermediate graphs depends order rejections, .e., sequence hypotheses rejected. default order defined increasing adjusted p-values earlier hypothesis numbering case ties adjusted p-values. Among rejected hypotheses, one smallest adjusted p-value first rejected one largest adjusted p-value last rejected. example, default order rejection \\(H_2\\rightarrow H_1\\rightarrow H_4\\). obtain intermediate graphs based order rejection, one can specify test_results_verbose$details$results. example, graph \\(H_1\\) \\(H_2\\) rejected given test_results_verbose$details$results[[3]]. order rejections may unique orders valid. example, rejected hypotheses \\(H_1\\), \\(H_2\\) \\(H_4\\). default order rejections \\(H_2 \\rightarrow H_1 \\rightarrow H_4\\). Another valid order rejections \\(H_2 \\rightarrow H_4 \\rightarrow H_1\\). However, first rejected hypothesis can \\(H_1\\) \\(H_4\\). obtain possible rejection orders, one can use function graph_rejection_orderings. Initial, intermediate final graphs can obtained using function graph_update particular order rejections. order intermediate graphs determined deleting hypothesis lowest adjusted p-values step (smaller hypothesis sequentially case adjusted p-value ties). cases, order valid order deletion. example, minimum adjusted p-value method results deleting order \\(H_2 \\rightarrow H_1 \\rightarrow H_4\\). \\(H_2 \\rightarrow H_4 \\rightarrow H_1\\) also valid deletion order. Deletion order can customized using [custom_order_function_once_it_is_written()].","code":"test_results_verbose <-   graph_test_shortcut(g, p = p_values, alpha = 0.025, verbose = TRUE) test_results_verbose$details$results[[3]] # Intermediate graph after H1 and H2 rejected #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.0 #> H2: 0.0 #> H3: 0.5 #> H4: 0.5 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  0  0 #>  H2  0  0  0  0 #>  H3  0  0  0  1 #>  H4  0  0  1  0 orders <- graph_rejection_orderings(test_results)$valid_orderings # Obtain all valid orders of rejections orders #> [[1]] #> H2 H4 H1  #>  2  4  1  #>  #> [[2]] #> H2 H1 H4  #>  2  1  4 graph_update(g, delete = orders[[2]]) # Graphs following the order of H2, H4, and H1 #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>      H1  H2  H3  H4 #>  H1 0.0 0.5 0.5 0.0 #>  H2 0.5 0.0 0.0 0.5 #>  H3 0.0 1.0 0.0 0.0 #>  H4 1.0 0.0 0.0 0.0 #>  #> Updated graph after deleting hypotheses 2, 1, 4 #>  #> --- Hypothesis weights --- #> H1: NA #> H2: NA #> H3:  1 #> H4: NA #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1 NA NA NA NA #>  H2 NA NA NA NA #>  H3 NA NA  0 NA #>  H4 NA NA NA NA #>  #> Deletion sequence ($intermediate_graphs) --------------------------------------- #>  #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>       H1  H2  H3  H4 #>   H1 0.0 0.5 0.5 0.0 #>   H2 0.5 0.0 0.0 0.5 #>   H3 0.0 1.0 0.0 0.0 #>   H4 1.0 0.0 0.0 0.0 #>  #>     Step 1: Updated graph after removing hypothesis 2 #>  #>     --- Hypothesis weights --- #>     H1: 0.75 #>     H2:   NA #>     H3: 0.00 #>     H4: 0.25 #>  #>     --- Transition weights --- #>              H1       H2       H3       H4 #>     H1 0.000000       NA 0.666667 0.333333 #>     H2       NA       NA       NA       NA #>     H3 0.500000       NA 0.000000 0.500000 #>     H4 1.000000       NA 0.000000 0.000000 #>  #>       Step 2: Updated graph after removing hypotheses 2, 1 #>  #>       --- Hypothesis weights --- #>       H1:  NA #>       H2:  NA #>       H3: 0.5 #>       H4: 0.5 #>  #>       --- Transition weights --- #>          H1 H2 H3 H4 #>       H1 NA NA NA NA #>       H2 NA NA NA NA #>       H3 NA NA  0  1 #>       H4 NA NA  1  0 #>  #>         Step 3: Updated graph after removing hypotheses 2, 1, 4 #>  #>         --- Hypothesis weights --- #>         H1: NA #>         H2: NA #>         H3:  1 #>         H4: NA #>  #>         --- Transition weights --- #>            H1 H2 H3 H4 #>         H1 NA NA NA NA #>         H2 NA NA NA NA #>         H3 NA NA  0 NA #>         H4 NA NA NA NA #>  #>   Final updated graph after removing deleted hypotheses #>  #>   --- Hypothesis weights --- #>   H1: NA #>   H2: NA #>   H3:  1 #>   H4: NA #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1 NA NA NA NA #>   H2 NA NA NA NA #>   H3 NA NA  0 NA #>   H4 NA NA NA NA"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"obtain-adjusted-significance-levels","dir":"Articles","previous_headings":"Perform the graphical multiple comparison procedure","what":"Obtain adjusted significance levels","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"equivalent way obtain rejections adjusting significance levels. hypothesis rejected p-value less equal adjusted significance level. adjusted significance levels calculated order adjusted p-values: \\(H_2 \\rightarrow H_1 \\rightarrow H_4\\), four steps checking rejections. First, \\(H_2\\) rejected adjusted significance level 0.5 * alpha. Second, \\(H_1\\) rejected adjusted significance level 0.75 * alpha, \\(H_2\\) rejected. Third, \\(H_4\\) rejected adjusted significance level 0.5 * alpha, \\(H_1\\) \\(H_2\\) rejected. Fourth finally, \\(H_3\\) rejected adjusted significance level alpha, \\(H_1\\), \\(H_2\\) \\(H_4\\) rejected. results can obtained specifying test_values = TRUE.","code":"test_results_test_values <-   graph_test_shortcut(g, p = p_values, alpha = 0.025, test_values = TRUE) test_results_test_values$test_values$results #>    Step Hypothesis     p <= Weight * Alpha Inequality_holds #> H2    1         H2 0.012 <=   0.50 * 0.025             TRUE #> H1    2         H1 0.013 <=   0.75 * 0.025             TRUE #> H4    3         H4 0.005 <=   0.50 * 0.025             TRUE #> H3    4         H3 0.105 <=   1.00 * 0.025            FALSE"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"power-calculation","dir":"Articles","previous_headings":"","what":"Power calculation","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"Given graph, trial team often interested power trial. single null hypothesis, power probability reject null hypothesis significance level alpha alternative hypothesis true. multiple null hypotheses, multiple versions power. example, power reject least one hypothesis power reject hypotheses, given alternative hypotheses true. graphical multiple comparison procedures, also important understand power reject hypothesis, given multiplicity adjustment. Sometimes, team may want customize definitions power define success. Thus power calculation important aspect trial design.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"input-marginal-power-for-primary-hypotheses","dir":"Articles","previous_headings":"Power calculation","what":"Input: Marginal power for primary hypotheses","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"Assume primary endpoint occurrence unfavorable clinical event. evaluate treatment effect, proportion patients event calculated, lower proportion preferred. Assume proportions 0.181 low high doses, 0.3 control. Using equal randomization among three treatment groups, clinical trial team chooses total sample size 600 200 per group. leads marginal power 80% \\(H_1\\) \\(H_2\\), respectively, using two-sample test difference proportions unpooled variance one-sided significance level 0.025. calculation, use marginal power combine information treatment effect, nuisance parameter, sample sizes hypothesis. Note significance level used marginal power calculation must alpha, used power calculation significance level FWER control. addition, marginal power one--one relationship noncentrality parameter, illustrated .","code":"alpha <- 0.025 prop <- c(0.3, 0.181, 0.181) sample_size <- rep(200, 3)  unpooled_variance <- prop[-1] * (1 - prop[-1]) / sample_size[-1] +   prop[1] * (1 - prop[1]) / sample_size[1]  noncentrality_parameter_primary <-   -(prop[-1] - prop[1]) / sqrt(unpooled_variance)  power_marginal_primary <- pnorm(qnorm(alpha, lower.tail = FALSE),   noncentrality_parameter_primary, 1,   lower.tail = FALSE )  names(power_marginal_primary) <- c(\"H1\", \"H2\") power_marginal_primary #>        H1        H2  #> 0.8028315 0.8028315"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"input-marginal-power-for-secondary-hypotheses","dir":"Articles","previous_headings":"Power calculation","what":"Input: Marginal power for secondary hypotheses","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"Assume secondary endpoint change total medication score baseline, continuous outcome. evaluate treatment effect, mean change calculated, greater reduction preferred. Assume mean change baseline reduction 7.5 8.25, respectively low high doses, 5 control. assume known common standard deviation 10. Given sample size 200 per treatment group, marginal power 71% 90% \\(H_3\\) \\(H_4\\), respectively, using two-sample \\(z\\)-test difference means one-sided significance level 0.025.","code":"mean_change <- c(5, 7.5, 8.25) sd <- rep(10, 3) variance <- sd[-1]^2 / sample_size[-1] + sd[1]^2 / sample_size[1]  noncentrality_parameter_secondary <- (mean_change[-1] - mean_change[1]) /   sqrt(variance)  power_marginal_secondary <- pnorm(qnorm(alpha, lower.tail = FALSE),   noncentrality_parameter_secondary, 1,   lower.tail = FALSE )  names(power_marginal_secondary) <- c(\"H3\", \"H4\") power_marginal_secondary #>        H3        H4  #> 0.7054139 0.9014809"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"input-correlation-structure-to-simulate-test-statistics","dir":"Articles","previous_headings":"Power calculation","what":"Input: Correlation structure to simulate test statistics","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"addition marginal power, also need make assumptions joint distribution test statistics. example, assume follow multivariate normal distribution means defined noncentrality parameters correlation matrix \\(R\\). obtain correlations, helpful understand two types correlations example. correlation two dose-control comparisons endpoint correlation endpoints. former correlation can calculated function sample size. example, correlation test statistics \\(H_1\\) \\(H_2\\) \\(\\rho_{12}=\\left(\\frac{n_1}{n_1+n_0}\\right)^{1/2}\\left(\\frac{n_2}{n_3+n_0}\\right)^{1/2}\\). equal randomization, correlation 0.5. correlation test statistics \\(H_3\\) \\(H_4\\) . hand, correlation endpoints dose-control comparison often estimated based prior knowledge previous trials. Without information, assume \\(\\rho_{13}=\\rho_{24}=0.5\\). practice, one set correlation parameter try multiple values assess sensitivity assumption. Regarding correlation test statistics \\(H_1\\) \\(H_4\\) \\(H_2\\) \\(H_3\\), even difficult estimate. use simple product rule, means correlation product correlations two previously assumed correlations. example, \\(\\rho_{14}=\\rho_{12}*\\rho_{24}\\) \\(\\rho_{23}=\\rho_{12}*\\rho_{13}\\). practice, one may make assumptions instead using product rule.","code":"corr <- matrix(0, nrow = 4, ncol = 4)  corr[1, 2] <- corr[3, 4] <- sqrt(sample_size[2] / sum(sample_size[1:2]) *   sample_size[3] / sum(sample_size[c(1, 3)]))  rho <- 0.5 corr[1, 3] <- corr[2, 4] <- rho corr[1, 4] <- corr[2, 3] <- corr[1, 2] * rho corr <- corr + t(corr) diag(corr) <- 1 colnames(corr) <- hyp_names rownames(corr) <- hyp_names corr #>      H1   H2   H3   H4 #> H1 1.00 0.50 0.50 0.25 #> H2 0.50 1.00 0.25 0.50 #> H3 0.50 0.25 1.00 0.50 #> H4 0.25 0.50 0.50 1.00"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"user-defined-success-criteria","dir":"Articles","previous_headings":"Power calculation","what":"User-defined success criteria","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"mentioned earlier, multiple versions “power” multiple hypotheses. Commonly used “power” definitions include: Local power: probability hypothesis rejected (multiplicity adjustment) Expected . rejections: expected number rejections Power reject 1 : probability reject least one hypothesis Power reject : probability reject hypotheses default outputs graph_calculate_power function. addition, user can customize success criteria define versions “power”.","code":"# Probability to reject H1 H1 <- function(x) x[1]  # Expected number of rejections `Expected no. of rejections` <- function(x) x[1] + x[2] + x[3] + x[4]  # Probability to reject at least one hypothesis `AtLeast1` <- function(x) x[1] | x[2] | x[3] | x[4]  # Probability to reject all hypotheses `All` <- function(x) x[1] & x[2] & x[3] & x[4]  # Probability to reject both H1 and H2 `H1andH2` <- function(x) x[1] & x[2]  # Probability to reject both hypotheses for the low dose or the high dose `(H1andH3)or(H2andH4)` <- function(x) (x[1] & x[3]) | (x[2] & x[4]) succ <- list(   H1, `Expected no. of rejections`, `AtLeast1`, `All`, `H1andH2`,   `(H1andH3)or(H2andH4)` )  succ <- setNames(   succ,   c(     \"H1\", \"Expected no. of rejections\", \"AtLeast1\", \"All\", \"H1andH2\",     \"(H1andH3)or(H2andH4)\"   ) )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"output-calculate-power","dir":"Articles","previous_headings":"Power calculation","what":"Output: Calculate power","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"Given inputs, can calculate power via simulation graphical multiple comparison procedure one-sided significance level alpha=0.025 using sim_n = 1e5 simulations seed sim_seed = 1234. local power 0.758, 0.765, 0.689, 0.570, respectively \\(H_1, \\ldots, H_4\\). Note local power lower marginal power former adjusted multiplicity. power reject least one hypothesis 0.856 power reject hypotheses 0.512. expected number rejections 2.782. last two user-defined success criteria, probability reject \\(H_1\\) \\(H_2\\) 0.667, probability reject least one pair \\((H_1\\) \\(H_3)\\) \\((H_2\\) \\(H_4)\\) 0.747. see detailed outputs simulated p-values rejection decisions hypotheses, specify verbose = TRUE. produce lot outputs. allow flexible printing functions, user can change following: indented space default setting indent = 2 precision numeric values (.e., number significant digits) default setting precision = 4","code":"output <- graph_calculate_power(g,   alpha = .025,   sim_corr = corr,   sim_n = 1e5,   power_marginal = c(power_marginal_primary, power_marginal_secondary),   sim_success = succ,   sim_seed = 1234 ) output #>  #> Test parameters ($inputs) ------------------------------------------------------ #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>       H1  H2  H3  H4 #>   H1 0.0 0.5 0.5 0.0 #>   H2 0.5 0.0 0.0 0.5 #>   H3 0.0 1.0 0.0 0.0 #>   H4 1.0 0.0 0.0 0.0 #>  #>   Alpha = 0.025 #>  #>   Test types #>   bonferroni: (H1, H2, H3, H4) #>  #> Simulation parameters ($inputs) ------------------------------------------------ #>   Testing 100,000 simulations - random seed 1234 & multivariate normal params: #>  #>                       H1     H2     H3     H4 #>   Marginal power: 0.8028 0.8028 0.7054 0.9015 #>  #>   Correlation:      H1   H2   H3   H4 #>                H1 1.00 0.50 0.50 0.25 #>                H2 0.50 1.00 0.25 0.50 #>                H3 0.50 0.25 1.00 0.50 #>                H4 0.25 0.50 0.50 1.00 #>  #> Power calculation ($power) ----------------------------------------------------- #>                                   H1     H2     H3     H4 #>                  Local power: 0.7640 0.7589 0.5677 0.6913 #>  #>   Expected no. of rejections: 2.782 #>    Power to reject 1 or more: 0.8556 #>          Power to reject all: 0.5121 #>  #>              Power to reject:      H1   Expected no. of rejections   AtLeast1 #>                                0.7640                       2.7818     0.8556 #>     All   H1andH2   (H1andH3)or(H2andH4) #>  0.5121    0.6673                 0.7470 verbose_output <- graph_calculate_power(g,   alpha = .025,   sim_corr = corr,   sim_n = 1e5,   power_marginal = c(power_marginal_primary, power_marginal_secondary),   sim_success = succ,   sim_seed = 1234,   verbose = TRUE )  print(verbose_output) #>  #> Test parameters ($inputs) ------------------------------------------------------ #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>       H1  H2  H3  H4 #>   H1 0.0 0.5 0.5 0.0 #>   H2 0.5 0.0 0.0 0.5 #>   H3 0.0 1.0 0.0 0.0 #>   H4 1.0 0.0 0.0 0.0 #>  #>   Alpha = 0.025 #>  #>   Test types #>   bonferroni: (H1, H2, H3, H4) #>  #> Simulation parameters ($inputs) ------------------------------------------------ #>   Testing 100,000 simulations - random seed 1234 & multivariate normal params: #>  #>                       H1     H2     H3     H4 #>   Marginal power: 0.8028 0.8028 0.7054 0.9015 #>  #>   Correlation:      H1   H2   H3   H4 #>                H1 1.00 0.50 0.50 0.25 #>                H2 0.50 1.00 0.25 0.50 #>                H3 0.50 0.25 1.00 0.50 #>                H4 0.25 0.50 0.50 1.00 #>  #> Power calculation ($power) ----------------------------------------------------- #>                                   H1     H2     H3     H4 #>                  Local power: 0.7640 0.7589 0.5677 0.6913 #>  #>   Expected no. of rejections: 2.782 #>    Power to reject 1 or more: 0.8556 #>          Power to reject all: 0.5121 #>  #>              Power to reject:      H1   Expected no. of rejections   AtLeast1 #>                                0.7640                       2.7818     0.8556 #>     All   H1andH2   (H1andH3)or(H2andH4) #>  0.5121    0.6673                 0.7470 #>  #> Simulation details ($details) -------------------------------------------------- #>     p_sim_H1  p_sim_H2  p_sim_H3  p_sim_H4 rej_H1 rej_H2 rej_H3 rej_H4 #>    3.082e-02 1.207e-02 4.119e-03 9.324e-02  FALSE   TRUE  FALSE  FALSE #>    7.934e-04 6.499e-04 2.452e-02 2.966e-03   TRUE   TRUE   TRUE   TRUE #>    3.030e-02 5.954e-02 5.431e-02 2.626e-02  FALSE  FALSE  FALSE  FALSE #>    9.743e-03 3.319e-03 7.417e-04 4.025e-04   TRUE   TRUE   TRUE   TRUE #>    1.971e-02 8.616e-03 1.642e-02 2.418e-07   TRUE   TRUE   TRUE   TRUE #>    3.121e-03 6.702e-03 1.374e-02 2.752e-04   TRUE   TRUE   TRUE   TRUE #>    3.022e-02 1.424e-01 6.038e-03 2.117e-02  FALSE  FALSE  FALSE  FALSE #>    2.498e-03 2.940e-02 4.143e-04 2.208e-03   TRUE  FALSE   TRUE  FALSE #>    6.190e-02 3.873e-02 3.166e-01 5.700e-02  FALSE  FALSE  FALSE  FALSE #>    3.678e-01 1.896e-01 7.023e-02 1.190e-02  FALSE  FALSE  FALSE  FALSE #>   ... (Use `print(x, rows = <nn>)` for more)  print(verbose_output, indent = 4, precision = 6) #>  #> Test parameters ($inputs) ------------------------------------------------------ #>     Initial graph #>  #>     --- Hypothesis weights --- #>     H1: 0.5 #>     H2: 0.5 #>     H3: 0.0 #>     H4: 0.0 #>  #>     --- Transition weights --- #>         H1  H2  H3  H4 #>     H1 0.0 0.5 0.5 0.0 #>     H2 0.5 0.0 0.0 0.5 #>     H3 0.0 1.0 0.0 0.0 #>     H4 1.0 0.0 0.0 0.0 #>  #>     Alpha = 0.025 #>  #>     Test types #>     bonferroni: (H1, H2, H3, H4) #>  #> Simulation parameters ($inputs) ------------------------------------------------ #>     Testing 100,000 simulations - random seed 1234 & multivariate normal params: #>  #>                           H1       H2       H3       H4 #>     Marginal power: 0.802831 0.802831 0.705414 0.901481 #>  #>     Correlation:      H1   H2   H3   H4 #>                  H1 1.00 0.50 0.50 0.25 #>                  H2 0.50 1.00 0.25 0.50 #>                  H3 0.50 0.25 1.00 0.50 #>                  H4 0.25 0.50 0.50 1.00 #>  #> Power calculation ($power) ----------------------------------------------------- #>                                      H1      H2      H3      H4 #>                    Local power: 0.76396 0.75887 0.56767 0.69133 #>  #>     Expected no. of rejections: 2.78183 #>      Power to reject 1 or more: 0.85557 #>            Power to reject all: 0.51205 #>  #>                Power to reject:       H1   Expected no. of rejections #>                                  0.76396                      2.78183 #>    AtLeast1     All   H1andH2   (H1andH3)or(H2andH4) #>     0.85557 0.51205   0.66726                0.74695 #>  #> Simulation details ($details) -------------------------------------------------- #>         p_sim_H1    p_sim_H2    p_sim_H3    p_sim_H4 rej_H1 rej_H2 rej_H3 rej_H4 #>      3.08204e-02 1.20654e-02 4.11858e-03 9.32434e-02  FALSE   TRUE  FALSE  FALSE #>      7.93372e-04 6.49905e-04 2.45178e-02 2.96560e-03   TRUE   TRUE   TRUE   TRUE #>      3.02992e-02 5.95396e-02 5.43083e-02 2.62583e-02  FALSE  FALSE  FALSE  FALSE #>      9.74332e-03 3.31857e-03 7.41721e-04 4.02469e-04   TRUE   TRUE   TRUE   TRUE #>      1.97135e-02 8.61618e-03 1.64183e-02 2.41833e-07   TRUE   TRUE   TRUE   TRUE #>      3.12066e-03 6.70231e-03 1.37441e-02 2.75170e-04   TRUE   TRUE   TRUE   TRUE #>      3.02208e-02 1.42376e-01 6.03828e-03 2.11740e-02  FALSE  FALSE  FALSE  FALSE #>      2.49757e-03 2.94026e-02 4.14273e-04 2.20779e-03   TRUE  FALSE   TRUE  FALSE #>      6.18994e-02 3.87257e-02 3.16613e-01 5.69979e-02  FALSE  FALSE  FALSE  FALSE #>      3.67792e-01 1.89598e-01 7.02265e-02 1.18965e-02  FALSE  FALSE  FALSE  FALSE #>     ... (Use `print(x, rows = <nn>)` for more)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/authors.html","id":null,"dir":"","previous_headings":"","what":"Authors","title":"Authors and Citation","text":"Dong Xi. Author, maintainer. Ethan Brockmann. Author. Gilead Biostatistics. Copyright holder, funder. Atorus Research LLC. Copyright holder.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/authors.html","id":"citation","dir":"","previous_headings":"","what":"Citation","title":"Authors and Citation","text":"Xi, D.; Brockmann, E. (2023). graphicalMCP: Graph-based multiple comparison procedures. version 0.1.0. Gilead Sciences, Inc. Foster City, California. https://github.com/Gilead-BioStats/graphicalMCP Frank Bretz, Martin Posch, Ekkehard Glimm, Florian Klinglmueller, Willi Maurer, Kornelius Rohmeyer (2011), Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes parametric tests. Biometrical Journal 53 (6), pages 894--913, Wiley.","code":"@Manual{,   title = {{graphicalMCP}: Graph-based multiple comparison procedures.},   author = {Dong Xi and Ethan Brockmann},   organization = {Gilead Sciences, Inc},   note = {version 0.1.0},   year = {2023},   url = {https://github.com/Gilead-BioStats/graphicalMCP}, } @Article{,   title = {Graphical approaches for multiple comparison procedures using weighted Bonferroni, Simes or parametric tests},   author = {Frank Bretz and Martin Posch and Ekkehard Glimm and Florian Klinglmueller and Willi Maurer and Kornelius Rohmeyer},   journal = {Biometrical Journal},   year = {2011},   publisher = {Wiley},   volume = {53},   number = {6},   pages = {894--913},   url = {http://onlinelibrary.wiley.com/doi/10.1002/bimj.201000239/full}, }"},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"introduction","dir":"","previous_headings":"","what":"Introduction","title":"Graphical Approach for Multiple Comparison Procedures","text":"multiple comparison procedure (MCP) statistical analysis method allows assessing efficacy multiple endpoints, dependent , single clinical trial. Endpoints can different doses, treatment different conditions, combined superiority & non-inferiority testing, many others. key feature MCPs strongly control type-error rate level alpha. Bretz et al (2011), graphical method MCPs described, separates weighting clinical endpoints particular statistical test used assess endpoint. graphical approach can also helpful communicating study design clinical teams. package low-dependency implementation methods.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"installation","dir":"","previous_headings":"","what":"Installation","title":"Graphical Approach for Multiple Comparison Procedures","text":"graphicalMCP CRAN, install GitHub ","code":"# install.packages(\"pak\") pak::pak(\"Gilead-BioStats/graphicalMCP@dev\")"},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"initial-graph","dir":"","previous_headings":"Basic usage","what":"Initial graph","title":"Graphical Approach for Multiple Comparison Procedures","text":"base object graphicalMCP initial_graph, weighted, directed graph represented matrix transition (edge) weights, vector hypothesis (vertex) weights.","code":"library(graphicalMCP)  # A graphical multiple comparison procedure with two primary hypotheses (H1 # and H2) and two secondary hypotheses (H3 and H4) # See Figure 1 in Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, # W., & Rohmeyer, K. (2011). Graphical approaches for multiple comparison # procedures using weighted Bonferroni, Simes, or parametric tests. Biometrical # Journal, 53(6), 894-913. hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   c(0, 0, 1, 0),   c(0, 0, 0, 1),   c(0, 1, 0, 0),   c(1, 0, 0, 0) )  hyp_names <- c(\"H1\", \"H2\", \"H3\", \"H4\") example_graph <- graph_create(hypotheses, transitions, hyp_names)  plot(example_graph, layout = \"grid\")"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"update-graph","dir":"","previous_headings":"Basic usage","what":"Update graph","title":"Graphical Approach for Multiple Comparison Procedures","text":"Hypotheses can deleted MCP using graph_update(). Updated weights transitions calculated according weighting strategy Algorithm 1 Bretz et al (2011). distinguish deleting rejecting glossary.","code":"graph_update(example_graph, delete = c(FALSE, TRUE, TRUE, FALSE))"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"generate-weights","dir":"","previous_headings":"Basic usage","what":"Generate weights","title":"Graphical Approach for Multiple Comparison Procedures","text":"weights sub-graphs can calculated graph_generate_weights(). uses efficient code hood graph_update() order performant larger graphs. information closure can found [Link closure vignette].","code":"graph_generate_weights(example_graph)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"test-hypotheses","dir":"","previous_headings":"Basic usage","what":"Test hypotheses","title":"Graphical Approach for Multiple Comparison Procedures","text":"Bonferroni testing via shortcut method supported graphicalMCP. test can performed graph_test_shortcut(), generates report showing graph & test results. See insight hypothesis rejected setting verbose test_values flags. details shortcut testing can found [link shortcut vignette]. graph can also tested using Simes- parametric-based testing using closure test. types tests added time, combination tests can used groups hypotheses. Additional details closure testing can found [link closed test vignette].","code":"graph_test_shortcut(example_graph, p = c(.01, .03, .02, .01), alpha = .025) graph_test_closure(   example_graph,   p = c(.01, .03, .02, .01),   alpha = .025,   test_types = \"bonferroni\",   test_groups = list(1:4) )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"power-simulations","dir":"","previous_headings":"Basic usage","what":"Power simulations","title":"Graphical Approach for Multiple Comparison Procedures","text":"’s always obvious graph structure easy difficult reject hypothesis. One way understand better run power simulation, many p-values simulated, graph tested . valid test types & hypothesis groupings valid power simulations well. Power simulations discussed [shortcut testing vignette] [closure testing vignette].","code":"graph_calculate_power(   example_graph,   sim_n = 1e5,   power_marginal = c(.9, .9, .8, .8) )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"related-work","dir":"","previous_headings":"","what":"Related work","title":"Graphical Approach for Multiple Comparison Procedures","text":"methods originally implemented gMCP package, still available CRAN today. also lighter version gMCP implemented gMCPmini successor, gMCPLite. two contain subset original functionality, remove rJava dependency add plotting functionality based ggplot2. However, development ceased original package, hope re-implement methods general testing framework; fewer dependencies, particular shedding Java dependency; simpler, transparent S3 classes; improvements accuracy parametric Simes test methods. portion Simes testing also implemented lrstat package.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"citation","dir":"","previous_headings":"","what":"Citation","title":"Graphical Approach for Multiple Comparison Procedures","text":"","code":"citation(\"graphicalMCP\") #> To cite graphicalMCP in publications use: #>  #>   Xi, D.; Brockmann, E. (2023). graphicalMCP: Graph-based multiple #>   comparison procedures. version 0.1.0. Gilead Sciences, Inc. Foster #>   City, California. https://github.com/Gilead-BioStats/graphicalMCP #>  #>   Frank Bretz, Martin Posch, Ekkehard Glimm, Florian Klinglmueller, #>   Willi Maurer, Kornelius Rohmeyer (2011), Graphical approaches for #>   multiple comparison procedures using weighted Bonferroni, Simes or #>   parametric tests. Biometrical Journal 53 (6), pages 894--913, Wiley. #>  #> To see these entries in BibTeX format, use 'print(<citation>, #> bibtex=TRUE)', 'toBibtex(.)', or set #> 'options(citation.bibtex.max=999)'."},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"glossary-of-terms","dir":"","previous_headings":"","what":"Glossary of terms","title":"Graphical Approach for Multiple Comparison Procedures","text":"package seeks accurate performant, course. beyond , much thought put readability code. Whether read user validating methods, developer extending package new uses, contributor helping enhance core functionality, hope code contained can serve educational document grow people’s understanding graphical approach multiple comparison procedures. end, several entities encountered world graphical MCPs define . used internal code package, inputs output one exported functions. believe definitions clarified improved, please contact package authors submit issue GitHub repository. set weights, adjusted according testing algorithm: Bonferroni: change Simes: Sum weights hypotheses smaller p-values Parametric: Multiply weights c-value, based joint distribution power reject null hypothesis full alpha. Closely related non-centrality parameter, mean null hypothesis underlying multivariate normal distribution: ncp = qnorm(1 - alpha) - qnorm(1 - marginal_power)","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjust_p.html","id":null,"dir":"Reference","previous_headings":"","what":"Calculate adjusted p-values — adjust_p_bonferroni","title":"Calculate adjusted p-values — adjust_p_bonferroni","text":"Calculate adjusted p-values","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjust_p.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Calculate adjusted p-values — adjust_p_bonferroni","text":"","code":"adjust_p_bonferroni(p, hypotheses)  adjust_p_parametric(p, hypotheses, test_corr = NULL)  adjust_p_simes(p, hypotheses)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjust_p.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Calculate adjusted p-values — adjust_p_bonferroni","text":"p numeric vector p-values hypotheses numeric vector hypothesis weights test_corr (Optional) numeric matrix correlations hypotheses' test statistics","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjust_p.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Calculate adjusted p-values — adjust_p_bonferroni","text":"single adjusted p-value given group","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjust_p.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Calculate adjusted p-values — adjust_p_bonferroni","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjust_p.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Calculate adjusted p-values — adjust_p_bonferroni","text":"","code":"set.seed(22723)  w <- c(\"H1\" = .75, \"H2\" = .25, \"H3\" = 0) p <- c(\"H1\" = .019, \"H2\" = .025, \"H3\" = .05)  graphicalMCP:::adjust_p_bonferroni(p, w) #> [1] 0.02533333 graphicalMCP:::adjust_p_simes(p, w) #> [1] 0.025  corr1 <- diag(3) corr2 <- corr1 corr2[lower.tri(corr2)] <- corr2[upper.tri(corr2)] <- runif(3, -1, 1)  # No correlation graphicalMCP:::adjust_p_parametric(p, w, corr1) #> [1] 0.025213  # Uniform random pairwise correlations graphicalMCP:::adjust_p_parametric(p, w, corr2) #> [1] 0.02196006"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjusted-weights.html","id":null,"dir":"Reference","previous_headings":"","what":"Calculate updated hypothesis weights for the closure of a graph — adjust_weights_parametric","title":"Calculate updated hypothesis weights for the closure of a graph — adjust_weights_parametric","text":"weights created graph_generate_weights() work immediately Bonferroni testing, parametric Simes testing require additional calculations. adjust_weights_*() functions apply parametric Simes weight increases get updated weights testing. also subset weights columns appropriate groups","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjusted-weights.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Calculate updated hypothesis weights for the closure of a graph — adjust_weights_parametric","text":"","code":"adjust_weights_parametric(   matrix_weights,   matrix_intersections,   test_corr,   alpha,   test_groups )  adjust_weights_simes(matrix_weights, p, groups)  c_value_function(x, hypotheses, test_corr, alpha)  solve_c_parametric(hypotheses, test_corr, alpha)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjusted-weights.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Calculate updated hypothesis weights for the closure of a graph — adjust_weights_parametric","text":"matrix_weights second half columns graph_generate_weights() output, indicating weights intersection matrix_intersections first half columns graph_generate_weights() output, indicating hypotheses contained intersection test_corr numeric matrix correlations hypotheses' test statistics alpha numeric scalar specifying global significance level testing test_groups list numeric vectors specifying hypotheses test together p numeric vector p-values x root solve stats::uniroot() hypotheses numeric vector hypothesis weights","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjusted-weights.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Calculate updated hypothesis weights for the closure of a graph — adjust_weights_parametric","text":"Outputs: adjust_weights_*(), matrix shape weighting_strategy, weights adjusted according specified adjustment method c_value_function(), \\(c_{J_h}\\) value given group, according Formula 6 Xi et al. (2017).","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjusted-weights.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Calculate updated hypothesis weights for the closure of a graph — adjust_weights_parametric","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjusted-weights.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Calculate updated hypothesis weights for the closure of a graph — adjust_weights_parametric","text":"","code":"p <- 1:6 / 200  g <- bonferroni_holm(6) gw_large <- graph_generate_weights(g)  gw_0 <- gw_large[, 7:12] gw <- ifelse(gw_large[, 1:6], gw_0, NA)  graphicalMCP:::adjust_weights_parametric(   gw_0,   gw_large[, 1:6],   diag(6),   .05,   list(1:3) ) #>           H1        H2        H3 #> 1  0.1680752 0.1680752 0.1680752 #> 2  0.2020340 0.2020340 0.2020340 #> 3  0.2020340 0.2020340 0.2020340 #> 4  0.2531918 0.2531918 0.2531918 #> 5  0.2020340 0.2020340 0.2020340 #> 6  0.2531918 0.2531918 0.2531918 #> 7  0.2531918 0.2531918 0.2531918 #> 8  0.3390485 0.3390485 0.3390485 #> 9  0.2010082 0.2010082 0.0000000 #> 10 0.2515823 0.2515823 0.0000000 #> 11 0.2515823 0.2515823 0.0000000 #> 12 0.3361584 0.3361584 0.0000000 #> 13 0.2515823 0.2515823 0.0000000 #> 14 0.3361584 0.3361584 0.0000000 #> 15 0.3361584 0.3361584 0.0000000 #> 16 0.5064113 0.5064113 0.0000000 #> 17 0.2010082 0.0000000 0.2010082 #> 18 0.2515823 0.0000000 0.2515823 #> 19 0.2515823 0.0000000 0.2515823 #> 20 0.3361584 0.0000000 0.3361584 #> 21 0.2515823 0.0000000 0.2515823 #> 22 0.3361584 0.0000000 0.3361584 #> 23 0.3361584 0.0000000 0.3361584 #> 24 0.5064113 0.0000000 0.5064113 #> 25 0.2500000 0.0000000 0.0000000 #> 26 0.3333333 0.0000000 0.0000000 #> 27 0.3333333 0.0000000 0.0000000 #> 28 0.5000000 0.0000000 0.0000000 #> 29 0.3333333 0.0000000 0.0000000 #> 30 0.5000000 0.0000000 0.0000000 #> 31 0.5000000 0.0000000 0.0000000 #> 32 1.0000000 0.0000000 0.0000000 #> 33 0.0000000 0.2010082 0.2010082 #> 34 0.0000000 0.2515823 0.2515823 #> 35 0.0000000 0.2515823 0.2515823 #> 36 0.0000000 0.3361584 0.3361584 #> 37 0.0000000 0.2515823 0.2515823 #> 38 0.0000000 0.3361584 0.3361584 #> 39 0.0000000 0.3361584 0.3361584 #> 40 0.0000000 0.5064113 0.5064113 #> 41 0.0000000 0.2500000 0.0000000 #> 42 0.0000000 0.3333333 0.0000000 #> 43 0.0000000 0.3333333 0.0000000 #> 44 0.0000000 0.5000000 0.0000000 #> 45 0.0000000 0.3333333 0.0000000 #> 46 0.0000000 0.5000000 0.0000000 #> 47 0.0000000 0.5000000 0.0000000 #> 48 0.0000000 1.0000000 0.0000000 #> 49 0.0000000 0.0000000 0.2500000 #> 50 0.0000000 0.0000000 0.3333333 #> 51 0.0000000 0.0000000 0.3333333 #> 52 0.0000000 0.0000000 0.5000000 #> 53 0.0000000 0.0000000 0.3333333 #> 54 0.0000000 0.0000000 0.5000000 #> 55 0.0000000 0.0000000 0.5000000 #> 56 0.0000000 0.0000000 1.0000000 #> 57 0.0000000 0.0000000 0.0000000 #> 58 0.0000000 0.0000000 0.0000000 #> 59 0.0000000 0.0000000 0.0000000 #> 60 0.0000000 0.0000000 0.0000000 #> 61 0.0000000 0.0000000 0.0000000 #> 62 0.0000000 0.0000000 0.0000000 #> 63 0.0000000 0.0000000 0.0000000  graphicalMCP:::adjust_weights_simes(gw_0, p, list(4:6)) #>           H4        H5        H6 #> 1  0.1666667 0.3333333 0.5000000 #> 2  0.2000000 0.4000000 0.4000000 #> 3  0.2000000 0.2000000 0.4000000 #> 4  0.2500000 0.2500000 0.2500000 #> 5  0.0000000 0.2000000 0.4000000 #> 6  0.0000000 0.2500000 0.2500000 #> 7  0.0000000 0.0000000 0.2500000 #> 8  0.0000000 0.0000000 0.0000000 #> 9  0.2000000 0.4000000 0.6000000 #> 10 0.2500000 0.5000000 0.5000000 #> 11 0.2500000 0.2500000 0.5000000 #> 12 0.3333333 0.3333333 0.3333333 #> 13 0.0000000 0.2500000 0.5000000 #> 14 0.0000000 0.3333333 0.3333333 #> 15 0.0000000 0.0000000 0.3333333 #> 16 0.0000000 0.0000000 0.0000000 #> 17 0.2000000 0.4000000 0.6000000 #> 18 0.2500000 0.5000000 0.5000000 #> 19 0.2500000 0.2500000 0.5000000 #> 20 0.3333333 0.3333333 0.3333333 #> 21 0.0000000 0.2500000 0.5000000 #> 22 0.0000000 0.3333333 0.3333333 #> 23 0.0000000 0.0000000 0.3333333 #> 24 0.0000000 0.0000000 0.0000000 #> 25 0.2500000 0.5000000 0.7500000 #> 26 0.3333333 0.6666667 0.6666667 #> 27 0.3333333 0.3333333 0.6666667 #> 28 0.5000000 0.5000000 0.5000000 #> 29 0.0000000 0.3333333 0.6666667 #> 30 0.0000000 0.5000000 0.5000000 #> 31 0.0000000 0.0000000 0.5000000 #> 32 0.0000000 0.0000000 0.0000000 #> 33 0.2000000 0.4000000 0.6000000 #> 34 0.2500000 0.5000000 0.5000000 #> 35 0.2500000 0.2500000 0.5000000 #> 36 0.3333333 0.3333333 0.3333333 #> 37 0.0000000 0.2500000 0.5000000 #> 38 0.0000000 0.3333333 0.3333333 #> 39 0.0000000 0.0000000 0.3333333 #> 40 0.0000000 0.0000000 0.0000000 #> 41 0.2500000 0.5000000 0.7500000 #> 42 0.3333333 0.6666667 0.6666667 #> 43 0.3333333 0.3333333 0.6666667 #> 44 0.5000000 0.5000000 0.5000000 #> 45 0.0000000 0.3333333 0.6666667 #> 46 0.0000000 0.5000000 0.5000000 #> 47 0.0000000 0.0000000 0.5000000 #> 48 0.0000000 0.0000000 0.0000000 #> 49 0.2500000 0.5000000 0.7500000 #> 50 0.3333333 0.6666667 0.6666667 #> 51 0.3333333 0.3333333 0.6666667 #> 52 0.5000000 0.5000000 0.5000000 #> 53 0.0000000 0.3333333 0.6666667 #> 54 0.0000000 0.5000000 0.5000000 #> 55 0.0000000 0.0000000 0.5000000 #> 56 0.0000000 0.0000000 0.0000000 #> 57 0.3333333 0.6666667 1.0000000 #> 58 0.5000000 1.0000000 1.0000000 #> 59 0.5000000 0.5000000 1.0000000 #> 60 1.0000000 1.0000000 1.0000000 #> 61 0.0000000 0.5000000 1.0000000 #> 62 0.0000000 1.0000000 1.0000000 #> 63 0.0000000 0.0000000 1.0000000"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/as-graph.html","id":null,"dir":"Reference","previous_headings":"","what":"Convert between gMCP and graphicalMCP graph classes — as_initial_graph","title":"Convert between gMCP and graphicalMCP graph classes — as_initial_graph","text":"Convert gMCP graphicalMCP graph classes","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/as-graph.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Convert between gMCP and graphicalMCP graph classes — as_initial_graph","text":"","code":"as_initial_graph(graph)  # S3 method for graphMCP as_initial_graph(graph)  # S3 method for igraph as_initial_graph(graph)  as_graphMCP(graph)  # S3 method for initial_graph as_graphMCP(graph)  as_igraph(graph)  # S3 method for initial_graph as_igraph(graph)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/as-graph.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Convert between gMCP and graphicalMCP graph classes — as_initial_graph","text":"graph initial_graph object graphicalMCP, graphMCP object gMCP, igraph object igraph, depending direction converting","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/as-graph.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Convert between gMCP and graphicalMCP graph classes — as_initial_graph","text":"as_graphMCP(), gMCP graph object, as_igraph(), igraph object, as_initial_graph(), graphicalMCP graph object","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/as-graph.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Convert between gMCP and graphicalMCP graph classes — as_initial_graph","text":"","code":"g1 <- random_graph(5)  if (requireNamespace(\"gMCP\", quietly = TRUE)) {   g2 <- as_graphMCP(g1)    all.equal(g1, as_initial_graph(g2)) } #> [1] TRUE  if (requireNamespace(\"igraph\", quietly = TRUE)) {   g3 <- as_igraph(g1)    all.equal(g1, as_initial_graph(g3)) } #> igraph method #>  [1] \"names for target but not for current\"                              #>  [2] \"Attributes: < Component “class”: 1 string mismatch >\"              #>  [3] \"Length mismatch: comparison on first 2 components\"                 #>  [4] \"Component 1: names for target but not for current\"                 #>  [5] \"Component 1: Numeric: lengths (5, 1) differ\"                       #>  [6] \"Component 2: Modes: numeric, logical\"                              #>  [7] \"Component 2: Lengths: 25, 1\"                                       #>  [8] \"Component 2: Attributes: < Modes: list, NULL >\"                    #>  [9] \"Component 2: Attributes: < Lengths: 2, 0 >\"                        #> [10] \"Component 2: Attributes: < names for target but not for current >\" #> [11] \"Component 2: Attributes: < current is not list-like >\"             #> [12] \"Component 2: target is matrix, current is logical\""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/calc-test_values.html","id":null,"dir":"Reference","previous_headings":"","what":"Test hypotheses with the adjusted significance method — test_values_bonferroni","title":"Test hypotheses with the adjusted significance method — test_values_bonferroni","text":"Test hypotheses adjusted significance method","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/calc-test_values.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Test hypotheses with the adjusted significance method — test_values_bonferroni","text":"","code":"test_values_bonferroni(p, hypotheses, alpha, intersection = NA)  test_values_parametric(p, hypotheses, alpha, intersection = NA, test_corr)  test_values_simes(p, hypotheses, alpha, intersection = NA)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/calc-test_values.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Test hypotheses with the adjusted significance method — test_values_bonferroni","text":"p numeric vector p-values hypotheses numeric vector hypothesis hypotheses alpha numeric scalar specifying global significance level testing intersection numeric scalar used track intersection test values test_corr numeric matrix correlations hypotheses' test statistics","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/calc-test_values.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Test hypotheses with the adjusted significance method — test_values_bonferroni","text":"data frame columns specifying values used calculate hypothesis test","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/calc-test_values.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Test hypotheses with the adjusted significance method — test_values_bonferroni","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/calc-test_values.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Test hypotheses with the adjusted significance method — test_values_bonferroni","text":"","code":"w <- c(H1 = .5, H2 = .5, H3 = 0, H4 = 0)  p <- c(.024, .01, .026, .027)  graphicalMCP:::test_values_bonferroni(p, w, .05) #>    Intersection Hypothesis       Test     p <= c_value * Weight * Alpha #> H1           NA         H1 bonferroni 0.024 <=              0.5 *  0.05 #> H2           NA         H2 bonferroni 0.010 <=              0.5 *  0.05 #> H3           NA         H3 bonferroni 0.026 <=              0.0 *  0.05 #> H4           NA         H4 bonferroni 0.027 <=              0.0 *  0.05 #>    Inequality_holds #> H1             TRUE #> H2             TRUE #> H3            FALSE #> H4            FALSE graphicalMCP:::test_values_parametric(p, w, .05, test_corr = diag(4)) #>    Intersection Hypothesis       Test     p <=  c_value * Weight * Alpha #> H1           NA         H1 parametric 0.024 <= 1.012823 *    0.5 *  0.05 #> H2           NA         H2 parametric 0.010 <= 1.012823 *    0.5 *  0.05 #> H3           NA         H3 parametric 0.026 <= 1.012823 *    0.0 *  0.05 #> H4           NA         H4 parametric 0.027 <= 1.012823 *    0.0 *  0.05 #>    Inequality_holds #> H1             TRUE #> H2             TRUE #> H3            FALSE #> H4            FALSE graphicalMCP:::test_values_simes(p, w, .05) #>   Intersection Hypothesis  Test     p <= c_value * Weight * Alpha #> 1           NA         H1 simes 0.024 <=              1.0 *  0.05 #> 2           NA         H2 simes 0.010 <=              0.5 *  0.05 #> 3           NA         H3 simes 0.026 <=              1.0 *  0.05 #> 4           NA         H4 simes 0.027 <=              1.0 *  0.05 #>   Inequality_holds #> 1             TRUE #> 2             TRUE #> 3             TRUE #> 4             TRUE"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/example-graphs.html","id":null,"dir":"Reference","previous_headings":"","what":"Example graphs from a selection of papers — bonferroni_holm","title":"Example graphs from a selection of papers — bonferroni_holm","text":"Example graphs selection papers","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/example-graphs.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Example graphs from a selection of papers — bonferroni_holm","text":"","code":"bonferroni_holm(num_hyps, hyp_names = NULL)  huque_alosh_bhore_2011(hyp_names = NULL)  wiens_dmitrienko_2005(hyp_names = NULL)  fixed_sequence(num_hyps = 3, hyp_names = NULL)  fallback(hypotheses = c(1, 0, 0), hyp_names = NULL)  simple_successive_1(hyp_names = NULL)  simple_successive_2(hyp_names = NULL)  random_graph(num_hyps, hyp_names = NULL)  complex_example_1(hyp_names = NULL)  complex_example_2(hyp_names = NULL)  complex_example_2(hyp_names = NULL)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/example-graphs.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Example graphs from a selection of papers — bonferroni_holm","text":"num_hyps Number vertices graph hyp_names (Optional) character vector hypothesis names. names specified, hypotheses named sequentially H1, H2, ... hypotheses Hypothesis weights fallback procedure","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/example-graphs.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Example graphs from a selection of papers — bonferroni_holm","text":"S3 object returned graph_create()","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/example-graphs.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Example graphs from a selection of papers — bonferroni_holm","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/example-graphs.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Example graphs from a selection of papers — bonferroni_holm","text":"","code":"bonferroni_holm(3, hyp_names = paste(\"dose\", letters[1:3])) #> Initial graph #>  #> --- Hypothesis weights --- #> dose a: 0.3333 #> dose b: 0.3333 #> dose c: 0.3333 #>  #> --- Transition weights --- #>         dose a dose b dose c #>  dose a    0.0    0.5    0.5 #>  dose b    0.5    0.0    0.5 #>  dose c    0.5    0.5    0.0"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":null,"dir":"Reference","previous_headings":"","what":"Obtain hypothesis rejection probabilities — graph_calculate_power","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"often difficult tell likely given hypothesis rejected. power simulations useful. set distribution parameters, many p-values generated, graph tested one. testing strategy can used. probabilities calculated hypothesis rejected, well additional probabilities expected rejections probability rejecting hypothesis","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"","code":"graph_calculate_power(   graph,   alpha = 0.025,   power_marginal = rep(alpha, length(graph$hypotheses)),   test_groups = list(seq_along(graph$hypotheses)),   test_types = c(\"bonferroni\"),   test_corr = rep(list(NA), length(test_types)),   sim_n = 100,   sim_corr = diag(length(graph$hypotheses)),   sim_success = NULL,   sim_seed = NULL,   force_closure = FALSE,   verbose = FALSE )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"graph initial graph returned graph_create() alpha numeric scalar specifying global significance level testing power_marginal numeric vector mean values use simulating p-values. Exactly one mean per hypothesis needed, p-values sampled multivariate normal distribution. See Details test_groups list numeric vectors specifying hypotheses test together test_types character vector tests apply given groups test_corr Optional test_types parametric. numeric matrix correlations hypotheses' test statistics sim_n integer scalar specifying many simulations run sim_corr numeric matrix correlations hypotheses used sample multivariate normal distribution generate p-values sim_success list user-defined functions apply power results. Functions must take one simulation's logical vector results input, return length-one logical vector. instance, \"success\" means rejecting hypotheses 1 2, use sim_success = list(\"1 2\" = function(x) x[1] && x[2]). list named, function body used name. Lambda functions also work starting R 4.1, e.g. sim_success = list(\\(x) x[3] || x[4]) sim_seed (Optional) Random seed set simulating p-values. Set use consistent set p simulations across power calculations force_closure logical scalar used determine whether full closure test used Bonferroni testing. Ignored tests non-Bonferroni verbose logical scalar specifying whether full matrix simulations test results included output ","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"list five elements power_local - rejection proportion hypothesis individually power_expected - average number hypotheses rejected single simulation power_at_least_1 - proportion simulations reject hypothesis power_all - proportion simulations reject hypotheses power_success - proportion simulations reject hypotheses specified sim_success","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"parameters normal distribution set power_marginal (means) sim_corr (correlation test statistics). mean hypothesis set marginal power $$d_i=P_{\\xi_i}(p_i\\leq\\alpha)$$ \\(\\xi_i\\) non-centrality parameter. correlation test statistics induced study design.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":"success","dir":"Reference","previous_headings":"","what":"Success","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"Success mean something different trial, lot flexibility sim_success parameter. However, flexibility means little validation inputs. user make sure function(s) passed mean think. implementation perspective, function applied row-wise matrix test results simulation, resulting sim_n length vector. mean vector returned \"Probability success\"","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"","code":"par_gate <- simple_successive_1()  # The default is to test all hypotheses with: Bonferroni testing at alpha # level .025, 0 mean under the alternative, and 0 correlation between # hypotheses under the alternative # The default of 100 simulations will usually need to be increased graph_calculate_power(par_gate, sim_n = 1e5) #>  #> Test parameters ($inputs) ------------------------------------------------------ #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1  0  0  1  0 #>   H2  0  0  0  1 #>   H3  0  1  0  0 #>   H4  1  0  0  0 #>  #>   Alpha = 0.025 #>  #>   Test types #>   bonferroni: (H1, H2, H3, H4) #>  #> Simulation parameters ($inputs) ------------------------------------------------ #>   Testing 100,000 simulations with multivariate normal params: #>  #>                      H1    H2    H3    H4 #>   Marginal power: 0.025 0.025 0.025 0.025 #>  #>   Correlation:    H1 H2 H3 H4 #>                H1  1  0  0  0 #>                H2  0  1  0  0 #>                H3  0  0  1  0 #>                H4  0  0  0  1 #>  #> Power calculation ($power) ----------------------------------------------------- #>                                    H1      H2      H3      H4 #>                  Local power: 0.01201 0.01166 0.00016 0.00019 #>  #>   Expected no. of rejections: 0.02402 #>    Power to reject 1 or more: 0.02349 #>          Power to reject all: 0 #>   # But any test group/type combination that works for [graph_test_closure()] # can be used graph_calculate_power(   par_gate,   alpha = .025,   test_groups = list(1:2, 3:4),   test_types = c(\"s\", \"p\"),   test_corr = list(NA, diag(2)),   sim_n = 1e5,   sim_success = list(     function(.) .[1] || .[2],     function(.) .[1] && .[2]   ) ) #>  #> Test parameters ($inputs) ------------------------------------------------------ #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1  0  0  1  0 #>   H2  0  0  0  1 #>   H3  0  1  0  0 #>   H4  1  0  0  0 #>  #>   Alpha = 0.025 #>  #>   Parametric testing correlation:    H3 H4 #>                                   H3  1  0 #>                                   H4  0  1 #>  #>   Test types #>        simes: (H1, H2) #>   parametric: (H3, H4) #>  #> Simulation parameters ($inputs) ------------------------------------------------ #>   Testing 100,000 simulations with multivariate normal params: #>  #>                      H1    H2    H3    H4 #>   Marginal power: 0.025 0.025 0.025 0.025 #>  #>   Correlation:    H1 H2 H3 H4 #>                H1  1  0  0  0 #>                H2  0  1  0  0 #>                H3  0  0  1  0 #>                H4  0  0  0  1 #>  #> Power calculation ($power) ----------------------------------------------------- #>                                    H1      H2      H3      H4 #>                  Local power: 0.01203 0.01215 0.00018 0.00021 #>  #>   Expected no. of rejections: 0.02457 #>    Power to reject 1 or more: 0.02403 #>          Power to reject all: 0 #>  #>              Power to reject:    .[1] || .[2]   .[1] && .[2] #>                                       0.02403        0.00015 #>"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_create.html","id":null,"dir":"Reference","previous_headings":"","what":"Define a graph representing a multiple comparison procedure — graph_create","title":"Define a graph representing a multiple comparison procedure — graph_create","text":"multiple comparison procedure graph can represented 1) vector initial hypothesis weights, 2) matrix initial transition weights. function creates initial graph object using vector matrix. also validates inputs make sure combine form valid graph.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_create.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Define a graph representing a multiple comparison procedure — graph_create","text":"","code":"graph_create(hypotheses, transitions, hyp_names = NULL)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_create.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Define a graph representing a multiple comparison procedure — graph_create","text":"hypotheses numeric vector hypothesis weights initial graphical multiple comparison procedure. Must vector values 0 & 1 (inclusive). length match row column lengths transitions. sum hypothesis weights exceed 1 transitions numeric matrix transition weights hypotheses initial graphical multiple comparison procedure. Must square matrix values 0 & 1 (inclusive). row column lengths match length hypotheses. row (Transition weights leaving hypothesis) can sum 1. diagonal (Transition weights hypothesis ) must 0s hyp_names (Optional) character vector hypothesis names. provided, names hypotheses transitions used. names specified, hypotheses named sequentially H1, H2, ...","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_create.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Define a graph representing a multiple comparison procedure — graph_create","text":"S3 object class initial_graph. underlying structure list elements hypotheses transitions","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_create.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Define a graph representing a multiple comparison procedure — graph_create","text":"validations performed : Weights numeric Length hypotheses dimensions transitions Explicit hyp_names override names hypotheses transitions Hypothesis weights must non-negative sum 1 Transition weights matrix: Diagonal must 1 Rows must sum 1 Values must non-negative","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_create.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Define a graph representing a multiple comparison procedure — graph_create","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_create.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Define a graph representing a multiple comparison procedure — graph_create","text":"","code":"# A graphical multiple comparison procedure with two primary hypotheses (H1 # and H2) and two secondary hypotheses (H3 and H4) # See Figure 1 in Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, # W., & Rohmeyer, K. (2011). Graphical approaches for multiple comparison # procedures using weighted Bonferroni, Simes, or parametric tests. # Biometrical Journal, 53(6), 894-913. hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   c(0, 0, 1, 0),   c(0, 0, 0, 1),   c(0, 1, 0, 0),   c(1, 0, 0, 0) ) hyp_names <- c(\"H1\", \"H2\", \"H3\", \"H4\") g <- graph_create(hypotheses, transitions, hyp_names) g #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  1  0 #>  H2  0  0  0  1 #>  H3  0  1  0  0 #>  H4  1  0  0  0  # Explicit names override names in `hypotheses` (with a warning) hypotheses <- c(h1 = 0.5, h2 = 0.5, h3 = 0, h4 = 0) transitions <- rbind(   c(0, 0, 1, 0),   c(0, 0, 0, 1),   c(0, 1, 0, 0),   c(1, 0, 0, 0) )  g <- graph_create(hypotheses, transitions, hyp_names) #> Warning: Hypothesis names specified - overriding names in #>                     `hypotheses` and `transitions` g #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  1  0 #>  H2  0  0  0  1 #>  H3  0  1  0  0 #>  H4  1  0  0  0  # Explicit names override names in `transitions` (with a warning) hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   h1 = c(0, 0, 1, 0),   h2 = c(0, 0, 0, 1),   h3 = c(0, 1, 0, 0),   h4 = c(1, 0, 0, 0) )  g <- graph_create(hypotheses, transitions, hyp_names) #> Warning: Hypothesis names specified - overriding names in #>                     `hypotheses` and `transitions` g #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  1  0 #>  H2  0  0  0  1 #>  H3  0  1  0  0 #>  H4  1  0  0  0  # Use names in `hypotheses` hypotheses <- c(H1 = 0.5, H2 = 0.5, H3 = 0, H4 = 0) transitions <- rbind(   c(0, 0, 1, 0),   c(0, 0, 0, 1),   c(0, 1, 0, 0),   c(1, 0, 0, 0) ) g <- graph_create(hypotheses, transitions) g #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  1  0 #>  H2  0  0  0  1 #>  H3  0  1  0  0 #>  H4  1  0  0  0  # Use names in `transitions` hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   H1 = c(0, 0, 1, 0),   H2 = c(0, 0, 0, 1),   H3 = c(0, 1, 0, 0),   H4 = c(1, 0, 0, 0) ) g <- graph_create(hypotheses, transitions) g #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  1  0 #>  H2  0  0  0  1 #>  H3  0  1  0  0 #>  H4  1  0  0  0  # When names are not specified, hypotheses are numbered sequentially as # H1, H2, ... hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   c(0, 0, 1, 0),   c(0, 0, 0, 1),   c(0, 1, 0, 0),   c(1, 0, 0, 0) ) g <- graph_create(hypotheses, transitions) g #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  1  0 #>  H2  0  0  0  1 #>  H3  0  1  0  0 #>  H4  1  0  0  0"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_generate_weights.html","id":null,"dir":"Reference","previous_headings":"","what":"Calculate weights for the closure of a graph — graph_generate_weights","title":"Calculate weights for the closure of a graph — graph_generate_weights","text":"closure graph set sub-graphs, intersections hypotheses, graph. Weights sub-graph calculated using weighting strategy defined Bretz et al (2011).","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_generate_weights.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Calculate weights for the closure of a graph — graph_generate_weights","text":"","code":"graph_generate_weights(graph)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_generate_weights.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Calculate weights for the closure of a graph — graph_generate_weights","text":"graph initial graph returned graph_create()","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_generate_weights.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Calculate weights for the closure of a graph — graph_generate_weights","text":"numeric matrix intersection hypothesis weights. row corresponds single intersection hypothesis. first half columns indicate hypotheses included given intersection hypothesis, second half columns weights","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_generate_weights.html","id":"performance","dir":"Reference","previous_headings":"","what":"Performance","title":"Calculate weights for the closure of a graph — graph_generate_weights","text":"Much thought given performance code, memory time usage can grow quickly graph size grows. systems used testing, size 10 graph median run time 20-60 ms. Run time increases rate O(2 ^ n), e.g. size 5 graph takes approximately twice long run size 4 graph. See vignette(\"generate-weights-performance\") detailed analysis explanation","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_generate_weights.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Calculate weights for the closure of a graph — graph_generate_weights","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_generate_weights.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Calculate weights for the closure of a graph — graph_generate_weights","text":"","code":"par_gate <- graph_create(   hypotheses = c(.5, .5, 0, 0),   transitions = rbind(     c(0, 0, 1, 0),     c(0, 0, 0, 1),     c(0, 1, 0, 0),     c(1, 0, 0, 0)   ) )  graph_generate_weights(par_gate) #>    H1 H2 H3 H4  H1  H2  H3  H4 #> 1   1  1  1  1 0.5 0.5 0.0 0.0 #> 2   1  1  1  0 0.5 0.5 0.0 0.0 #> 3   1  1  0  1 0.5 0.5 0.0 0.0 #> 4   1  1  0  0 0.5 0.5 0.0 0.0 #> 5   1  0  1  1 0.5 0.0 0.0 0.5 #> 6   1  0  1  0 1.0 0.0 0.0 0.0 #> 7   1  0  0  1 0.5 0.0 0.0 0.5 #> 8   1  0  0  0 1.0 0.0 0.0 0.0 #> 9   0  1  1  1 0.0 0.5 0.5 0.0 #> 10  0  1  1  0 0.0 0.5 0.5 0.0 #> 11  0  1  0  1 0.0 1.0 0.0 0.0 #> 12  0  1  0  0 0.0 1.0 0.0 0.0 #> 13  0  0  1  1 0.0 0.0 0.5 0.5 #> 14  0  0  1  0 0.0 0.0 1.0 0.0 #> 15  0  0  0  1 0.0 0.0 0.0 1.0"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_rejection_orderings.html","id":null,"dir":"Reference","previous_headings":"","what":"Find alternate rejection orderings for shortcut testing — graph_rejection_orderings","title":"Find alternate rejection orderings for shortcut testing — graph_rejection_orderings","text":"using graph_test_shortcut(), may multiple hypotheses can rejected step. specific hypothesis chosen decided based minimum adjusted p-value. function shows every order deletes hypotheses, hypothesis still valid deletion step chosen.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_rejection_orderings.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Find alternate rejection orderings for shortcut testing — graph_rejection_orderings","text":"","code":"graph_rejection_orderings(shortcut_test_result)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_rejection_orderings.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Find alternate rejection orderings for shortcut testing — graph_rejection_orderings","text":"shortcut_test_result graph_report object returned graph_test_shortcut()","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_rejection_orderings.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Find alternate rejection orderings for shortcut testing — graph_rejection_orderings","text":"modified graph_report object containing valid orderings deleting significant hypotheses","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_rejection_orderings.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Find alternate rejection orderings for shortcut testing — graph_rejection_orderings","text":"","code":"graph <- simple_successive_2()  short_res <- graph_test_shortcut(graph, c(.018, .01, .03, .004))  # Reject H1, H2, and H4 short_res$outputs$rejected #>    H1    H2    H3    H4  #>  TRUE  TRUE FALSE  TRUE   # But these cannot be rejected in any order - For instance, H4 has 0 weight # in the initial graph and cannot be rejected first graph_rejection_orderings(short_res) #>  #> Test parameters ($inputs) ------------------------------------------------------ #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>       H1  H2  H3  H4 #>   H1 0.0 0.5 0.5 0.0 #>   H2 0.5 0.0 0.0 0.5 #>   H3 0.0 1.0 0.0 0.0 #>   H4 1.0 0.0 0.0 0.0 #>  #>   Alpha = 0.025 #>  #>                           H1    H2    H3    H4 #>   Unadjusted p-values: 0.018 0.010 0.030 0.004 #>  #>   Test types #>   bonferroni: (H1, H2, H3, H4) #>  #> Test summary ($outputs) -------------------------------------------------------- #>   Hypothesis Adj. P-value Reject #>           H1         0.02   TRUE #>           H2         0.02   TRUE #>           H3         0.03  FALSE #>           H4         0.02   TRUE #>  #>   Final updated graph after removing rejected hypotheses #>  #>   --- Hypothesis weights --- #>   H1: NA #>   H2: NA #>   H3:  1 #>   H4: NA #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1 NA NA NA NA #>   H2 NA NA NA NA #>   H3 NA NA  0 NA #>   H4 NA NA NA NA #>  #> Alternate rejection orderings ($valid_rejection_orderings) --------------------- #> H2 H4 H1  #>  2  4  1  #>  #> H2 H1 H4  #>  2  1  4  #>   # Finally, intermediate graphs can be obtained by putting one of the # orderings outputs into `graph_update()` graph_update(graph, delete = c(2, 1, 4)) #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>      H1  H2  H3  H4 #>  H1 0.0 0.5 0.5 0.0 #>  H2 0.5 0.0 0.0 0.5 #>  H3 0.0 1.0 0.0 0.0 #>  H4 1.0 0.0 0.0 0.0 #>  #> Updated graph after deleting hypotheses 2, 1, 4 #>  #> --- Hypothesis weights --- #> H1: NA #> H2: NA #> H3:  1 #> H4: NA #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1 NA NA NA NA #>  H2 NA NA NA NA #>  H3 NA NA  0 NA #>  H4 NA NA NA NA #>  #> Deletion sequence ($intermediate_graphs) --------------------------------------- #>  #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>       H1  H2  H3  H4 #>   H1 0.0 0.5 0.5 0.0 #>   H2 0.5 0.0 0.0 0.5 #>   H3 0.0 1.0 0.0 0.0 #>   H4 1.0 0.0 0.0 0.0 #>  #>     Step 1: Updated graph after removing hypothesis 2 #>  #>     --- Hypothesis weights --- #>     H1: 0.75 #>     H2:   NA #>     H3: 0.00 #>     H4: 0.25 #>  #>     --- Transition weights --- #>              H1       H2       H3       H4 #>     H1 0.000000       NA 0.666667 0.333333 #>     H2       NA       NA       NA       NA #>     H3 0.500000       NA 0.000000 0.500000 #>     H4 1.000000       NA 0.000000 0.000000 #>  #>       Step 2: Updated graph after removing hypotheses 2, 1 #>  #>       --- Hypothesis weights --- #>       H1:  NA #>       H2:  NA #>       H3: 0.5 #>       H4: 0.5 #>  #>       --- Transition weights --- #>          H1 H2 H3 H4 #>       H1 NA NA NA NA #>       H2 NA NA NA NA #>       H3 NA NA  0  1 #>       H4 NA NA  1  0 #>  #>         Step 3: Updated graph after removing hypotheses 2, 1, 4 #>  #>         --- Hypothesis weights --- #>         H1: NA #>         H2: NA #>         H3:  1 #>         H4: NA #>  #>         --- Transition weights --- #>            H1 H2 H3 H4 #>         H1 NA NA NA NA #>         H2 NA NA NA NA #>         H3 NA NA  0 NA #>         H4 NA NA NA NA #>  #>   Final updated graph after removing deleted hypotheses #>  #>   --- Hypothesis weights --- #>   H1: NA #>   H2: NA #>   H3:  1 #>   H4: NA #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1 NA NA NA NA #>   H2 NA NA NA NA #>   H3 NA NA  0 NA #>   H4 NA NA NA NA #>"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_update.html","id":null,"dir":"Reference","previous_headings":"","what":"Delete hypotheses from a graph — graph_update","title":"Delete hypotheses from a graph — graph_update","text":"always obvious graph's weights look like deleting one hypotheses. graph_generate_weights() calculates sub-graphs' hypothesis weights, graph_update() gives detailed view single set deletions, including transition weights well.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_update.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Delete hypotheses from a graph — graph_update","text":"","code":"graph_update(graph, delete)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_update.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Delete hypotheses from a graph — graph_update","text":"graph initial graph returned graph_create() delete logical integer vector, denoting hypotheses keep/delete. logical vector must match size graph, one entry per hypothesis, results \"unordered mode,\" selected hypotheses deleted sequential order. integer vector can length, must contain valid hypothesis numbers (greater 0, less equal size graph). trigger \"ordered mode,\" selected hypotheses deleted order appear delete","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_update.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Delete hypotheses from a graph — graph_update","text":"object class updated_graph 3 elements initial graph object Boolean vector indicating hypotheses kept/deleted updated graph object specific hypotheses deleted","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_update.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Delete hypotheses from a graph — graph_update","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_update.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Delete hypotheses from a graph — graph_update","text":"","code":"hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   c(0, 0, 1, 0),   c(0, 0, 0, 1),   c(0, 1, 0, 0),   c(1, 0, 0, 0) ) g <- graph_create(hypotheses, transitions)  # Delete the second hypothesis graph_update(g, c(TRUE, FALSE, TRUE, TRUE)) #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  1  0 #>  H2  0  0  0  1 #>  H3  0  1  0  0 #>  H4  1  0  0  0 #>  #> Updated graph after deleting hypotheses 1, 3, 4 #>  #> --- Hypothesis weights --- #> H1: NA #> H2:  1 #> H3: NA #> H4: NA #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1 NA NA NA NA #>  H2 NA  0 NA NA #>  H3 NA NA NA NA #>  H4 NA NA NA NA # Equivalent # graph_update(g, 2)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graphicalMCP-package.html","id":null,"dir":"Reference","previous_headings":"","what":"graphicalMCP: Graphical Approach for Multiple Comparison Procedures — graphicalMCP-package","title":"graphicalMCP: Graphical Approach for Multiple Comparison Procedures — graphicalMCP-package","text":"multiple comparison procedure (multiple test procedure) statistical analysis method determining efficacy multiple drugs, multiple doses drug, single clinical trial. (Bretz et al., 2011) laid graph-based approach multiple comparison procedures, weights vertices edges graph determined independently particular statistical test used assess results. low-dependency implementation methods described subsequent papers.","code":""},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graphicalMCP-package.html","id":"author","dir":"Reference","previous_headings":"","what":"Author","title":"graphicalMCP: Graphical Approach for Multiple Comparison Procedures — graphicalMCP-package","text":"Maintainer: Dong Xi dong.xi@gilead.com Authors: Ethan Brockmann ethan.brockmann@atorusresearch.com contributors: Gilead Biostatistics [copyright holder, funder] Atorus Research LLC [copyright holder]","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.initial_graph.html","id":null,"dir":"Reference","previous_headings":"","what":"S3 plot method for the class initial_graph — plot.initial_graph","title":"S3 plot method for the class initial_graph — plot.initial_graph","text":"S3 plot method class initial_graph","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.initial_graph.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"S3 plot method for the class initial_graph — plot.initial_graph","text":"","code":"# S3 method for initial_graph plot(   x,   ...,   layout = \"grid\",   nrow = NULL,   ncol = NULL,   edge_curves = NULL,   precision = 4,   eps = NULL,   background_color = \"white\",   margins = c(0, 0, 0, 0) )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.initial_graph.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"S3 plot method for the class initial_graph — plot.initial_graph","text":"x initial graph returned graph_create() ... arguments passed igraph::plot.igraph() layout igraph layout specification (See ?igraph.plotting), \"grid\", lays hypotheses left--right, top--bottom nrow rows ncol columns. nrow integer scalar specifying number rows vertex grid. row columns counts specified, vertices laid close square possible. ncol integer scalar specifying number columns vertex grid. row columns counts specified, vertices laid close square possible. edge_curves named numeric vector specifying curvature edges. Edge pairs (two vertices share edge possible direction) detected automatically get 0.25 curvature. Adjust edges adding entry name \"vertex1|vertex2. Adjust default edge pairs curvature adding entry name \"pairs\". precision integer scalar specifying many decimal places displayed weights eps numeric scalar. Edge weights 0 eps displayed \\(\\epsilon\\), edge weights 1 - eps 1 displayed \\(1 - \\epsilon\\) background_color character scalar specifying background color whole plotting area. Passed directly graphics::par() (bg) margins length 4 numeric vector specifying margins plot. Defaults 0, since igraph plots tend large margins. passed directly graphics::par() (mar)","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.initial_graph.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"S3 plot method for the class initial_graph — plot.initial_graph","text":"NULL, plotting graph","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.initial_graph.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"S3 plot method for the class initial_graph — plot.initial_graph","text":"","code":"plot(simple_successive_2(), layout = \"grid\")"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.updated_graph.html","id":null,"dir":"Reference","previous_headings":"","what":"S3 plot method for the class updated_graph — plot.updated_graph","title":"S3 plot method for the class updated_graph — plot.updated_graph","text":"S3 plot method class updated_graph","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.updated_graph.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"S3 plot method for the class updated_graph — plot.updated_graph","text":"","code":"# S3 method for updated_graph plot(x, ...)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.updated_graph.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"S3 plot method for the class updated_graph — plot.updated_graph","text":"x updated_graph object returned graph_update() ... Arguments passed plot.initial_graph layout igraph layout specification (See ?igraph.plotting), \"grid\", lays hypotheses left--right, top--bottom nrow rows ncol columns. nrow integer scalar specifying number rows vertex grid. row columns counts specified, vertices laid close square possible. ncol integer scalar specifying number columns vertex grid. row columns counts specified, vertices laid close square possible. edge_curves named numeric vector specifying curvature edges. Edge pairs (two vertices share edge possible direction) detected automatically get 0.25 curvature. Adjust edges adding entry name \"vertex1|vertex2. Adjust default edge pairs curvature adding entry name \"pairs\". precision integer scalar specifying many decimal places displayed weights eps numeric scalar. Edge weights 0 eps displayed \\(\\epsilon\\), edge weights 1 - eps 1 displayed \\(1 - \\epsilon\\) background_color character scalar specifying background color whole plotting area. Passed directly graphics::par() (bg) margins length 4 numeric vector specifying margins plot. Defaults 0, since igraph plots tend large margins. passed directly graphics::par() (mar)","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.updated_graph.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"S3 plot method for the class updated_graph — plot.updated_graph","text":"NULL, plotting graph","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.updated_graph.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"S3 plot method for the class updated_graph — plot.updated_graph","text":"","code":"plot(   graph_update(     simple_successive_2(),     c(TRUE, FALSE, TRUE, TRUE)   ),   layout = \"grid\" )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.graph_report.html","id":null,"dir":"Reference","previous_headings":"","what":"S3 print method for class graph_report — print.graph_report","title":"S3 print method for class graph_report — print.graph_report","text":"graph report displays initial graph tested, p-values & alpha used tests, hypotheses can rejected, Detailed test results matrix, including results graph_generate_weights() & test results intersection hypothesis","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.graph_report.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"S3 print method for class graph_report — print.graph_report","text":"","code":"# S3 method for graph_report print(x, ..., precision = 4, indent = 2, rows = 10)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.graph_report.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"S3 print method for class graph_report — print.graph_report","text":"x object class graph_report print ... values passed methods (currently unused) precision integer scalar indicating maximum number decimals include numeric values indent integer scalar indicating many spaces indent results rows integer scalar indicating many rows verbose test values output print","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.initial_graph.html","id":null,"dir":"Reference","previous_headings":"","what":"S3 print method for the class initial_graph — print.initial_graph","title":"S3 print method for the class initial_graph — print.initial_graph","text":"printed initial_graph displays header stating object , hypothesis weights, transition weights.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.initial_graph.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"S3 print method for the class initial_graph — print.initial_graph","text":"","code":"# S3 method for initial_graph print(x, ..., precision = 4, indent = 0, title = \"Initial graph\")"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.initial_graph.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"S3 print method for the class initial_graph — print.initial_graph","text":"x object class initial_graph print ... values passed methods (currently unused) precision integer scalar indicating maximum number decimals include numeric values indent integer scalar indicating many spaces indent results title First line printing, used internally distinguish initial graphs updated graphs","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.power_report.html","id":null,"dir":"Reference","previous_headings":"","what":"S3 print method for class power_report — print.power_report","title":"S3 print method for class power_report — print.power_report","text":"power report displays initial graph tested, Testing simulation options, Final power calculations, (Partial) Detailed p-values test results - underlying object contains full tables","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.power_report.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"S3 print method for class power_report — print.power_report","text":"","code":"# S3 method for power_report print(x, ..., precision = 4, indent = 2, rows = 10)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.power_report.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"S3 print method for class power_report — print.power_report","text":"x object class power_report print ... values passed methods (currently unused) precision integer scalar indicating maximum number decimals include numeric values indent integer scalar indicating many spaces indent results rows integer scalar indicating many rows verbose output print","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.updated_graph.html","id":null,"dir":"Reference","previous_headings":"","what":"S3 print method for class updated_graph — print.updated_graph","title":"S3 print method for class updated_graph — print.updated_graph","text":"printed updated_graph displays initial graph, vector keep/delete hypotheses, updated graph","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.updated_graph.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"S3 print method for class updated_graph — print.updated_graph","text":"","code":"# S3 method for updated_graph print(x, ..., precision = 6, indent = 2)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.updated_graph.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"S3 print method for class updated_graph — print.updated_graph","text":"x object class updated_graph print ... values passed methods (currently unused) precision integer scalar indicating maximum number decimals include numeric values indent integer scalar indicating many spaces indent results","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing-fast.html","id":null,"dir":"Reference","previous_headings":"","what":"Calculate hypothesis rejection results efficiently — graph_test_closure_fast","title":"Calculate hypothesis rejection results efficiently — graph_test_closure_fast","text":"insight nice reporting, prefer graph_test_closure() graph_test_shortcut() options. reasonably fast interactive use. However order minimize power run time, efficient testing functions available. power simulation can segmented certain parts, like generating weights calculating adjusted weights, can done single time. closure testing function stripped just vectorized lines efficiency. separate optimized function available testing graph Bonferroni sequential shortcut.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing-fast.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Calculate hypothesis rejection results efficiently — graph_test_closure_fast","text":"","code":"graph_test_closure_fast(p, alpha, adjusted_weights, matrix_intersections)  graph_test_shortcut_fast(p, alpha, adjusted_weights)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing-fast.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Calculate hypothesis rejection results efficiently — graph_test_closure_fast","text":"p numeric vector p-values alpha numeric scalar specifying global significance level testing adjusted_weights weights (second half columns) graph_generate_weights() output, adjusted appropriate testing algorithm (Bonferroni, Simes, parametric) graph initial graph returned graph_create() intersections first half columns graph_generate_weights() output, indicating hypotheses contained intersection","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing-fast.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Calculate hypothesis rejection results efficiently — graph_test_closure_fast","text":"logical integer vector results indicating whether hypothesis can accepted rejected globally.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing-fast.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Calculate hypothesis rejection results efficiently — graph_test_closure_fast","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing-fast.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Calculate hypothesis rejection results efficiently — graph_test_closure_fast","text":"","code":"par_gate <- simple_successive_1() num_hyps <- length(par_gate$hypotheses)  p <- c(.001, .02, .002, .03)  weighting_strategy <- graph_generate_weights(par_gate) intersections <- weighting_strategy[, seq_len(num_hyps), drop = FALSE] adjusted_weights <-   weighting_strategy[, seq_len(num_hyps) + num_hyps, drop = FALSE]  graphicalMCP:::graph_test_closure_fast(   p,   .025,   adjusted_weights,   intersections ) #>    H1    H2    H3    H4  #>  TRUE  TRUE  TRUE FALSE  graphicalMCP:::graph_test_shortcut_fast(   p,   .025,   adjusted_weights ) #>    H1    H2    H3    H4  #>  TRUE  TRUE  TRUE FALSE"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing.html","id":null,"dir":"Reference","previous_headings":"","what":"Report details of hypothesis rejections — graph_test_closure","title":"Report details of hypothesis rejections — graph_test_closure","text":"slower graph testing functions design choices made favor ease interpreting results speed. Results include hypothesis rejection decisions, also test values led final result. functions include options reporting details using adjusted p-value method adjusted weight method.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Report details of hypothesis rejections — graph_test_closure","text":"","code":"graph_test_closure(   graph,   p,   alpha = 0.025,   test_groups = list(seq_along(graph$hypotheses)),   test_types = c(\"bonferroni\"),   test_corr = rep(list(NA), length(test_types)),   verbose = FALSE,   test_values = FALSE )  graph_test_shortcut(   graph,   p,   alpha = 0.025,   verbose = FALSE,   test_values = FALSE )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Report details of hypothesis rejections — graph_test_closure","text":"graph initial graph returned graph_create() p numeric vector p-values alpha numeric scalar specifying global significance level testing test_groups list numeric vectors specifying hypotheses test together test_types character vector tests apply given groups test_corr (Optional) numeric matrix correlations hypotheses' test statistics verbose logical scalar specifying whether results intersection hypothesis included test_values logical scalar specifying whether hypothesis-level detail included results, including calculating adjusted weights parametric tests","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Report details of hypothesis rejections — graph_test_closure","text":"graph_report object, list 4 elements: inputs, outputs, details, test_values Inputs - list input parameters used run test Outputs - list global test results Details - matrix detailed adjusted p-value results (graph deletion sequence shortcut testing) Test values - data frame hypothesis-level test details intersection (step shortcut testing)","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Report details of hypothesis rejections — graph_test_closure","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Report details of hypothesis rejections — graph_test_closure","text":"","code":"hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   c(0, 0, 1, 0),   c(0, 0, 0, 1),   c(0, 1, 0, 0),   c(1, 0, 0, 0) )  g <- graph_create(hypotheses, transitions) p <- c(.01, .005, .015, .022)  test_corr <- list(NA, matrix(c(1, .5, .5, 1), nrow = 2, byrow = TRUE))  # The default is all Bonferroni with alpha = .025 graph_test_closure(g, p) #>  #> Test parameters ($inputs) ------------------------------------------------------ #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1  0  0  1  0 #>   H2  0  0  0  1 #>   H3  0  1  0  0 #>   H4  1  0  0  0 #>  #>   Alpha = 0.025 #>  #>                           H1    H2    H3    H4 #>   Unadjusted p-values: 0.010 0.005 0.015 0.022 #>  #>   Test types #>   bonferroni: (H1, H2, H3, H4) #>  #> Test summary ($outputs) -------------------------------------------------------- #>   Hypothesis Adj. P-value Reject #>           H1         0.02   TRUE #>           H2         0.01   TRUE #>           H3         0.03  FALSE #>           H4         0.03  FALSE #>  #>   Final updated graph after removing rejected hypotheses #>  #>   --- Hypothesis weights --- #>   H1:  NA #>   H2:  NA #>   H3: 0.5 #>   H4: 0.5 #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1 NA NA NA NA #>   H2 NA NA NA NA #>   H3 NA NA  0  1 #>   H4 NA NA  1  0 #>   # But tests can be specified at the hypothesis-level graph_test_closure(   graph = g,   p = p,   alpha = .025,   test_groups = list(1:2, 3:4),   test_types = c(\"bonferroni\", \"parametric\"),   test_corr = test_corr ) #>  #> Test parameters ($inputs) ------------------------------------------------------ #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1  0  0  1  0 #>   H2  0  0  0  1 #>   H3  0  1  0  0 #>   H4  1  0  0  0 #>  #>   Alpha = 0.025 #>  #>                           H1    H2    H3    H4 #>   Unadjusted p-values: 0.010 0.005 0.015 0.022 #>  #>   Correlation matrix:     H3  H4 #>                       H3 1.0 0.5 #>                       H4 0.5 1.0 #>  #>   Test types #>   bonferroni: (H1, H2) #>   parametric: (H3, H4) #>  #> Test summary ($outputs) -------------------------------------------------------- #>   Hypothesis Adj. P-value Reject #>           H1      0.02000   TRUE #>           H2      0.01000   TRUE #>           H3      0.02773  FALSE #>           H4      0.02773  FALSE #>  #>   Final updated graph after removing rejected hypotheses #>  #>   --- Hypothesis weights --- #>   H1:  NA #>   H2:  NA #>   H3: 0.5 #>   H4: 0.5 #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1 NA NA NA NA #>   H2 NA NA NA NA #>   H3 NA NA  0  1 #>   H4 NA NA  1  0 #>"}]
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If not, see <http://www.gnu.org/licenses/>. <program>  Copyright (C) <year>  <name of author> This program comes with ABSOLUTELY NO WARRANTY; for details type 'show w'. This is free software, and you are welcome to redistribute it under certain conditions; type 'show c' for details."},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"what-is-the-closure","dir":"Articles","previous_headings":"","what":"What is the closure?","title":"Generating the closure of a graph","text":"closure graph set sub-graphs, along weights calculated according algorithm 1 Bretz et al (2011). primarily used closed testing, sub-graphs tested significance, results aggregated determine null hypotheses significant globally. Throughout article common example used demonstrations - simple successive graph. two primary hypotheses, \\(H_1\\) \\(H_2\\), initial weight evenly split . secondary hypotheses, \\(H_3\\) \\(H_4\\), weight propagated \\(H_1\\) \\(H_2\\) deleted, respectively.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"components-of-the-closure","dir":"Articles","previous_headings":"What is the closure?","what":"Components of the closure","title":"Generating the closure of a graph","text":"graphicalMCP, closure represented matrix, row represents sub-graph (also called intersection hypothesis), column corresponds individual hypothesis. matrix can created graph_generate_weights(), two parts: indicator matrix showing hypotheses contained sub-graph (-called powerset set), weights matrix containing induced weights sub-graph.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"properties-of-the-closure","dir":"Articles","previous_headings":"What is the closure?","what":"Properties of the closure","title":"Generating the closure of a graph","text":"rows closure generated particular way order give useful properties.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"repeating-recursive-blocks","dir":"Articles","previous_headings":"What is the closure? > Properties of the closure","what":"Repeating recursive blocks","title":"Generating the closure of a graph","text":"First, notice row can obtained row higher matrix flipping single 1 0. example, go row 1 row 3 flipping \\(H_3\\), go row 10 row 14 flipping \\(H_2\\). upper row pairing like can thought “parent” sub-graph, lower row “child” sub-graph. Flipping 0 1 moving matrix called “finding sub-graph’s parent.” Now consider parent-finding strategy left-0 row flipped. reveals pattern bottom half top half, row’s parent bottom half corresponding row top half, eight rows . pattern repeats within box recursively, top half box matching bottom half, first missing hypothesis flipped 0 1.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"binary-counting","dir":"Articles","previous_headings":"What is the closure? > Properties of the closure","what":"Binary counting","title":"Generating the closure of a graph","text":"second useful property somewhat re-framing first, perhaps viewed reason first true. Starting bottom row, powerset particular order counts 1 binary, incrementing 1 per row. means row number can directly calculated vector showing hypotheses currently deleted graph: row_number == number_of_rows - incl_excl_vec_converted_to_base_10 + 1. example, intersection number 6 hypothesis vector 1010. interpreted binary, 1 * 8 + 0 * 4 + 1 * 2 + 0 * 1 = 10 base 10, 6 == 15 - 10 + 1.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"strategies","dir":"Articles","previous_headings":"","what":"Strategies","title":"Generating the closure of a graph","text":"size closure grows quickly graph size increases (n-graph 2^n - 1 sub-graphs), calculating full closure large graphs can computationally intensive. Optimizing process led three main strategies: simplest approach, uses full graph starting point every sub-graph, deletes appropriate hypotheses recursive method, traverses closure tree, deleting one hypothesis time step graphs formulaic shortcut using order graphs generated recursive method Note discussion methods focuses primarily weights side rather powerset side closure. fastest methods discovered generate powerset implicitly missing values weights side. methods simple approach, rely powerset order generate weights, powerset can created efficiently. num_hyps refers number hypotheses initial graph.","code":"powerset <- as.matrix(rev(expand.grid(rep(list(1:0), num_hyps))[-2^num_hyps, ]))"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"simple-approach","dir":"Articles","previous_headings":"Strategies","what":"Simple approach","title":"Generating the closure of a graph","text":"simplest approach generate weights closure apply graph_update() initial graph sub-graph. short sweet write, ’s inefficient hypothesis gets deleted graph multiple times. code simple method.","code":"ggw_simple <- function(graph) {   num_hyps <- length(graph$hypotheses)    matrix_intersections <-     as.matrix(rev(expand.grid(rep(list(1:0), num_hyps))[-2^num_hyps, ]))   colnames(matrix_intersections) <- names(graph$hypotheses)    matrix_weights <- apply(     matrix_intersections,     1,     function(h) graph_update(graph, !h)$updated_graph$hypotheses,     simplify = FALSE   )    cbind(matrix_intersections, do.call(rbind, matrix_weights)) }"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"recursive","dir":"Articles","previous_headings":"Strategies","what":"Recursive","title":"Generating the closure of a graph","text":"recursive method treats space sub-graphs tree, initial graph root, sub-graphs decreasing size going branches. essence recursive step delete hypothesis current graph. every hypothesis every sub-graph tree result taking multiple paths many graphs. key part recursive step memory - graph tree delete hypotheses come hypothesis just deleted reach current graph. base case also needs memory - reached graph one hypothesis left, last-deleted hypothesis number larger current hypotheses. memory recursion enables tree traversal reach unique graph state exactly . implementation recursion, followed wrapper processing sub-graph list standard matrix form.","code":"delete_nodes_recursive <- function(graph, last = 0) {   init_hypotheses <- hypotheses <- graph$hypotheses   init_transitions <- transitions <- graph$transitions    ### base case   int_hyp <- as.integer(names(hypotheses))    is_single_node <- length(hypotheses) == 1   last_is_bigger <- last > max(int_hyp)    if (is_single_node || last_is_bigger) {     return(list(graph))   }    ### recursive step   children <- list()    for (orig_hyp_num in int_hyp[int_hyp > last]) {     del_index <- match(orig_hyp_num, int_hyp)     hyp_nums <- seq_along(hypotheses)[seq_along(hypotheses) != del_index]      for (hyp_num in hyp_nums) {       hypotheses[[hyp_num]] <-         init_hypotheses[[hyp_num]] +         init_hypotheses[[del_index]] * init_transitions[[del_index, hyp_num]]        denominator <- 1 - init_transitions[[hyp_num, del_index]] *         init_transitions[[del_index, hyp_num]]        for (end_num in hyp_nums) {         if (hyp_num == end_num || denominator <= 0) {           transitions[[hyp_num, end_num]] <- 0         } else {           transitions[[hyp_num, end_num]] <- (             init_transitions[[hyp_num, end_num]] +               init_transitions[[hyp_num, del_index]] *                 init_transitions[[del_index, end_num]]           ) / denominator         }       }     }      smaller_graph <- structure(       list(         hypotheses = hypotheses[-del_index],         transitions = as.matrix(transitions[-del_index, -del_index])       ),       class = \"initial_graph\"     )      children[[del_index]] <- delete_nodes_recursive(       smaller_graph,       orig_hyp_num     )   }    c(     unlist(children, recursive = FALSE),     list(graph)   ) } ggw_recursive <- function(graph) {   # The recursion requires the hypotheses to be named sequentially as actual   # numbers for the memory property to work   hyp_names <- names(graph$hypotheses)   names(graph$hypotheses) <- seq_along(graph$hypotheses)   colnames(graph$transitions) <- names(graph$hypotheses)   rownames(graph$transitions) <- names(graph$hypotheses)    # Recursively generate a list of all sub-graphs   list_subgraphs <- delete_nodes_recursive(graph)    # Process the list of graphs into the normal matrix form   matrix_weights <- structure(     do.call(       rbind,       lapply(         list_subgraphs,         function(graph) graph$hypotheses[as.character(seq_along(hyp_names))]       )     ),     dimnames = list(1:(2^length(hyp_names) - 1), hyp_names)   )    matrix_intersections <- !is.na(matrix_weights)   matrix_weights[is.na(matrix_weights)] <- 0    cbind(matrix_intersections, matrix_weights) }"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"formula-shortcut","dir":"Articles","previous_headings":"Strategies","what":"Formula shortcut","title":"Generating the closure of a graph","text":"Finally, fastest method found far - formula shortcut. recursion can save lot time first method, still bit overhead get list sub-graphs matrix form standard. “repeating block” property closure mentioned earlier useful. Instead using recursion connect parent sub-graphs children, pair formulas can used. One formula generates parent graph obtained flipping left-0 1: .call(c, lapply(2^(seq_len(num_hyps) - 1), seq_len)). Note rows 2 (non-existent) row 16, empty graph. Row 1 parent graph. left-0 child graph easy find, parent graph’s perspective, formula calculates hypothesis delete: rep(rev(seq_len(num_hyps)), 2^(seq_len(num_hyps) - 1)). also applies rows 2 16. formula shortcut also results simpler code recursive solution.","code":""},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"generating-the-closure","dir":"Articles","previous_headings":"Performance gains","what":"Generating the closure","title":"Generating the closure of a graph","text":"formula method reduces step generating closure single deletion prior graph, almost additional overhead. ’s different methods fare, including version gMCP package reference. Also worthy note lrstat package, contains MCP-related functions, including generating weights closure. uses excellent C++ code perform even faster, since speed current formula-based method acceptable, adding Rcpp dependency considered worth additional time savings.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"power-simulations","dir":"Articles","previous_headings":"Performance gains","what":"Power simulations","title":"Generating the closure of a graph","text":"time savings closure nice, graphs savings make big difference compared longer run-times e.g. power simulations. However, paying attention closure important reasons , improving power algorithm.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"standard-algorithm","dir":"Articles","previous_headings":"Performance gains > Power simulations","what":"Standard algorithm","title":"Generating the closure of a graph","text":"typical method running Bonferroni shortcut procedure graph : Search graph single hypothesis can rejected Delete rejected hypothesis update weights Repeat significant hypotheses Running process either R low-level language like C fast single procedure, ’s run 100,000 times power simulation, R version becomes onerous. However, ’s lot duplication power simulation using method. many simulations, steps taken, means re-calculating set weights many times.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/generate-closure.html","id":"closure-shortcut","dir":"Articles","previous_headings":"Performance gains > Power simulations","what":"Closure shortcut","title":"Generating the closure of a graph","text":"binary counting property closure admits shortcut can implemented R get scalable competitor typical algorithm: Generate closure single time get sub-graph weights efficiently simulation: 1. Search graph hypotheses can rejected - fast vectorization 1. Using binary counting property, index row closure corresponding hypotheses rejected far get updated weights - substantially faster updating graph re-calculating weights, especially larger graphs 1. Repeat using updated weights method fast gMCP’s C implementation, still runs substantial power simulations matter seconds. biggest drawback can scale differently different graph structures. instance fixed sequence procedure can take longer balanced graph, like Holm procedure, takes steps reject possible hypotheses.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"introduction","dir":"Articles","previous_headings":"","what":"Introduction","title":"Common procedures","text":"confirmatory clinical trials multiple endpoints, regulations mandate strong control family-wise error rate reduce risk false positives. Multiple comparison procedures (MCPs) used spread total significance multiple hypotheses. article ’ll demonstrate go standard procedures advanced procedures using graphical approach.","code":""},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"bonferroni","dir":"Articles","previous_headings":"Simple procedures","what":"Bonferroni","title":"Common procedures","text":"Probably common MCP Bonferroni correction. simply divides alpha level evenly hypotheses. significance adjustment made hypothesis rejected. Bonferroni correction can performed easily without graphical approach, graphical example .","code":"transitions <- matrix(0, 5, 5)  bonferroni_graph <- graph_create(rep(1 / 5, 5), transitions)  plot(bonferroni_graph, layout = igraph::layout_in_circle, vertex.size = 30)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"holm","dir":"Articles","previous_headings":"Simple procedures","what":"Holm","title":"Common procedures","text":"Holm procedure starts identical weights Bonferroni, weights updated uniformly hypothesis rejected. updating, always least powerful Bonferroni procedure.","code":"transitions <- matrix(1 / 4, 5, 5) diag(transitions) <- 0  holm_graph <- graph_create(rep(1 / 5, 5), transitions)  plot(holm_graph, layout = igraph::layout_in_circle, vertex.size = 30)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"fixed-sequence","dir":"Articles","previous_headings":"Simple procedures","what":"Fixed sequence","title":"Common procedures","text":"Another common procedure simplicity fixed sequence procedure. places emphasis rejecting hypotheses hypotheses expected p-values slightly alpha, reduced weights Holm procedure cause hypotheses rejected start.  procedures popular control FWER simple, allowing easily understood communicated . However, ’re always procedures fit study best, using complex procedures can make communication difficult. graphical approach can enable powerful procedures used enabling clinical teams understand communicate procedures.","code":"fixed_sequence_graph <- graph_create(   c(1, 0, 0, 0),   rbind(     c(0, 1, 0, 0),     c(0, 0, 1, 0),     c(0, 0, 0, 1),     c(0, 0, 0, 0)   ) )  plot(fixed_sequence_graph, nrow = 1, asp = .1)"},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"simple-successive","dir":"Articles","previous_headings":"Complex procedures","what":"Simple successive","title":"Common procedures","text":"simple successive procedure introduced [names] [year] paper, [title]. designed experiments involving four hypotheses: Two pairs hypotheses, containing primary related secondary hypothesis. Initial weight split equally pairs passed one pair hypotheses pair rejected.  General successive graphs good example multiple variations may useful slight differences starting edge weights. Variable edge weights currently supported, can done particular graph light wrapper. multiple variations can created compared easily. similar approach used deviating equal split hypothesis weights well.","code":"simple_successive_graph <- graph_create(   c(.5, .5, 0, 0),   rbind(     c(0, 0, 1, 0),     c(0, 0, 0, 1),     c(0, 1, 0, 0),     c(1, 0, 0, 0)   ) )  plot(simple_successive_graph, layout = \"grid\", nrow = 2, vertex.size = 30) simple_successive_var <- function(gamma) {   graph_create(     c(.5, .5, 0, 0),     rbind(       c(0, gamma, 1 - gamma, 0),       c(gamma, 0, 0, 1 - gamma),       c(0, 1, 0, 0),       c(1, 0, 0, 0)     )   ) } plot(   simple_successive_var(.75),   layout = \"grid\",   nrow = 2,   vertex.label.cex = 1.4,   edge.label.cex = 1.4,   vertex.size = 30 )  plot(   simple_successive_var(.9),   layout = \"grid\",   nrow = 2,   vertex.label.cex = 1.4,   edge.label.cex = 1.4,   vertex.size = 30 )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"huque-alosh-bhore-2011","dir":"Articles","previous_headings":"Complex procedures","what":"Huque-Alosh-Bhore (2011)","title":"Common procedures","text":"2011, Huque, Alosh, Bhore proposed another size 4 procedure. ’s one particularly meant ?","code":"hab_2011_graph <- graph_create(   c(1, 0, 0, 0),   rbind(     c(0, .5, .5, 0),     c(0, 0, 0, 1),     c(0, .5, 0, .5),     c(0, 1, 0, 0)   ) )  plot(hab_2011_graph, layout = \"grid\", vertex.size = 30)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"wiens-dmitrienko-2005","dir":"Articles","previous_headings":"Complex procedures","what":"Wiens-Dmitrienko (2005)","title":"Common procedures","text":"","code":"wd_2005_graph <- graph_create(   c(1 / 3, 1 / 3, 1 / 3),   rbind(     c(0, 1, 0),     c(0, 0, 1),     c(.5, .5, 0)   ) )  plot(   wd_2005_graph,   layout = \"grid\",   nrow = 1,   edge_curves = c(pairs = -4, \"H3|H1\" = -4),   vertex.size = 30,   asp = .1 )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"parallel-gate-keeping","dir":"Articles","previous_headings":"Complex procedures","what":"Parallel gate-keeping","title":"Common procedures","text":"[year], [names] introduced parallel gate-keeping procedure. useful …?","code":"par_gate_graph <- graph_create(   c(.5, .5, 0, 0),   rbind(     c(0, 0, .5, .5),     c(0, 0, .5, .5),     c(0, 0, 0, 1),     c(0, 0, 1, 0)   ) )   plot(par_gate_graph, layout = \"grid\", nrow = 2, vertex.size = 30)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"improved-parallel-gate-keeping","dir":"Articles","previous_headings":"Complex procedures","what":"Improved parallel gate-keeping","title":"Common procedures","text":"One shortcoming parallel gate-keeping approach case e.g. H2, H3, H4 found significant. three deleted, H1 left 0.5 weight. makes graph sub-optimal.  graph can made optimal adding epsilon edges going back secondary primary hypotheses. Epsilon edges infinitesimal weight creates connection hypotheses effectively propagates weight ’s edge left exiting hypothesis. yields improved parallel gate-keeping procedure.","code":"plot(graph_update(par_gate_graph, 2:4), vertex.size = 30) imp_par_gate_graph <- graph_create(   c(.5, .5, 0, 0),   rbind(     c(0, 0, .5, .5),     c(0, 0, .5, .5),     c(.0001, 0, 0, .9999),     c(0, .0001, .9999, 0)   ) )  plot(   imp_par_gate_graph,   layout = \"grid\",   nrow = 2,   eps = .001,   vertex.size = 30 ) plot(   graph_update(imp_par_gate_graph, c(2)),   vertex.size = 60,   vertex.label.cex = 2.5,   edge.label.cex = 2.5,   eps = 1e-3 )  plot(   graph_update(imp_par_gate_graph, c(2, 4)),   vertex.size = 60,   vertex.label.cex = 2.5,   edge.label.cex = 2.5 )  plot(   graph_update(imp_par_gate_graph, c(2, 4, 3)),   vertex.size = 60,   vertex.label.cex = 2.5,   edge.label.cex = 2.5 )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/graph-examples.html","id":"conclusion","dir":"Articles","previous_headings":"","what":"Conclusion","title":"Common procedures","text":"examples just unlimited possibilities enabled graphical approach MCPs. Graphs can customized based priorities trial design, marginal power endpoints, assumptions, expected known correlation endpoints. graphical approach allows complex procedures maximize chances successful trial enabling better communication insight procedures.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"multiple-comparison-problem","dir":"Articles","previous_headings":"","what":"Multiple comparison problem","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"Consider confirmatory clinical trial comparing test treatment (treatment) control treatment (control) disease. two doses treatment: low dose high dose. two endpoints included multiplicity adjustment strategy, primary endpoint (PE) secondary endpoint (SE). total, four null hypotheses: \\(H_1\\) \\(H_3\\) primary secondary hypotheses respectively low dose versus control; \\(H_2\\) \\(H_4\\) primary secondary hypotheses respectively high dose versus control. discussing appropriate multiple comparison procedures, clinical considerations, adds structure multiple comparison procedures, can flexibly incorporated using graphical approaches. First, low high doses considered equally important, means rejecting primary hypothesis either dose versus control leads successful trial. Regarding secondary hypotheses, one tested corresponding primary hypothesis rejected. means \\(H_3\\) tested \\(H_1\\) rejected; \\(H_4\\) tested \\(H_2\\) rejected. addition, statistical considerations complete graph. primary hypotheses \\(H_1\\) \\(H_2\\) equal hypothesis weight 0.5. secondary hypotheses hypothesis weight 0. primary hypothesis rejected, weight propagate along two outgoing edges: one primary hypothesis one descendant secondary hypothesis. two edges equal transition weight 0.5. primary secondary hypotheses rejected dose-control comparison, hypothesis weights propagated primary hypothesis dose-control comparison. specifications, can create following graph.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"create-a-graph","dir":"Articles","previous_headings":"","what":"Create a graph","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"","code":"hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   c(0, 0.5, 0.5, 0),   c(0.5, 0, 0, 0.5),   c(0, 1, 0, 0),   c(1, 0, 0, 0) ) hyp_names <- c(\"H1\", \"H2\", \"H3\", \"H4\")  g <- graph_create(hypotheses, transitions, hyp_names)  plot(g, layout = \"grid\", vertex.size = 20)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"perform-the-graphical-multiple-comparison-procedure","dir":"Articles","previous_headings":"","what":"Perform the graphical multiple comparison procedure","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"Given set p-values \\(H_1, \\ldots, H_4\\), graphical multiple comparison procedure can performed control familywise error rate (FWER) significance level alpha. graph_test_shortcut function agnostic one-sided two-sided tests. one-sided p-values, alpha often set 0.025 (default); two-sided p-values, alpha often set 0.05. consider one-sided tests . hypothesis rejected adjusted p-value less equal alpha. running procedure, hypotheses \\(H_1\\), \\(H_2\\), \\(H_4\\) rejected adjusted p-value calculated.","code":"p_values <- c(.013, .012, .105, .005) test_results <- graph_test_shortcut(g, p = p_values, alpha = 0.025) test_results$outputs # Adjusted p-values and rejections #> $adjusted_p #>    H1    H2    H3    H4  #> 0.024 0.024 0.105 0.024  #>  #> $rejected #>    H1    H2    H3    H4  #>  TRUE  TRUE FALSE  TRUE  #>  #> $graph #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0 #> H2: 0 #> H3: 1 #> H4: 0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  0  0 #>  H2  0  0  0  0 #>  H3  0  0  0  0 #>  H4  0  0  0  0"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"obtain-final-and-intermediate-graphs-after-rejections","dir":"Articles","previous_headings":"Perform the graphical multiple comparison procedure","what":"Obtain final and intermediate graphs after rejections","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"addition, two graphs provided: one initial graph one final graph removing rejected hypotheses \\(H_1\\), \\(H_2\\), \\(H_4\\). also interested intermediate graphs - example, graph \\(H_1\\) \\(H_2\\) rejected - can specify verbose = TRUE. Note intermediate graphs depends order rejections, .e., sequence hypotheses rejected. default order defined increasing adjusted p-values earlier hypothesis numbering case ties adjusted p-values. Among rejected hypotheses, one smallest adjusted p-value first rejected one largest adjusted p-value last rejected. example, default order rejection \\(H_2\\rightarrow H_1\\rightarrow H_4\\). obtain intermediate graphs based order rejection, one can specify test_results_verbose$details$results. example, graph \\(H_1\\) \\(H_2\\) rejected given test_results_verbose$details$results[[3]]. order rejections may unique orders valid. example, rejected hypotheses \\(H_1\\), \\(H_2\\) \\(H_4\\). default order rejections \\(H_2 \\rightarrow H_1 \\rightarrow H_4\\). Another valid order rejections \\(H_2 \\rightarrow H_4 \\rightarrow H_1\\). However, first rejected hypothesis can \\(H_1\\) \\(H_4\\). obtain possible rejection orders, one can use function graph_rejection_orderings. Initial, intermediate final graphs can obtained using function graph_update particular order rejections. order intermediate graphs determined deleting hypothesis lowest adjusted p-values step (smaller hypothesis sequentially case adjusted p-value ties). cases, order valid order deletion. example, minimum adjusted p-value method results deleting order \\(H_2 \\rightarrow H_1 \\rightarrow H_4\\). \\(H_2 \\rightarrow H_4 \\rightarrow H_1\\) also valid deletion order. Deletion order can customized using [custom_order_function_once_it_is_written()].","code":"test_results_verbose <-   graph_test_shortcut(g, p = p_values, alpha = 0.025, verbose = TRUE) test_results_verbose$details$results[[3]] # Intermediate graph after H1 and H2 rejected #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.0 #> H2: 0.0 #> H3: 0.5 #> H4: 0.5 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  0  0 #>  H2  0  0  0  0 #>  H3  0  0  0  1 #>  H4  0  0  1  0 orders <- graph_rejection_orderings(test_results)$valid_orderings # Obtain all valid orders of rejections orders #> [[1]] #> H2 H4 H1  #>  2  4  1  #>  #> [[2]] #> H2 H1 H4  #>  2  1  4 graph_update(g, delete = orders[[2]]) # Graphs following the order of H2, H4, and H1 #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>      H1  H2  H3  H4 #>  H1 0.0 0.5 0.5 0.0 #>  H2 0.5 0.0 0.0 0.5 #>  H3 0.0 1.0 0.0 0.0 #>  H4 1.0 0.0 0.0 0.0 #>  #> Updated graph after deleting hypotheses 2, 1, 4 #>  #> --- Hypothesis weights --- #> H1: NA #> H2: NA #> H3:  1 #> H4: NA #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1 NA NA NA NA #>  H2 NA NA NA NA #>  H3 NA NA  0 NA #>  H4 NA NA NA NA #>  #> Deletion sequence ($intermediate_graphs) --------------------------------------- #>  #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>       H1  H2  H3  H4 #>   H1 0.0 0.5 0.5 0.0 #>   H2 0.5 0.0 0.0 0.5 #>   H3 0.0 1.0 0.0 0.0 #>   H4 1.0 0.0 0.0 0.0 #>  #>     Step 1: Updated graph after removing hypothesis 2 #>  #>     --- Hypothesis weights --- #>     H1: 0.75 #>     H2:   NA #>     H3: 0.00 #>     H4: 0.25 #>  #>     --- Transition weights --- #>              H1       H2       H3       H4 #>     H1 0.000000       NA 0.666667 0.333333 #>     H2       NA       NA       NA       NA #>     H3 0.500000       NA 0.000000 0.500000 #>     H4 1.000000       NA 0.000000 0.000000 #>  #>       Step 2: Updated graph after removing hypotheses 2, 1 #>  #>       --- Hypothesis weights --- #>       H1:  NA #>       H2:  NA #>       H3: 0.5 #>       H4: 0.5 #>  #>       --- Transition weights --- #>          H1 H2 H3 H4 #>       H1 NA NA NA NA #>       H2 NA NA NA NA #>       H3 NA NA  0  1 #>       H4 NA NA  1  0 #>  #>         Step 3: Updated graph after removing hypotheses 2, 1, 4 #>  #>         --- Hypothesis weights --- #>         H1: NA #>         H2: NA #>         H3:  1 #>         H4: NA #>  #>         --- Transition weights --- #>            H1 H2 H3 H4 #>         H1 NA NA NA NA #>         H2 NA NA NA NA #>         H3 NA NA  0 NA #>         H4 NA NA NA NA #>  #>   Final updated graph after removing deleted hypotheses #>  #>   --- Hypothesis weights --- #>   H1: NA #>   H2: NA #>   H3:  1 #>   H4: NA #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1 NA NA NA NA #>   H2 NA NA NA NA #>   H3 NA NA  0 NA #>   H4 NA NA NA NA"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"obtain-adjusted-significance-levels","dir":"Articles","previous_headings":"Perform the graphical multiple comparison procedure","what":"Obtain adjusted significance levels","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"equivalent way obtain rejections adjusting significance levels. hypothesis rejected p-value less equal adjusted significance level. adjusted significance levels calculated order adjusted p-values: \\(H_2 \\rightarrow H_1 \\rightarrow H_4\\), four steps checking rejections. First, \\(H_2\\) rejected adjusted significance level 0.5 * alpha. Second, \\(H_1\\) rejected adjusted significance level 0.75 * alpha, \\(H_2\\) rejected. Third, \\(H_4\\) rejected adjusted significance level 0.5 * alpha, \\(H_1\\) \\(H_2\\) rejected. Fourth finally, \\(H_3\\) rejected adjusted significance level alpha, \\(H_1\\), \\(H_2\\) \\(H_4\\) rejected. results can obtained specifying test_values = TRUE.","code":"test_results_test_values <-   graph_test_shortcut(g, p = p_values, alpha = 0.025, test_values = TRUE) test_results_test_values$test_values$results #>    Step Hypothesis     p <= Weight * Alpha Inequality_holds #> H2    1         H2 0.012 <=   0.50 * 0.025             TRUE #> H1    2         H1 0.013 <=   0.75 * 0.025             TRUE #> H4    3         H4 0.005 <=   0.50 * 0.025             TRUE #> H3    4         H3 0.105 <=   1.00 * 0.025            FALSE"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"power-calculation","dir":"Articles","previous_headings":"","what":"Power calculation","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"Given graph, trial team often interested power trial. single null hypothesis, power probability reject null hypothesis significance level alpha alternative hypothesis true. multiple null hypotheses, multiple versions power. example, power reject least one hypothesis power reject hypotheses, given alternative hypotheses true. graphical multiple comparison procedures, also important understand power reject hypothesis, given multiplicity adjustment. Sometimes, team may want customize definitions power define success. Thus power calculation important aspect trial design.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"input-marginal-power-for-primary-hypotheses","dir":"Articles","previous_headings":"Power calculation","what":"Input: Marginal power for primary hypotheses","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"Assume primary endpoint occurrence unfavorable clinical event. evaluate treatment effect, proportion patients event calculated, lower proportion preferred. Assume proportions 0.181 low high doses, 0.3 control. Using equal randomization among three treatment groups, clinical trial team chooses total sample size 600 200 per group. leads marginal power 80% \\(H_1\\) \\(H_2\\), respectively, using two-sample test difference proportions unpooled variance one-sided significance level 0.025. calculation, use marginal power combine information treatment effect, nuisance parameter, sample sizes hypothesis. Note significance level used marginal power calculation must alpha, used power calculation significance level FWER control. addition, marginal power one--one relationship noncentrality parameter, illustrated .","code":"alpha <- 0.025 prop <- c(0.3, 0.181, 0.181) sample_size <- rep(200, 3)  unpooled_variance <- prop[-1] * (1 - prop[-1]) / sample_size[-1] +   prop[1] * (1 - prop[1]) / sample_size[1]  noncentrality_parameter_primary <-   -(prop[-1] - prop[1]) / sqrt(unpooled_variance)  power_marginal_primary <- pnorm(qnorm(alpha, lower.tail = FALSE),   noncentrality_parameter_primary, 1,   lower.tail = FALSE )  names(power_marginal_primary) <- c(\"H1\", \"H2\") power_marginal_primary #>        H1        H2  #> 0.8028315 0.8028315"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"input-marginal-power-for-secondary-hypotheses","dir":"Articles","previous_headings":"Power calculation","what":"Input: Marginal power for secondary hypotheses","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"Assume secondary endpoint change total medication score baseline, continuous outcome. evaluate treatment effect, mean change calculated, greater reduction preferred. Assume mean change baseline reduction 7.5 8.25, respectively low high doses, 5 control. assume known common standard deviation 10. Given sample size 200 per treatment group, marginal power 71% 90% \\(H_3\\) \\(H_4\\), respectively, using two-sample \\(z\\)-test difference means one-sided significance level 0.025.","code":"mean_change <- c(5, 7.5, 8.25) sd <- rep(10, 3) variance <- sd[-1]^2 / sample_size[-1] + sd[1]^2 / sample_size[1]  noncentrality_parameter_secondary <- (mean_change[-1] - mean_change[1]) /   sqrt(variance)  power_marginal_secondary <- pnorm(qnorm(alpha, lower.tail = FALSE),   noncentrality_parameter_secondary, 1,   lower.tail = FALSE )  names(power_marginal_secondary) <- c(\"H3\", \"H4\") power_marginal_secondary #>        H3        H4  #> 0.7054139 0.9014809"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"input-correlation-structure-to-simulate-test-statistics","dir":"Articles","previous_headings":"Power calculation","what":"Input: Correlation structure to simulate test statistics","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"addition marginal power, also need make assumptions joint distribution test statistics. example, assume follow multivariate normal distribution means defined noncentrality parameters correlation matrix \\(R\\). obtain correlations, helpful understand two types correlations example. correlation two dose-control comparisons endpoint correlation endpoints. former correlation can calculated function sample size. example, correlation test statistics \\(H_1\\) \\(H_2\\) \\(\\rho_{12}=\\left(\\frac{n_1}{n_1+n_0}\\right)^{1/2}\\left(\\frac{n_2}{n_3+n_0}\\right)^{1/2}\\). equal randomization, correlation 0.5. correlation test statistics \\(H_3\\) \\(H_4\\) . hand, correlation endpoints dose-control comparison often estimated based prior knowledge previous trials. Without information, assume \\(\\rho_{13}=\\rho_{24}=0.5\\). practice, one set correlation parameter try multiple values assess sensitivity assumption. Regarding correlation test statistics \\(H_1\\) \\(H_4\\) \\(H_2\\) \\(H_3\\), even difficult estimate. use simple product rule, means correlation product correlations two previously assumed correlations. example, \\(\\rho_{14}=\\rho_{12}*\\rho_{24}\\) \\(\\rho_{23}=\\rho_{12}*\\rho_{13}\\). practice, one may make assumptions instead using product rule.","code":"corr <- matrix(0, nrow = 4, ncol = 4)  corr[1, 2] <- corr[3, 4] <- sqrt(sample_size[2] / sum(sample_size[1:2]) *   sample_size[3] / sum(sample_size[c(1, 3)]))  rho <- 0.5 corr[1, 3] <- corr[2, 4] <- rho corr[1, 4] <- corr[2, 3] <- corr[1, 2] * rho corr <- corr + t(corr) diag(corr) <- 1 colnames(corr) <- hyp_names rownames(corr) <- hyp_names corr #>      H1   H2   H3   H4 #> H1 1.00 0.50 0.50 0.25 #> H2 0.50 1.00 0.25 0.50 #> H3 0.50 0.25 1.00 0.50 #> H4 0.25 0.50 0.50 1.00"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"user-defined-success-criteria","dir":"Articles","previous_headings":"Power calculation","what":"User-defined success criteria","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"mentioned earlier, multiple versions “power” multiple hypotheses. Commonly used “power” definitions include: Local power: probability hypothesis rejected (multiplicity adjustment) Expected . rejections: expected number rejections Power reject 1 : probability reject least one hypothesis Power reject : probability reject hypotheses default outputs graph_calculate_power function. addition, user can customize success criteria define versions “power”.","code":"# Probability to reject H1 H1 <- function(x) x[1]  # Expected number of rejections `Expected no. of rejections` <- function(x) x[1] + x[2] + x[3] + x[4]  # Probability to reject at least one hypothesis `AtLeast1` <- function(x) x[1] | x[2] | x[3] | x[4]  # Probability to reject all hypotheses `All` <- function(x) x[1] & x[2] & x[3] & x[4]  # Probability to reject both H1 and H2 `H1andH2` <- function(x) x[1] & x[2]  # Probability to reject both hypotheses for the low dose or the high dose `(H1andH3)or(H2andH4)` <- function(x) (x[1] & x[3]) | (x[2] & x[4]) succ <- list(   H1, `Expected no. of rejections`, `AtLeast1`, `All`, `H1andH2`,   `(H1andH3)or(H2andH4)` )  succ <- setNames(   succ,   c(     \"H1\", \"Expected no. of rejections\", \"AtLeast1\", \"All\", \"H1andH2\",     \"(H1andH3)or(H2andH4)\"   ) )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/articles/shortuct-testing.html","id":"output-calculate-power","dir":"Articles","previous_headings":"Power calculation","what":"Output: Calculate power","title":"Sequentially rejective graphical approaches based on Bonferroni tests","text":"Given inputs, can calculate power via simulation graphical multiple comparison procedure one-sided significance level alpha=0.025 using sim_n = 1e5 simulations seed sim_seed = 1234. local power 0.758, 0.765, 0.689, 0.570, respectively \\(H_1, \\ldots, H_4\\). Note local power lower marginal power former adjusted multiplicity. power reject least one hypothesis 0.856 power reject hypotheses 0.512. expected number rejections 2.782. last two user-defined success criteria, probability reject \\(H_1\\) \\(H_2\\) 0.667, probability reject least one pair \\((H_1\\) \\(H_3)\\) \\((H_2\\) \\(H_4)\\) 0.747. see detailed outputs simulated p-values rejection decisions hypotheses, specify verbose = TRUE. produce lot outputs. allow flexible printing functions, user can change following: indented space default setting indent = 2 precision numeric values (.e., number significant digits) default setting precision = 4","code":"output <- graph_calculate_power(g,   alpha = .025,   sim_corr = corr,   sim_n = 1e5,   power_marginal = c(power_marginal_primary, power_marginal_secondary),   sim_success = succ,   sim_seed = 1234 ) output #>  #> Test parameters ($inputs) ------------------------------------------------------ #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>       H1  H2  H3  H4 #>   H1 0.0 0.5 0.5 0.0 #>   H2 0.5 0.0 0.0 0.5 #>   H3 0.0 1.0 0.0 0.0 #>   H4 1.0 0.0 0.0 0.0 #>  #>   Alpha = 0.025 #>  #>   Test types #>   bonferroni: (H1, H2, H3, H4) #>  #> Simulation parameters ($inputs) ------------------------------------------------ #>   Testing 100,000 simulations - random seed 1234 & multivariate normal params: #>  #>                       H1     H2     H3     H4 #>   Marginal power: 0.8028 0.8028 0.7054 0.9015 #>  #>   Correlation:      H1   H2   H3   H4 #>                H1 1.00 0.50 0.50 0.25 #>                H2 0.50 1.00 0.25 0.50 #>                H3 0.50 0.25 1.00 0.50 #>                H4 0.25 0.50 0.50 1.00 #>  #> Power calculation ($power) ----------------------------------------------------- #>                                   H1     H2     H3     H4 #>                  Local power: 0.7640 0.7589 0.5677 0.6913 #>  #>   Expected no. of rejections: 2.782 #>    Power to reject 1 or more: 0.8556 #>          Power to reject all: 0.5121 #>  #>              Power to reject:      H1   Expected no. of rejections   AtLeast1 #>                                0.7640                       2.7818     0.8556 #>     All   H1andH2   (H1andH3)or(H2andH4) #>  0.5121    0.6673                 0.7470 verbose_output <- graph_calculate_power(g,   alpha = .025,   sim_corr = corr,   sim_n = 1e5,   power_marginal = c(power_marginal_primary, power_marginal_secondary),   sim_success = succ,   sim_seed = 1234,   verbose = TRUE )  print(verbose_output) #>  #> Test parameters ($inputs) ------------------------------------------------------ #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>       H1  H2  H3  H4 #>   H1 0.0 0.5 0.5 0.0 #>   H2 0.5 0.0 0.0 0.5 #>   H3 0.0 1.0 0.0 0.0 #>   H4 1.0 0.0 0.0 0.0 #>  #>   Alpha = 0.025 #>  #>   Test types #>   bonferroni: (H1, H2, H3, H4) #>  #> Simulation parameters ($inputs) ------------------------------------------------ #>   Testing 100,000 simulations - random seed 1234 & multivariate normal params: #>  #>                       H1     H2     H3     H4 #>   Marginal power: 0.8028 0.8028 0.7054 0.9015 #>  #>   Correlation:      H1   H2   H3   H4 #>                H1 1.00 0.50 0.50 0.25 #>                H2 0.50 1.00 0.25 0.50 #>                H3 0.50 0.25 1.00 0.50 #>                H4 0.25 0.50 0.50 1.00 #>  #> Power calculation ($power) ----------------------------------------------------- #>                                   H1     H2     H3     H4 #>                  Local power: 0.7640 0.7589 0.5677 0.6913 #>  #>   Expected no. of rejections: 2.782 #>    Power to reject 1 or more: 0.8556 #>          Power to reject all: 0.5121 #>  #>              Power to reject:      H1   Expected no. of rejections   AtLeast1 #>                                0.7640                       2.7818     0.8556 #>     All   H1andH2   (H1andH3)or(H2andH4) #>  0.5121    0.6673                 0.7470 #>  #> Simulation details ($details) -------------------------------------------------- #>     p_sim_H1  p_sim_H2  p_sim_H3  p_sim_H4 rej_H1 rej_H2 rej_H3 rej_H4 #>    3.082e-02 1.207e-02 4.119e-03 9.324e-02  FALSE   TRUE  FALSE  FALSE #>    7.934e-04 6.499e-04 2.452e-02 2.966e-03   TRUE   TRUE   TRUE   TRUE #>    3.030e-02 5.954e-02 5.431e-02 2.626e-02  FALSE  FALSE  FALSE  FALSE #>    9.743e-03 3.319e-03 7.417e-04 4.025e-04   TRUE   TRUE   TRUE   TRUE #>    1.971e-02 8.616e-03 1.642e-02 2.418e-07   TRUE   TRUE   TRUE   TRUE #>    3.121e-03 6.702e-03 1.374e-02 2.752e-04   TRUE   TRUE   TRUE   TRUE #>    3.022e-02 1.424e-01 6.038e-03 2.117e-02  FALSE  FALSE  FALSE  FALSE #>    2.498e-03 2.940e-02 4.143e-04 2.208e-03   TRUE  FALSE   TRUE  FALSE #>    6.190e-02 3.873e-02 3.166e-01 5.700e-02  FALSE  FALSE  FALSE  FALSE #>    3.678e-01 1.896e-01 7.023e-02 1.190e-02  FALSE  FALSE  FALSE  FALSE #>   ... (Use `print(x, rows = <nn>)` for more)  print(verbose_output, indent = 4, precision = 6) #>  #> Test parameters ($inputs) ------------------------------------------------------ #>     Initial graph #>  #>     --- Hypothesis weights --- #>     H1: 0.5 #>     H2: 0.5 #>     H3: 0.0 #>     H4: 0.0 #>  #>     --- Transition weights --- #>         H1  H2  H3  H4 #>     H1 0.0 0.5 0.5 0.0 #>     H2 0.5 0.0 0.0 0.5 #>     H3 0.0 1.0 0.0 0.0 #>     H4 1.0 0.0 0.0 0.0 #>  #>     Alpha = 0.025 #>  #>     Test types #>     bonferroni: (H1, H2, H3, H4) #>  #> Simulation parameters ($inputs) ------------------------------------------------ #>     Testing 100,000 simulations - random seed 1234 & multivariate normal params: #>  #>                           H1       H2       H3       H4 #>     Marginal power: 0.802831 0.802831 0.705414 0.901481 #>  #>     Correlation:      H1   H2   H3   H4 #>                  H1 1.00 0.50 0.50 0.25 #>                  H2 0.50 1.00 0.25 0.50 #>                  H3 0.50 0.25 1.00 0.50 #>                  H4 0.25 0.50 0.50 1.00 #>  #> Power calculation ($power) ----------------------------------------------------- #>                                      H1      H2      H3      H4 #>                    Local power: 0.76396 0.75887 0.56767 0.69133 #>  #>     Expected no. of rejections: 2.78183 #>      Power to reject 1 or more: 0.85557 #>            Power to reject all: 0.51205 #>  #>                Power to reject:       H1   Expected no. of rejections #>                                  0.76396                      2.78183 #>    AtLeast1     All   H1andH2   (H1andH3)or(H2andH4) #>     0.85557 0.51205   0.66726                0.74695 #>  #> Simulation details ($details) -------------------------------------------------- #>         p_sim_H1    p_sim_H2    p_sim_H3    p_sim_H4 rej_H1 rej_H2 rej_H3 rej_H4 #>      3.08204e-02 1.20654e-02 4.11858e-03 9.32434e-02  FALSE   TRUE  FALSE  FALSE #>      7.93372e-04 6.49905e-04 2.45178e-02 2.96560e-03   TRUE   TRUE   TRUE   TRUE #>      3.02992e-02 5.95396e-02 5.43083e-02 2.62583e-02  FALSE  FALSE  FALSE  FALSE #>      9.74332e-03 3.31857e-03 7.41721e-04 4.02469e-04   TRUE   TRUE   TRUE   TRUE #>      1.97135e-02 8.61618e-03 1.64183e-02 2.41833e-07   TRUE   TRUE   TRUE   TRUE #>      3.12066e-03 6.70231e-03 1.37441e-02 2.75170e-04   TRUE   TRUE   TRUE   TRUE #>      3.02208e-02 1.42376e-01 6.03828e-03 2.11740e-02  FALSE  FALSE  FALSE  FALSE #>      2.49757e-03 2.94026e-02 4.14273e-04 2.20779e-03   TRUE  FALSE   TRUE  FALSE #>      6.18994e-02 3.87257e-02 3.16613e-01 5.69979e-02  FALSE  FALSE  FALSE  FALSE #>      3.67792e-01 1.89598e-01 7.02265e-02 1.18965e-02  FALSE  FALSE  FALSE  FALSE #>     ... (Use `print(x, rows = <nn>)` for more)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/authors.html","id":null,"dir":"","previous_headings":"","what":"Authors","title":"Authors and Citation","text":"Dong Xi. Author, maintainer. Ethan Brockmann. Author. Gilead Biostatistics. Copyright holder, funder. Atorus Research LLC. Copyright holder.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/authors.html","id":"citation","dir":"","previous_headings":"","what":"Citation","title":"Authors and Citation","text":"Xi, D.; Brockmann, E. (2023). graphicalMCP: Graph-based multiple comparison procedures. version 0.1.0. Gilead Sciences, Inc. Foster City, California. https://github.com/Gilead-BioStats/graphicalMCP Frank Bretz, Martin Posch, Ekkehard Glimm, Florian Klinglmueller, Willi Maurer, Kornelius Rohmeyer (2011), Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes parametric tests. Biometrical Journal 53 (6), pages 894--913, Wiley.","code":"@Manual{,   title = {{graphicalMCP}: Graph-based multiple comparison procedures.},   author = {Dong Xi and Ethan Brockmann},   organization = {Gilead Sciences, Inc},   note = {version 0.1.0},   year = {2023},   url = {https://github.com/Gilead-BioStats/graphicalMCP}, } @Article{,   title = {Graphical approaches for multiple comparison procedures using weighted Bonferroni, Simes or parametric tests},   author = {Frank Bretz and Martin Posch and Ekkehard Glimm and Florian Klinglmueller and Willi Maurer and Kornelius Rohmeyer},   journal = {Biometrical Journal},   year = {2011},   publisher = {Wiley},   volume = {53},   number = {6},   pages = {894--913},   url = {http://onlinelibrary.wiley.com/doi/10.1002/bimj.201000239/full}, }"},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"introduction","dir":"","previous_headings":"","what":"Introduction","title":"Graphical Approach for Multiple Comparison Procedures","text":"multiple comparison procedure (MCP) statistical analysis method allows assessing efficacy multiple endpoints, dependent , single clinical trial. Endpoints can different doses, treatment different conditions, combined superiority & non-inferiority testing, many others. key feature MCPs strongly control type-error rate level alpha. Bretz et al (2011), graphical method MCPs described, separates weighting clinical endpoints particular statistical test used assess endpoint. graphical approach can also helpful communicating study design clinical teams. package low-dependency implementation methods.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"installation","dir":"","previous_headings":"","what":"Installation","title":"Graphical Approach for Multiple Comparison Procedures","text":"graphicalMCP CRAN, install GitHub ","code":"# install.packages(\"pak\") pak::pak(\"Gilead-BioStats/graphicalMCP@dev\")"},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"initial-graph","dir":"","previous_headings":"Basic usage","what":"Initial graph","title":"Graphical Approach for Multiple Comparison Procedures","text":"base object graphicalMCP initial_graph, weighted, directed graph represented matrix transition (edge) weights, vector hypothesis (vertex) weights.","code":"library(graphicalMCP)  # A graphical multiple comparison procedure with two primary hypotheses (H1 # and H2) and two secondary hypotheses (H3 and H4) # See Figure 1 in Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, # W., & Rohmeyer, K. (2011). Graphical approaches for multiple comparison # procedures using weighted Bonferroni, Simes, or parametric tests. Biometrical # Journal, 53(6), 894-913. hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   c(0, 0, 1, 0),   c(0, 0, 0, 1),   c(0, 1, 0, 0),   c(1, 0, 0, 0) )  hyp_names <- c(\"H1\", \"H2\", \"H3\", \"H4\") example_graph <- graph_create(hypotheses, transitions, hyp_names)  plot(example_graph, layout = \"grid\")"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"update-graph","dir":"","previous_headings":"Basic usage","what":"Update graph","title":"Graphical Approach for Multiple Comparison Procedures","text":"Hypotheses can deleted MCP using graph_update(). Updated weights transitions calculated according weighting strategy Algorithm 1 Bretz et al (2011). distinguish deleting rejecting glossary.","code":"graph_update(example_graph, delete = c(FALSE, TRUE, TRUE, FALSE))"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"generate-weights","dir":"","previous_headings":"Basic usage","what":"Generate weights","title":"Graphical Approach for Multiple Comparison Procedures","text":"weights sub-graphs can calculated graph_generate_weights(). uses efficient code hood graph_update() order performant larger graphs. information closure can found [Link closure vignette].","code":"graph_generate_weights(example_graph)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"test-hypotheses","dir":"","previous_headings":"Basic usage","what":"Test hypotheses","title":"Graphical Approach for Multiple Comparison Procedures","text":"Bonferroni testing via shortcut method supported graphicalMCP. test can performed graph_test_shortcut(), generates report showing graph & test results. See insight hypothesis rejected setting verbose test_values flags. details shortcut testing can found [link shortcut vignette]. graph can also tested using Simes- parametric-based testing using closure test. types tests added time, combination tests can used groups hypotheses. Additional details closure testing can found [link closed test vignette].","code":"graph_test_shortcut(example_graph, p = c(.01, .03, .02, .01), alpha = .025) graph_test_closure(   example_graph,   p = c(.01, .03, .02, .01),   alpha = .025,   test_types = \"bonferroni\",   test_groups = list(1:4) )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"power-simulations","dir":"","previous_headings":"Basic usage","what":"Power simulations","title":"Graphical Approach for Multiple Comparison Procedures","text":"’s always obvious graph structure easy difficult reject hypothesis. One way understand better run power simulation, many p-values simulated, graph tested . valid test types & hypothesis groupings valid power simulations well. Power simulations discussed [shortcut testing vignette] [closure testing vignette].","code":"graph_calculate_power(   example_graph,   sim_n = 1e5,   power_marginal = c(.9, .9, .8, .8) )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"related-work","dir":"","previous_headings":"","what":"Related work","title":"Graphical Approach for Multiple Comparison Procedures","text":"methods originally implemented gMCP package, still available CRAN today. also lighter version gMCP implemented gMCPmini successor, gMCPLite. two contain subset original functionality, remove rJava dependency add plotting functionality based ggplot2. However, development ceased original package, hope re-implement methods general testing framework; fewer dependencies, particular shedding Java dependency; simpler, transparent S3 classes; improvements accuracy parametric Simes test methods. portion Simes testing also implemented lrstat package.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"citation","dir":"","previous_headings":"","what":"Citation","title":"Graphical Approach for Multiple Comparison Procedures","text":"","code":"citation(\"graphicalMCP\") #> To cite graphicalMCP in publications use: #>  #>   Xi, D.; Brockmann, E. (2023). graphicalMCP: Graph-based multiple #>   comparison procedures. version 0.1.0. Gilead Sciences, Inc. Foster #>   City, California. https://github.com/Gilead-BioStats/graphicalMCP #>  #>   Frank Bretz, Martin Posch, Ekkehard Glimm, Florian Klinglmueller, #>   Willi Maurer, Kornelius Rohmeyer (2011), Graphical approaches for #>   multiple comparison procedures using weighted Bonferroni, Simes or #>   parametric tests. Biometrical Journal 53 (6), pages 894--913, Wiley. #>  #> To see these entries in BibTeX format, use 'print(<citation>, #> bibtex=TRUE)', 'toBibtex(.)', or set #> 'options(citation.bibtex.max=999)'."},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/index.html","id":"glossary-of-terms","dir":"","previous_headings":"","what":"Glossary of terms","title":"Graphical Approach for Multiple Comparison Procedures","text":"package seeks accurate performant, course. beyond , much thought put readability code. Whether read user validating methods, developer extending package new uses, contributor helping enhance core functionality, hope code contained can serve educational document grow people’s understanding graphical approach multiple comparison procedures. end, several entities encountered world graphical MCPs define . used internal code package, inputs output one exported functions. believe definitions clarified improved, please contact package authors submit issue GitHub repository. set weights, adjusted according testing algorithm: Bonferroni: change Simes: Sum weights hypotheses smaller p-values Parametric: Multiply weights c-value, based joint distribution power reject null hypothesis full alpha. Closely related non-centrality parameter, mean null hypothesis underlying multivariate normal distribution: ncp = qnorm(1 - alpha) - qnorm(1 - marginal_power)","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjust_p.html","id":null,"dir":"Reference","previous_headings":"","what":"Calculate adjusted p-values — adjust_p_bonferroni","title":"Calculate adjusted p-values — adjust_p_bonferroni","text":"Calculate adjusted p-values","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjust_p.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Calculate adjusted p-values — adjust_p_bonferroni","text":"","code":"adjust_p_bonferroni(p, hypotheses)  adjust_p_parametric(p, hypotheses, test_corr = NULL)  adjust_p_simes(p, hypotheses)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjust_p.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Calculate adjusted p-values — adjust_p_bonferroni","text":"p numeric vector p-values hypotheses numeric vector hypothesis weights test_corr (Optional) numeric matrix correlations hypotheses' test statistics","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjust_p.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Calculate adjusted p-values — adjust_p_bonferroni","text":"single adjusted p-value given group","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjust_p.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Calculate adjusted p-values — adjust_p_bonferroni","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjust_p.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Calculate adjusted p-values — adjust_p_bonferroni","text":"","code":"set.seed(22723)  w <- c(\"H1\" = .75, \"H2\" = .25, \"H3\" = 0) p <- c(\"H1\" = .019, \"H2\" = .025, \"H3\" = .05)  graphicalMCP:::adjust_p_bonferroni(p, w) #> [1] 0.02533333 graphicalMCP:::adjust_p_simes(p, w) #> [1] 0.025  corr1 <- diag(3) corr2 <- corr1 corr2[lower.tri(corr2)] <- corr2[upper.tri(corr2)] <- runif(3, -1, 1)  # No correlation graphicalMCP:::adjust_p_parametric(p, w, corr1) #> [1] 0.025213  # Uniform random pairwise correlations graphicalMCP:::adjust_p_parametric(p, w, corr2) #> [1] 0.02196006"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjusted-weights.html","id":null,"dir":"Reference","previous_headings":"","what":"Calculate updated hypothesis weights for the closure of a graph — adjust_weights_parametric","title":"Calculate updated hypothesis weights for the closure of a graph — adjust_weights_parametric","text":"weights created graph_generate_weights() work immediately Bonferroni testing, parametric Simes testing require additional calculations. adjust_weights_*() functions apply parametric Simes weight increases get updated weights testing. also subset weights columns appropriate groups","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjusted-weights.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Calculate updated hypothesis weights for the closure of a graph — adjust_weights_parametric","text":"","code":"adjust_weights_parametric(   matrix_weights,   matrix_intersections,   test_corr,   alpha,   test_groups )  adjust_weights_simes(matrix_weights, p, groups)  c_value_function(x, hypotheses, test_corr, alpha)  solve_c_parametric(hypotheses, test_corr, alpha)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjusted-weights.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Calculate updated hypothesis weights for the closure of a graph — adjust_weights_parametric","text":"matrix_weights second half columns graph_generate_weights() output, indicating weights intersection matrix_intersections first half columns graph_generate_weights() output, indicating hypotheses contained intersection test_corr numeric matrix correlations hypotheses' test statistics alpha numeric scalar specifying global significance level testing test_groups list numeric vectors specifying hypotheses test together p numeric vector p-values x root solve stats::uniroot() hypotheses numeric vector hypothesis weights","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjusted-weights.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Calculate updated hypothesis weights for the closure of a graph — adjust_weights_parametric","text":"Outputs: adjust_weights_*(), matrix shape weighting_strategy, weights adjusted according specified adjustment method c_value_function(), \\(c_{J_h}\\) value given group, according Formula 6 Xi et al. (2017).","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjusted-weights.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Calculate updated hypothesis weights for the closure of a graph — adjust_weights_parametric","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/adjusted-weights.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Calculate updated hypothesis weights for the closure of a graph — adjust_weights_parametric","text":"","code":"p <- 1:6 / 200  g <- bonferroni_holm(6) gw_large <- graph_generate_weights(g)  gw_0 <- gw_large[, 7:12] gw <- ifelse(gw_large[, 1:6], gw_0, NA)  graphicalMCP:::adjust_weights_parametric(   gw_0,   gw_large[, 1:6],   diag(6),   .05,   list(1:3) ) #>           H1        H2        H3 #> 1  0.1680752 0.1680752 0.1680752 #> 2  0.2020340 0.2020340 0.2020340 #> 3  0.2020340 0.2020340 0.2020340 #> 4  0.2531918 0.2531918 0.2531918 #> 5  0.2020340 0.2020340 0.2020340 #> 6  0.2531918 0.2531918 0.2531918 #> 7  0.2531918 0.2531918 0.2531918 #> 8  0.3390485 0.3390485 0.3390485 #> 9  0.2010082 0.2010082 0.0000000 #> 10 0.2515823 0.2515823 0.0000000 #> 11 0.2515823 0.2515823 0.0000000 #> 12 0.3361584 0.3361584 0.0000000 #> 13 0.2515823 0.2515823 0.0000000 #> 14 0.3361584 0.3361584 0.0000000 #> 15 0.3361584 0.3361584 0.0000000 #> 16 0.5064113 0.5064113 0.0000000 #> 17 0.2010082 0.0000000 0.2010082 #> 18 0.2515823 0.0000000 0.2515823 #> 19 0.2515823 0.0000000 0.2515823 #> 20 0.3361584 0.0000000 0.3361584 #> 21 0.2515823 0.0000000 0.2515823 #> 22 0.3361584 0.0000000 0.3361584 #> 23 0.3361584 0.0000000 0.3361584 #> 24 0.5064113 0.0000000 0.5064113 #> 25 0.2500000 0.0000000 0.0000000 #> 26 0.3333333 0.0000000 0.0000000 #> 27 0.3333333 0.0000000 0.0000000 #> 28 0.5000000 0.0000000 0.0000000 #> 29 0.3333333 0.0000000 0.0000000 #> 30 0.5000000 0.0000000 0.0000000 #> 31 0.5000000 0.0000000 0.0000000 #> 32 1.0000000 0.0000000 0.0000000 #> 33 0.0000000 0.2010082 0.2010082 #> 34 0.0000000 0.2515823 0.2515823 #> 35 0.0000000 0.2515823 0.2515823 #> 36 0.0000000 0.3361584 0.3361584 #> 37 0.0000000 0.2515823 0.2515823 #> 38 0.0000000 0.3361584 0.3361584 #> 39 0.0000000 0.3361584 0.3361584 #> 40 0.0000000 0.5064113 0.5064113 #> 41 0.0000000 0.2500000 0.0000000 #> 42 0.0000000 0.3333333 0.0000000 #> 43 0.0000000 0.3333333 0.0000000 #> 44 0.0000000 0.5000000 0.0000000 #> 45 0.0000000 0.3333333 0.0000000 #> 46 0.0000000 0.5000000 0.0000000 #> 47 0.0000000 0.5000000 0.0000000 #> 48 0.0000000 1.0000000 0.0000000 #> 49 0.0000000 0.0000000 0.2500000 #> 50 0.0000000 0.0000000 0.3333333 #> 51 0.0000000 0.0000000 0.3333333 #> 52 0.0000000 0.0000000 0.5000000 #> 53 0.0000000 0.0000000 0.3333333 #> 54 0.0000000 0.0000000 0.5000000 #> 55 0.0000000 0.0000000 0.5000000 #> 56 0.0000000 0.0000000 1.0000000 #> 57 0.0000000 0.0000000 0.0000000 #> 58 0.0000000 0.0000000 0.0000000 #> 59 0.0000000 0.0000000 0.0000000 #> 60 0.0000000 0.0000000 0.0000000 #> 61 0.0000000 0.0000000 0.0000000 #> 62 0.0000000 0.0000000 0.0000000 #> 63 0.0000000 0.0000000 0.0000000  graphicalMCP:::adjust_weights_simes(gw_0, p, list(4:6)) #>           H4        H5        H6 #> 1  0.1666667 0.3333333 0.5000000 #> 2  0.2000000 0.4000000 0.4000000 #> 3  0.2000000 0.2000000 0.4000000 #> 4  0.2500000 0.2500000 0.2500000 #> 5  0.0000000 0.2000000 0.4000000 #> 6  0.0000000 0.2500000 0.2500000 #> 7  0.0000000 0.0000000 0.2500000 #> 8  0.0000000 0.0000000 0.0000000 #> 9  0.2000000 0.4000000 0.6000000 #> 10 0.2500000 0.5000000 0.5000000 #> 11 0.2500000 0.2500000 0.5000000 #> 12 0.3333333 0.3333333 0.3333333 #> 13 0.0000000 0.2500000 0.5000000 #> 14 0.0000000 0.3333333 0.3333333 #> 15 0.0000000 0.0000000 0.3333333 #> 16 0.0000000 0.0000000 0.0000000 #> 17 0.2000000 0.4000000 0.6000000 #> 18 0.2500000 0.5000000 0.5000000 #> 19 0.2500000 0.2500000 0.5000000 #> 20 0.3333333 0.3333333 0.3333333 #> 21 0.0000000 0.2500000 0.5000000 #> 22 0.0000000 0.3333333 0.3333333 #> 23 0.0000000 0.0000000 0.3333333 #> 24 0.0000000 0.0000000 0.0000000 #> 25 0.2500000 0.5000000 0.7500000 #> 26 0.3333333 0.6666667 0.6666667 #> 27 0.3333333 0.3333333 0.6666667 #> 28 0.5000000 0.5000000 0.5000000 #> 29 0.0000000 0.3333333 0.6666667 #> 30 0.0000000 0.5000000 0.5000000 #> 31 0.0000000 0.0000000 0.5000000 #> 32 0.0000000 0.0000000 0.0000000 #> 33 0.2000000 0.4000000 0.6000000 #> 34 0.2500000 0.5000000 0.5000000 #> 35 0.2500000 0.2500000 0.5000000 #> 36 0.3333333 0.3333333 0.3333333 #> 37 0.0000000 0.2500000 0.5000000 #> 38 0.0000000 0.3333333 0.3333333 #> 39 0.0000000 0.0000000 0.3333333 #> 40 0.0000000 0.0000000 0.0000000 #> 41 0.2500000 0.5000000 0.7500000 #> 42 0.3333333 0.6666667 0.6666667 #> 43 0.3333333 0.3333333 0.6666667 #> 44 0.5000000 0.5000000 0.5000000 #> 45 0.0000000 0.3333333 0.6666667 #> 46 0.0000000 0.5000000 0.5000000 #> 47 0.0000000 0.0000000 0.5000000 #> 48 0.0000000 0.0000000 0.0000000 #> 49 0.2500000 0.5000000 0.7500000 #> 50 0.3333333 0.6666667 0.6666667 #> 51 0.3333333 0.3333333 0.6666667 #> 52 0.5000000 0.5000000 0.5000000 #> 53 0.0000000 0.3333333 0.6666667 #> 54 0.0000000 0.5000000 0.5000000 #> 55 0.0000000 0.0000000 0.5000000 #> 56 0.0000000 0.0000000 0.0000000 #> 57 0.3333333 0.6666667 1.0000000 #> 58 0.5000000 1.0000000 1.0000000 #> 59 0.5000000 0.5000000 1.0000000 #> 60 1.0000000 1.0000000 1.0000000 #> 61 0.0000000 0.5000000 1.0000000 #> 62 0.0000000 1.0000000 1.0000000 #> 63 0.0000000 0.0000000 1.0000000"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/as-graph.html","id":null,"dir":"Reference","previous_headings":"","what":"Convert between gMCP and graphicalMCP graph classes — as_initial_graph","title":"Convert between gMCP and graphicalMCP graph classes — as_initial_graph","text":"Convert gMCP graphicalMCP graph classes","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/as-graph.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Convert between gMCP and graphicalMCP graph classes — as_initial_graph","text":"","code":"as_initial_graph(graph)  # S3 method for graphMCP as_initial_graph(graph)  # S3 method for igraph as_initial_graph(graph)  as_graphMCP(graph)  # S3 method for initial_graph as_graphMCP(graph)  as_igraph(graph)  # S3 method for initial_graph as_igraph(graph)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/as-graph.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Convert between gMCP and graphicalMCP graph classes — as_initial_graph","text":"graph initial_graph object graphicalMCP, graphMCP object gMCP, igraph object igraph, depending direction converting","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/as-graph.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Convert between gMCP and graphicalMCP graph classes — as_initial_graph","text":"as_graphMCP(), gMCP graph object, as_igraph(), igraph object, as_initial_graph(), graphicalMCP graph object","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/as-graph.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Convert between gMCP and graphicalMCP graph classes — as_initial_graph","text":"","code":"g1 <- random_graph(5)  if (requireNamespace(\"gMCP\", quietly = TRUE)) {   g2 <- as_graphMCP(g1)    all.equal(g1, as_initial_graph(g2)) } #> [1] TRUE  if (requireNamespace(\"igraph\", quietly = TRUE)) {   g3 <- as_igraph(g1)    all.equal(g1, as_initial_graph(g3)) } #> igraph method #>  [1] \"names for target but not for current\"                              #>  [2] \"Attributes: < Component “class”: 1 string mismatch >\"              #>  [3] \"Length mismatch: comparison on first 2 components\"                 #>  [4] \"Component 1: names for target but not for current\"                 #>  [5] \"Component 1: Numeric: lengths (5, 1) differ\"                       #>  [6] \"Component 2: Modes: numeric, logical\"                              #>  [7] \"Component 2: Lengths: 25, 1\"                                       #>  [8] \"Component 2: Attributes: < Modes: list, NULL >\"                    #>  [9] \"Component 2: Attributes: < Lengths: 2, 0 >\"                        #> [10] \"Component 2: Attributes: < names for target but not for current >\" #> [11] \"Component 2: Attributes: < current is not list-like >\"             #> [12] \"Component 2: target is matrix, current is logical\""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/calc-test_values.html","id":null,"dir":"Reference","previous_headings":"","what":"Test hypotheses with the adjusted significance method — test_values_bonferroni","title":"Test hypotheses with the adjusted significance method — test_values_bonferroni","text":"Test hypotheses adjusted significance method","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/calc-test_values.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Test hypotheses with the adjusted significance method — test_values_bonferroni","text":"","code":"test_values_bonferroni(p, hypotheses, alpha, intersection = NA)  test_values_parametric(p, hypotheses, alpha, intersection = NA, test_corr)  test_values_simes(p, hypotheses, alpha, intersection = NA)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/calc-test_values.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Test hypotheses with the adjusted significance method — test_values_bonferroni","text":"p numeric vector p-values hypotheses numeric vector hypothesis hypotheses alpha numeric scalar specifying global significance level testing intersection numeric scalar used track intersection test values test_corr numeric matrix correlations hypotheses' test statistics","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/calc-test_values.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Test hypotheses with the adjusted significance method — test_values_bonferroni","text":"data frame columns specifying values used calculate hypothesis test","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/calc-test_values.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Test hypotheses with the adjusted significance method — test_values_bonferroni","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/calc-test_values.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Test hypotheses with the adjusted significance method — test_values_bonferroni","text":"","code":"w <- c(H1 = .5, H2 = .5, H3 = 0, H4 = 0)  p <- c(.024, .01, .026, .027)  graphicalMCP:::test_values_bonferroni(p, w, .05) #>    Intersection Hypothesis       Test     p <= c_value * Weight * Alpha #> H1           NA         H1 bonferroni 0.024 <=              0.5 *  0.05 #> H2           NA         H2 bonferroni 0.010 <=              0.5 *  0.05 #> H3           NA         H3 bonferroni 0.026 <=              0.0 *  0.05 #> H4           NA         H4 bonferroni 0.027 <=              0.0 *  0.05 #>    Inequality_holds #> H1             TRUE #> H2             TRUE #> H3            FALSE #> H4            FALSE graphicalMCP:::test_values_parametric(p, w, .05, test_corr = diag(4)) #>    Intersection Hypothesis       Test     p <=  c_value * Weight * Alpha #> H1           NA         H1 parametric 0.024 <= 1.012823 *    0.5 *  0.05 #> H2           NA         H2 parametric 0.010 <= 1.012823 *    0.5 *  0.05 #> H3           NA         H3 parametric 0.026 <= 1.012823 *    0.0 *  0.05 #> H4           NA         H4 parametric 0.027 <= 1.012823 *    0.0 *  0.05 #>    Inequality_holds #> H1             TRUE #> H2             TRUE #> H3            FALSE #> H4            FALSE graphicalMCP:::test_values_simes(p, w, .05) #>   Intersection Hypothesis  Test     p <= c_value * Weight * Alpha #> 1           NA         H1 simes 0.024 <=              1.0 *  0.05 #> 2           NA         H2 simes 0.010 <=              0.5 *  0.05 #> 3           NA         H3 simes 0.026 <=              1.0 *  0.05 #> 4           NA         H4 simes 0.027 <=              1.0 *  0.05 #>   Inequality_holds #> 1             TRUE #> 2             TRUE #> 3             TRUE #> 4             TRUE"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/example-graphs.html","id":null,"dir":"Reference","previous_headings":"","what":"Example graphs from a selection of papers — bonferroni_holm","title":"Example graphs from a selection of papers — bonferroni_holm","text":"Example graphs selection papers","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/example-graphs.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Example graphs from a selection of papers — bonferroni_holm","text":"","code":"bonferroni_holm(num_hyps, hyp_names = NULL)  huque_alosh_bhore_2011(hyp_names = NULL)  wiens_dmitrienko_2005(hyp_names = NULL)  fixed_sequence(num_hyps = 3, hyp_names = NULL)  fallback(hypotheses = c(1, 0, 0), hyp_names = NULL)  simple_successive_1(hyp_names = NULL)  simple_successive_2(hyp_names = NULL)  random_graph(num_hyps, hyp_names = NULL)  complex_example_1(hyp_names = NULL)  complex_example_2(hyp_names = NULL)  complex_example_2(hyp_names = NULL)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/example-graphs.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Example graphs from a selection of papers — bonferroni_holm","text":"num_hyps Number vertices graph hyp_names (Optional) character vector hypothesis names. names specified, hypotheses named sequentially H1, H2, ... hypotheses Hypothesis weights fallback procedure","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/example-graphs.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Example graphs from a selection of papers — bonferroni_holm","text":"S3 object returned graph_create()","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/example-graphs.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Example graphs from a selection of papers — bonferroni_holm","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/example-graphs.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Example graphs from a selection of papers — bonferroni_holm","text":"","code":"bonferroni_holm(3, hyp_names = paste(\"dose\", letters[1:3])) #> Initial graph #>  #> --- Hypothesis weights --- #> dose a: 0.3333 #> dose b: 0.3333 #> dose c: 0.3333 #>  #> --- Transition weights --- #>         dose a dose b dose c #>  dose a    0.0    0.5    0.5 #>  dose b    0.5    0.0    0.5 #>  dose c    0.5    0.5    0.0"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":null,"dir":"Reference","previous_headings":"","what":"Obtain hypothesis rejection probabilities — graph_calculate_power","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"often difficult tell likely given hypothesis rejected. power simulations useful. set distribution parameters, many p-values generated, graph tested one. testing strategy can used. probabilities calculated hypothesis rejected, well additional probabilities expected rejections probability rejecting hypothesis","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"","code":"graph_calculate_power(   graph,   alpha = 0.025,   power_marginal = rep(alpha, length(graph$hypotheses)),   test_groups = list(seq_along(graph$hypotheses)),   test_types = c(\"bonferroni\"),   test_corr = rep(list(NA), length(test_types)),   sim_n = 100,   sim_corr = diag(length(graph$hypotheses)),   sim_success = NULL,   sim_seed = NULL,   force_closure = FALSE,   verbose = FALSE )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"graph initial graph returned graph_create() alpha numeric scalar specifying global significance level testing power_marginal numeric vector mean values use simulating p-values. Exactly one mean per hypothesis needed, p-values sampled multivariate normal distribution. See Details test_groups list numeric vectors specifying hypotheses test together test_types character vector tests apply given groups test_corr Optional test_types parametric. numeric matrix correlations hypotheses' test statistics sim_n integer scalar specifying many simulations run sim_corr numeric matrix correlations hypotheses used sample multivariate normal distribution generate p-values sim_success list user-defined functions apply power results. Functions must take one simulation's logical vector results input, return length-one logical vector. instance, \"success\" means rejecting hypotheses 1 2, use sim_success = list(\"1 2\" = function(x) x[1] && x[2]). list named, function body used name. Lambda functions also work starting R 4.1, e.g. sim_success = list(\\(x) x[3] || x[4]) sim_seed (Optional) Random seed set simulating p-values. Set use consistent set p simulations across power calculations force_closure logical scalar used determine whether full closure test used Bonferroni testing. Ignored tests non-Bonferroni verbose logical scalar specifying whether full matrix simulations test results included output ","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"list five elements power_local - rejection proportion hypothesis individually power_expected - average number hypotheses rejected single simulation power_at_least_1 - proportion simulations reject hypothesis power_all - proportion simulations reject hypotheses power_success - proportion simulations reject hypotheses specified sim_success","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"parameters normal distribution set power_marginal (means) sim_corr (correlation test statistics). mean hypothesis set marginal power $$d_i=P_{\\xi_i}(p_i\\leq\\alpha)$$ \\(\\xi_i\\) non-centrality parameter. correlation test statistics induced study design.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":"success","dir":"Reference","previous_headings":"","what":"Success","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"Success mean something different trial, lot flexibility sim_success parameter. However, flexibility means little validation inputs. user make sure function(s) passed mean think. implementation perspective, function applied row-wise matrix test results simulation, resulting sim_n length vector. mean vector returned \"Probability success\"","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_calculate_power.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Obtain hypothesis rejection probabilities — graph_calculate_power","text":"","code":"par_gate <- simple_successive_1()  # The default is to test all hypotheses with: Bonferroni testing at alpha # level .025, 0 mean under the alternative, and 0 correlation between # hypotheses under the alternative # The default of 100 simulations will usually need to be increased graph_calculate_power(par_gate, sim_n = 1e5) #>  #> Test parameters ($inputs) ------------------------------------------------------ #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1  0  0  1  0 #>   H2  0  0  0  1 #>   H3  0  1  0  0 #>   H4  1  0  0  0 #>  #>   Alpha = 0.025 #>  #>   Test types #>   bonferroni: (H1, H2, H3, H4) #>  #> Simulation parameters ($inputs) ------------------------------------------------ #>   Testing 100,000 simulations with multivariate normal params: #>  #>                      H1    H2    H3    H4 #>   Marginal power: 0.025 0.025 0.025 0.025 #>  #>   Correlation:    H1 H2 H3 H4 #>                H1  1  0  0  0 #>                H2  0  1  0  0 #>                H3  0  0  1  0 #>                H4  0  0  0  1 #>  #> Power calculation ($power) ----------------------------------------------------- #>                                    H1      H2      H3      H4 #>                  Local power: 0.01201 0.01166 0.00016 0.00019 #>  #>   Expected no. of rejections: 0.02402 #>    Power to reject 1 or more: 0.02349 #>          Power to reject all: 0 #>   # But any test group/type combination that works for [graph_test_closure()] # can be used graph_calculate_power(   par_gate,   alpha = .025,   test_groups = list(1:2, 3:4),   test_types = c(\"s\", \"p\"),   test_corr = list(NA, diag(2)),   sim_n = 1e5,   sim_success = list(     function(.) .[1] || .[2],     function(.) .[1] && .[2]   ) ) #>  #> Test parameters ($inputs) ------------------------------------------------------ #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1  0  0  1  0 #>   H2  0  0  0  1 #>   H3  0  1  0  0 #>   H4  1  0  0  0 #>  #>   Alpha = 0.025 #>  #>   Parametric testing correlation:    H3 H4 #>                                   H3  1  0 #>                                   H4  0  1 #>  #>   Test types #>        simes: (H1, H2) #>   parametric: (H3, H4) #>  #> Simulation parameters ($inputs) ------------------------------------------------ #>   Testing 100,000 simulations with multivariate normal params: #>  #>                      H1    H2    H3    H4 #>   Marginal power: 0.025 0.025 0.025 0.025 #>  #>   Correlation:    H1 H2 H3 H4 #>                H1  1  0  0  0 #>                H2  0  1  0  0 #>                H3  0  0  1  0 #>                H4  0  0  0  1 #>  #> Power calculation ($power) ----------------------------------------------------- #>                                    H1      H2      H3      H4 #>                  Local power: 0.01203 0.01215 0.00018 0.00021 #>  #>   Expected no. of rejections: 0.02457 #>    Power to reject 1 or more: 0.02403 #>          Power to reject all: 0 #>  #>              Power to reject:    .[1] || .[2]   .[1] && .[2] #>                                       0.02403        0.00015 #>"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_create.html","id":null,"dir":"Reference","previous_headings":"","what":"Define a graph representing a multiple comparison procedure — graph_create","title":"Define a graph representing a multiple comparison procedure — graph_create","text":"multiple comparison procedure graph can represented 1) vector initial hypothesis weights, 2) matrix initial transition weights. function creates initial graph object using vector matrix. also validates inputs make sure combine form valid graph.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_create.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Define a graph representing a multiple comparison procedure — graph_create","text":"","code":"graph_create(hypotheses, transitions, hyp_names = NULL)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_create.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Define a graph representing a multiple comparison procedure — graph_create","text":"hypotheses numeric vector hypothesis weights initial graphical multiple comparison procedure. Must vector values 0 & 1 (inclusive). length match row column lengths transitions. sum hypothesis weights exceed 1 transitions numeric matrix transition weights hypotheses initial graphical multiple comparison procedure. Must square matrix values 0 & 1 (inclusive). row column lengths match length hypotheses. row (Transition weights leaving hypothesis) can sum 1. diagonal (Transition weights hypothesis ) must 0s hyp_names (Optional) character vector hypothesis names. provided, names hypotheses transitions used. names specified, hypotheses named sequentially H1, H2, ...","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_create.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Define a graph representing a multiple comparison procedure — graph_create","text":"S3 object class initial_graph. underlying structure list elements hypotheses transitions","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_create.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Define a graph representing a multiple comparison procedure — graph_create","text":"validations performed : Weights numeric Length hypotheses dimensions transitions Explicit hyp_names override names hypotheses transitions Hypothesis weights must non-negative sum 1 Transition weights matrix: Diagonal must 1 Rows must sum 1 Values must non-negative","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_create.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Define a graph representing a multiple comparison procedure — graph_create","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_create.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Define a graph representing a multiple comparison procedure — graph_create","text":"","code":"# A graphical multiple comparison procedure with two primary hypotheses (H1 # and H2) and two secondary hypotheses (H3 and H4) # See Figure 1 in Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, # W., & Rohmeyer, K. (2011). Graphical approaches for multiple comparison # procedures using weighted Bonferroni, Simes, or parametric tests. # Biometrical Journal, 53(6), 894-913. hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   c(0, 0, 1, 0),   c(0, 0, 0, 1),   c(0, 1, 0, 0),   c(1, 0, 0, 0) ) hyp_names <- c(\"H1\", \"H2\", \"H3\", \"H4\") g <- graph_create(hypotheses, transitions, hyp_names) g #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  1  0 #>  H2  0  0  0  1 #>  H3  0  1  0  0 #>  H4  1  0  0  0  # Explicit names override names in `hypotheses` (with a warning) hypotheses <- c(h1 = 0.5, h2 = 0.5, h3 = 0, h4 = 0) transitions <- rbind(   c(0, 0, 1, 0),   c(0, 0, 0, 1),   c(0, 1, 0, 0),   c(1, 0, 0, 0) )  g <- graph_create(hypotheses, transitions, hyp_names) #> Warning: Hypothesis names specified - overriding names in #>                     `hypotheses` and `transitions` g #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  1  0 #>  H2  0  0  0  1 #>  H3  0  1  0  0 #>  H4  1  0  0  0  # Explicit names override names in `transitions` (with a warning) hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   h1 = c(0, 0, 1, 0),   h2 = c(0, 0, 0, 1),   h3 = c(0, 1, 0, 0),   h4 = c(1, 0, 0, 0) )  g <- graph_create(hypotheses, transitions, hyp_names) #> Warning: Hypothesis names specified - overriding names in #>                     `hypotheses` and `transitions` g #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  1  0 #>  H2  0  0  0  1 #>  H3  0  1  0  0 #>  H4  1  0  0  0  # Use names in `hypotheses` hypotheses <- c(H1 = 0.5, H2 = 0.5, H3 = 0, H4 = 0) transitions <- rbind(   c(0, 0, 1, 0),   c(0, 0, 0, 1),   c(0, 1, 0, 0),   c(1, 0, 0, 0) ) g <- graph_create(hypotheses, transitions) g #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  1  0 #>  H2  0  0  0  1 #>  H3  0  1  0  0 #>  H4  1  0  0  0  # Use names in `transitions` hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   H1 = c(0, 0, 1, 0),   H2 = c(0, 0, 0, 1),   H3 = c(0, 1, 0, 0),   H4 = c(1, 0, 0, 0) ) g <- graph_create(hypotheses, transitions) g #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  1  0 #>  H2  0  0  0  1 #>  H3  0  1  0  0 #>  H4  1  0  0  0  # When names are not specified, hypotheses are numbered sequentially as # H1, H2, ... hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   c(0, 0, 1, 0),   c(0, 0, 0, 1),   c(0, 1, 0, 0),   c(1, 0, 0, 0) ) g <- graph_create(hypotheses, transitions) g #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  1  0 #>  H2  0  0  0  1 #>  H3  0  1  0  0 #>  H4  1  0  0  0"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_generate_weights.html","id":null,"dir":"Reference","previous_headings":"","what":"Calculate weights for the closure of a graph — graph_generate_weights","title":"Calculate weights for the closure of a graph — graph_generate_weights","text":"closure graph set sub-graphs, intersections hypotheses, graph. Weights sub-graph calculated using weighting strategy defined Bretz et al (2011).","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_generate_weights.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Calculate weights for the closure of a graph — graph_generate_weights","text":"","code":"graph_generate_weights(graph)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_generate_weights.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Calculate weights for the closure of a graph — graph_generate_weights","text":"graph initial graph returned graph_create()","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_generate_weights.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Calculate weights for the closure of a graph — graph_generate_weights","text":"numeric matrix intersection hypothesis weights. row corresponds single intersection hypothesis. first half columns indicate hypotheses included given intersection hypothesis, second half columns weights","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_generate_weights.html","id":"performance","dir":"Reference","previous_headings":"","what":"Performance","title":"Calculate weights for the closure of a graph — graph_generate_weights","text":"Much thought given performance code, memory time usage can grow quickly graph size grows. systems used testing, size 10 graph median run time 20-60 ms. Run time increases rate O(2 ^ n), e.g. size 5 graph takes approximately twice long run size 4 graph. See vignette(\"generate-weights-performance\") detailed analysis explanation","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_generate_weights.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Calculate weights for the closure of a graph — graph_generate_weights","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_generate_weights.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Calculate weights for the closure of a graph — graph_generate_weights","text":"","code":"par_gate <- graph_create(   hypotheses = c(.5, .5, 0, 0),   transitions = rbind(     c(0, 0, 1, 0),     c(0, 0, 0, 1),     c(0, 1, 0, 0),     c(1, 0, 0, 0)   ) )  graph_generate_weights(par_gate) #>    H1 H2 H3 H4  H1  H2  H3  H4 #> 1   1  1  1  1 0.5 0.5 0.0 0.0 #> 2   1  1  1  0 0.5 0.5 0.0 0.0 #> 3   1  1  0  1 0.5 0.5 0.0 0.0 #> 4   1  1  0  0 0.5 0.5 0.0 0.0 #> 5   1  0  1  1 0.5 0.0 0.0 0.5 #> 6   1  0  1  0 1.0 0.0 0.0 0.0 #> 7   1  0  0  1 0.5 0.0 0.0 0.5 #> 8   1  0  0  0 1.0 0.0 0.0 0.0 #> 9   0  1  1  1 0.0 0.5 0.5 0.0 #> 10  0  1  1  0 0.0 0.5 0.5 0.0 #> 11  0  1  0  1 0.0 1.0 0.0 0.0 #> 12  0  1  0  0 0.0 1.0 0.0 0.0 #> 13  0  0  1  1 0.0 0.0 0.5 0.5 #> 14  0  0  1  0 0.0 0.0 1.0 0.0 #> 15  0  0  0  1 0.0 0.0 0.0 1.0"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_rejection_orderings.html","id":null,"dir":"Reference","previous_headings":"","what":"Find alternate rejection orderings for shortcut testing — graph_rejection_orderings","title":"Find alternate rejection orderings for shortcut testing — graph_rejection_orderings","text":"using graph_test_shortcut(), may multiple hypotheses can rejected step. specific hypothesis chosen decided based minimum adjusted p-value. function shows every order deletes hypotheses, hypothesis still valid deletion step chosen.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_rejection_orderings.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Find alternate rejection orderings for shortcut testing — graph_rejection_orderings","text":"","code":"graph_rejection_orderings(shortcut_test_result)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_rejection_orderings.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Find alternate rejection orderings for shortcut testing — graph_rejection_orderings","text":"shortcut_test_result graph_report object returned graph_test_shortcut()","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_rejection_orderings.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Find alternate rejection orderings for shortcut testing — graph_rejection_orderings","text":"modified graph_report object containing valid orderings deleting significant hypotheses","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_rejection_orderings.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Find alternate rejection orderings for shortcut testing — graph_rejection_orderings","text":"","code":"graph <- simple_successive_2()  short_res <- graph_test_shortcut(graph, c(.018, .01, .03, .004))  # Reject H1, H2, and H4 short_res$outputs$rejected #>    H1    H2    H3    H4  #>  TRUE  TRUE FALSE  TRUE   # But these cannot be rejected in any order - For instance, H4 has 0 weight # in the initial graph and cannot be rejected first graph_rejection_orderings(short_res) #>  #> Test parameters ($inputs) ------------------------------------------------------ #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>       H1  H2  H3  H4 #>   H1 0.0 0.5 0.5 0.0 #>   H2 0.5 0.0 0.0 0.5 #>   H3 0.0 1.0 0.0 0.0 #>   H4 1.0 0.0 0.0 0.0 #>  #>   Alpha = 0.025 #>  #>                           H1    H2    H3    H4 #>   Unadjusted p-values: 0.018 0.010 0.030 0.004 #>  #>   Test types #>   bonferroni: (H1, H2, H3, H4) #>  #> Test summary ($outputs) -------------------------------------------------------- #>   Hypothesis Adj. P-value Reject #>           H1         0.02   TRUE #>           H2         0.02   TRUE #>           H3         0.03  FALSE #>           H4         0.02   TRUE #>  #>   Final updated graph after removing rejected hypotheses #>  #>   --- Hypothesis weights --- #>   H1: NA #>   H2: NA #>   H3:  1 #>   H4: NA #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1 NA NA NA NA #>   H2 NA NA NA NA #>   H3 NA NA  0 NA #>   H4 NA NA NA NA #>  #> Alternate rejection orderings ($valid_rejection_orderings) --------------------- #> H2 H4 H1  #>  2  4  1  #>  #> H2 H1 H4  #>  2  1  4  #>   # Finally, intermediate graphs can be obtained by putting one of the # orderings outputs into `graph_update()` graph_update(graph, delete = c(2, 1, 4)) #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>      H1  H2  H3  H4 #>  H1 0.0 0.5 0.5 0.0 #>  H2 0.5 0.0 0.0 0.5 #>  H3 0.0 1.0 0.0 0.0 #>  H4 1.0 0.0 0.0 0.0 #>  #> Updated graph after deleting hypotheses 2, 1, 4 #>  #> --- Hypothesis weights --- #> H1: NA #> H2: NA #> H3:  1 #> H4: NA #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1 NA NA NA NA #>  H2 NA NA NA NA #>  H3 NA NA  0 NA #>  H4 NA NA NA NA #>  #> Deletion sequence ($intermediate_graphs) --------------------------------------- #>  #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>       H1  H2  H3  H4 #>   H1 0.0 0.5 0.5 0.0 #>   H2 0.5 0.0 0.0 0.5 #>   H3 0.0 1.0 0.0 0.0 #>   H4 1.0 0.0 0.0 0.0 #>  #>     Step 1: Updated graph after removing hypothesis 2 #>  #>     --- Hypothesis weights --- #>     H1: 0.75 #>     H2:   NA #>     H3: 0.00 #>     H4: 0.25 #>  #>     --- Transition weights --- #>              H1       H2       H3       H4 #>     H1 0.000000       NA 0.666667 0.333333 #>     H2       NA       NA       NA       NA #>     H3 0.500000       NA 0.000000 0.500000 #>     H4 1.000000       NA 0.000000 0.000000 #>  #>       Step 2: Updated graph after removing hypotheses 2, 1 #>  #>       --- Hypothesis weights --- #>       H1:  NA #>       H2:  NA #>       H3: 0.5 #>       H4: 0.5 #>  #>       --- Transition weights --- #>          H1 H2 H3 H4 #>       H1 NA NA NA NA #>       H2 NA NA NA NA #>       H3 NA NA  0  1 #>       H4 NA NA  1  0 #>  #>         Step 3: Updated graph after removing hypotheses 2, 1, 4 #>  #>         --- Hypothesis weights --- #>         H1: NA #>         H2: NA #>         H3:  1 #>         H4: NA #>  #>         --- Transition weights --- #>            H1 H2 H3 H4 #>         H1 NA NA NA NA #>         H2 NA NA NA NA #>         H3 NA NA  0 NA #>         H4 NA NA NA NA #>  #>   Final updated graph after removing deleted hypotheses #>  #>   --- Hypothesis weights --- #>   H1: NA #>   H2: NA #>   H3:  1 #>   H4: NA #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1 NA NA NA NA #>   H2 NA NA NA NA #>   H3 NA NA  0 NA #>   H4 NA NA NA NA #>"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_update.html","id":null,"dir":"Reference","previous_headings":"","what":"Delete hypotheses from a graph — graph_update","title":"Delete hypotheses from a graph — graph_update","text":"always obvious graph's weights look like deleting one hypotheses. graph_generate_weights() calculates sub-graphs' hypothesis weights, graph_update() gives detailed view single set deletions, including transition weights well.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_update.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Delete hypotheses from a graph — graph_update","text":"","code":"graph_update(graph, delete)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_update.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Delete hypotheses from a graph — graph_update","text":"graph initial graph returned graph_create() delete logical integer vector, denoting hypotheses keep/delete. logical vector must match size graph, one entry per hypothesis, results \"unordered mode,\" selected hypotheses deleted sequential order. integer vector can length, must contain valid hypothesis numbers (greater 0, less equal size graph). trigger \"ordered mode,\" selected hypotheses deleted order appear delete","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_update.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Delete hypotheses from a graph — graph_update","text":"object class updated_graph 3 elements initial graph object Boolean vector indicating hypotheses kept/deleted updated graph object specific hypotheses deleted","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_update.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Delete hypotheses from a graph — graph_update","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graph_update.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Delete hypotheses from a graph — graph_update","text":"","code":"hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   c(0, 0, 1, 0),   c(0, 0, 0, 1),   c(0, 1, 0, 0),   c(1, 0, 0, 0) ) g <- graph_create(hypotheses, transitions)  # Delete the second hypothesis graph_update(g, c(TRUE, FALSE, TRUE, TRUE)) #> Initial graph #>  #> --- Hypothesis weights --- #> H1: 0.5 #> H2: 0.5 #> H3: 0.0 #> H4: 0.0 #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1  0  0  1  0 #>  H2  0  0  0  1 #>  H3  0  1  0  0 #>  H4  1  0  0  0 #>  #> Updated graph after deleting hypotheses 1, 3, 4 #>  #> --- Hypothesis weights --- #> H1: NA #> H2:  1 #> H3: NA #> H4: NA #>  #> --- Transition weights --- #>     H1 H2 H3 H4 #>  H1 NA NA NA NA #>  H2 NA  0 NA NA #>  H3 NA NA NA NA #>  H4 NA NA NA NA # Equivalent # graph_update(g, 2)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graphicalMCP-package.html","id":null,"dir":"Reference","previous_headings":"","what":"graphicalMCP: Graphical Approach for Multiple Comparison Procedures — graphicalMCP-package","title":"graphicalMCP: Graphical Approach for Multiple Comparison Procedures — graphicalMCP-package","text":"multiple comparison procedure (multiple test procedure) statistical analysis method determining efficacy multiple drugs, multiple doses drug, single clinical trial. (Bretz et al., 2011) laid graph-based approach multiple comparison procedures, weights vertices edges graph determined independently particular statistical test used assess results. low-dependency implementation methods described subsequent papers.","code":""},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/graphicalMCP-package.html","id":"author","dir":"Reference","previous_headings":"","what":"Author","title":"graphicalMCP: Graphical Approach for Multiple Comparison Procedures — graphicalMCP-package","text":"Maintainer: Dong Xi dong.xi@gilead.com Authors: Ethan Brockmann ethan.brockmann@atorusresearch.com contributors: Gilead Biostatistics [copyright holder, funder] Atorus Research LLC [copyright holder]","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.initial_graph.html","id":null,"dir":"Reference","previous_headings":"","what":"S3 plot method for the class initial_graph — plot.initial_graph","title":"S3 plot method for the class initial_graph — plot.initial_graph","text":"S3 plot method class initial_graph","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.initial_graph.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"S3 plot method for the class initial_graph — plot.initial_graph","text":"","code":"# S3 method for initial_graph plot(   x,   ...,   layout = \"grid\",   nrow = NULL,   ncol = NULL,   edge_curves = NULL,   precision = 4,   eps = NULL,   background_color = \"white\",   margins = c(0, 0, 0, 0) )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.initial_graph.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"S3 plot method for the class initial_graph — plot.initial_graph","text":"x initial graph returned graph_create() ... arguments passed igraph::plot.igraph() layout igraph layout specification (See ?igraph.plotting), \"grid\", lays hypotheses left--right, top--bottom nrow rows ncol columns. nrow integer scalar specifying number rows vertex grid. row columns counts specified, vertices laid close square possible. ncol integer scalar specifying number columns vertex grid. row columns counts specified, vertices laid close square possible. edge_curves named numeric vector specifying curvature edges. Edge pairs (two vertices share edge possible direction) detected automatically get 0.25 curvature. Adjust edges adding entry name \"vertex1|vertex2. Adjust default edge pairs curvature adding entry name \"pairs\". precision integer scalar specifying many decimal places displayed weights eps numeric scalar. Edge weights 0 eps displayed \\(\\epsilon\\), edge weights 1 - eps 1 displayed \\(1 - \\epsilon\\) background_color character scalar specifying background color whole plotting area. Passed directly graphics::par() (bg) margins length 4 numeric vector specifying margins plot. Defaults 0, since igraph plots tend large margins. passed directly graphics::par() (mar)","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.initial_graph.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"S3 plot method for the class initial_graph — plot.initial_graph","text":"NULL, plotting graph","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.initial_graph.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"S3 plot method for the class initial_graph — plot.initial_graph","text":"","code":"plot(simple_successive_2(), layout = \"grid\")"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.updated_graph.html","id":null,"dir":"Reference","previous_headings":"","what":"S3 plot method for the class updated_graph — plot.updated_graph","title":"S3 plot method for the class updated_graph — plot.updated_graph","text":"S3 plot method class updated_graph","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.updated_graph.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"S3 plot method for the class updated_graph — plot.updated_graph","text":"","code":"# S3 method for updated_graph plot(x, ...)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.updated_graph.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"S3 plot method for the class updated_graph — plot.updated_graph","text":"x updated_graph object returned graph_update() ... Arguments passed plot.initial_graph layout igraph layout specification (See ?igraph.plotting), \"grid\", lays hypotheses left--right, top--bottom nrow rows ncol columns. nrow integer scalar specifying number rows vertex grid. row columns counts specified, vertices laid close square possible. ncol integer scalar specifying number columns vertex grid. row columns counts specified, vertices laid close square possible. edge_curves named numeric vector specifying curvature edges. Edge pairs (two vertices share edge possible direction) detected automatically get 0.25 curvature. Adjust edges adding entry name \"vertex1|vertex2. Adjust default edge pairs curvature adding entry name \"pairs\". precision integer scalar specifying many decimal places displayed weights eps numeric scalar. Edge weights 0 eps displayed \\(\\epsilon\\), edge weights 1 - eps 1 displayed \\(1 - \\epsilon\\) background_color character scalar specifying background color whole plotting area. Passed directly graphics::par() (bg) margins length 4 numeric vector specifying margins plot. Defaults 0, since igraph plots tend large margins. passed directly graphics::par() (mar)","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.updated_graph.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"S3 plot method for the class updated_graph — plot.updated_graph","text":"NULL, plotting graph","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/plot.updated_graph.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"S3 plot method for the class updated_graph — plot.updated_graph","text":"","code":"plot(   graph_update(     simple_successive_2(),     c(TRUE, FALSE, TRUE, TRUE)   ),   layout = \"grid\" )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.graph_report.html","id":null,"dir":"Reference","previous_headings":"","what":"S3 print method for class graph_report — print.graph_report","title":"S3 print method for class graph_report — print.graph_report","text":"graph report displays initial graph tested, p-values & alpha used tests, hypotheses can rejected, Detailed test results matrix, including results graph_generate_weights() & test results intersection hypothesis","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.graph_report.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"S3 print method for class graph_report — print.graph_report","text":"","code":"# S3 method for graph_report print(x, ..., precision = 4, indent = 2, rows = 10)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.graph_report.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"S3 print method for class graph_report — print.graph_report","text":"x object class graph_report print ... values passed methods (currently unused) precision integer scalar indicating maximum number decimals include numeric values indent integer scalar indicating many spaces indent results rows integer scalar indicating many rows verbose test values output print","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.initial_graph.html","id":null,"dir":"Reference","previous_headings":"","what":"S3 print method for the class initial_graph — print.initial_graph","title":"S3 print method for the class initial_graph — print.initial_graph","text":"printed initial_graph displays header stating object , hypothesis weights, transition weights.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.initial_graph.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"S3 print method for the class initial_graph — print.initial_graph","text":"","code":"# S3 method for initial_graph print(x, ..., precision = 4, indent = 0, title = \"Initial graph\")"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.initial_graph.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"S3 print method for the class initial_graph — print.initial_graph","text":"x object class initial_graph print ... values passed methods (currently unused) precision integer scalar indicating maximum number decimals include numeric values indent integer scalar indicating many spaces indent results title First line printing, used internally distinguish initial graphs updated graphs","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.power_report.html","id":null,"dir":"Reference","previous_headings":"","what":"S3 print method for class power_report — print.power_report","title":"S3 print method for class power_report — print.power_report","text":"power report displays initial graph tested, Testing simulation options, Final power calculations, (Partial) Detailed p-values test results - underlying object contains full tables","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.power_report.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"S3 print method for class power_report — print.power_report","text":"","code":"# S3 method for power_report print(x, ..., precision = 4, indent = 2, rows = 10)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.power_report.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"S3 print method for class power_report — print.power_report","text":"x object class power_report print ... values passed methods (currently unused) precision integer scalar indicating maximum number decimals include numeric values indent integer scalar indicating many spaces indent results rows integer scalar indicating many rows verbose output print","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.updated_graph.html","id":null,"dir":"Reference","previous_headings":"","what":"S3 print method for class updated_graph — print.updated_graph","title":"S3 print method for class updated_graph — print.updated_graph","text":"printed updated_graph displays initial graph, vector keep/delete hypotheses, updated graph","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.updated_graph.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"S3 print method for class updated_graph — print.updated_graph","text":"","code":"# S3 method for updated_graph print(x, ..., precision = 6, indent = 2)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/print.updated_graph.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"S3 print method for class updated_graph — print.updated_graph","text":"x object class updated_graph print ... values passed methods (currently unused) precision integer scalar indicating maximum number decimals include numeric values indent integer scalar indicating many spaces indent results","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing-fast.html","id":null,"dir":"Reference","previous_headings":"","what":"Calculate hypothesis rejection results efficiently — graph_test_closure_fast","title":"Calculate hypothesis rejection results efficiently — graph_test_closure_fast","text":"insight nice reporting, prefer graph_test_closure() graph_test_shortcut() options. reasonably fast interactive use. However order minimize power run time, efficient testing functions available. power simulation can segmented certain parts, like generating weights calculating adjusted weights, can done single time. closure testing function stripped just vectorized lines efficiency. separate optimized function available testing graph Bonferroni sequential shortcut.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing-fast.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Calculate hypothesis rejection results efficiently — graph_test_closure_fast","text":"","code":"graph_test_closure_fast(p, alpha, adjusted_weights, matrix_intersections)  graph_test_shortcut_fast(p, alpha, adjusted_weights)"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing-fast.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Calculate hypothesis rejection results efficiently — graph_test_closure_fast","text":"p numeric vector p-values alpha numeric scalar specifying global significance level testing adjusted_weights weights (second half columns) graph_generate_weights() output, adjusted appropriate testing algorithm (Bonferroni, Simes, parametric) graph initial graph returned graph_create() intersections first half columns graph_generate_weights() output, indicating hypotheses contained intersection","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing-fast.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Calculate hypothesis rejection results efficiently — graph_test_closure_fast","text":"logical integer vector results indicating whether hypothesis can accepted rejected globally.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing-fast.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Calculate hypothesis rejection results efficiently — graph_test_closure_fast","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing-fast.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Calculate hypothesis rejection results efficiently — graph_test_closure_fast","text":"","code":"par_gate <- simple_successive_1() num_hyps <- length(par_gate$hypotheses)  p <- c(.001, .02, .002, .03)  weighting_strategy <- graph_generate_weights(par_gate) intersections <- weighting_strategy[, seq_len(num_hyps), drop = FALSE] adjusted_weights <-   weighting_strategy[, seq_len(num_hyps) + num_hyps, drop = FALSE]  graphicalMCP:::graph_test_closure_fast(   p,   .025,   adjusted_weights,   intersections ) #>    H1    H2    H3    H4  #>  TRUE  TRUE  TRUE FALSE  graphicalMCP:::graph_test_shortcut_fast(   p,   .025,   adjusted_weights ) #>    H1    H2    H3    H4  #>  TRUE  TRUE  TRUE FALSE"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing.html","id":null,"dir":"Reference","previous_headings":"","what":"Report details of hypothesis rejections — graph_test_closure","title":"Report details of hypothesis rejections — graph_test_closure","text":"slower graph testing functions design choices made favor ease interpreting results speed. Results include hypothesis rejection decisions, also test values led final result. functions include options reporting details using adjusted p-value method adjusted weight method.","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Report details of hypothesis rejections — graph_test_closure","text":"","code":"graph_test_closure(   graph,   p,   alpha = 0.025,   test_groups = list(seq_along(graph$hypotheses)),   test_types = c(\"bonferroni\"),   test_corr = rep(list(NA), length(test_types)),   verbose = FALSE,   test_values = FALSE )  graph_test_shortcut(   graph,   p,   alpha = 0.025,   verbose = FALSE,   test_values = FALSE )"},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Report details of hypothesis rejections — graph_test_closure","text":"graph initial graph returned graph_create() p numeric vector p-values alpha numeric scalar specifying global significance level testing test_groups list numeric vectors specifying hypotheses test together test_types character vector tests apply given groups test_corr (Optional) numeric matrix correlations hypotheses' test statistics verbose logical scalar specifying whether results intersection hypothesis included test_values logical scalar specifying whether hypothesis-level detail included results, including calculating adjusted weights parametric tests","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Report details of hypothesis rejections — graph_test_closure","text":"graph_report object, list 4 elements: inputs, outputs, details, test_values Inputs - list input parameters used run test Outputs - list global test results Details - matrix detailed adjusted p-value results (graph deletion sequence shortcut testing) Test values - data frame hypothesis-level test details intersection (step shortcut testing)","code":""},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"Report details of hypothesis rejections — graph_test_closure","text":"Bretz, F., Maurer, W., Brannath, W., Posch, M. (2009). graphical approach sequentially rejective multiple test procedures. Statistics Medicine, 28(4), 586–604. https://doi.org/10.1002/sim.3495 Bretz, F., Maurer, W., Hommel, G. (2011). Test power considerations multiple endpoint analyses using sequentially rejective graphical procedures. Statistics Medicine, 30(13), 1489–1501. https://doi.org/10.1002/sim.3988 Bretz, F., Posch, M., Glimm, E., Klinglmueller, F., Maurer, W., Rohmeyer, K. (2011). Graphical approaches multiple comparison procedures using weighted Bonferroni, Simes, parametric tests. Biometrical Journal, 53(6), 894–913. https://doi.org/10.1002/bimj.201000239 Lu, K. (2016). Graphical approaches using Bonferroni mixture weighted Simes tests. Statistics Medicine, 35(22), 4041–4055. https://doi.org/10.1002/sim.6985 Xi, D., Glimm, E., Maurer, W., Bretz, F. (2017). unified framework weighted parametric multiple test procedures. Biometrical Journal, 59(5), 918–931. https://doi.org/10.1002/bimj.201600233 Xi, D., Bretz, F. (2019). Symmetric graphs equally weighted tests, application Hochberg procedure. Statistics Medicine, 38(27), 5268–5282. https://doi.org/10.1002/sim.8375 Rohmeyer K, Klinglmueller F (2020). gMCP: Graph Based Multiple Test Procedures. R package version 0.8-15, https://cran.r-project.org/package=gMCP.","code":""},{"path":[]},{"path":"https://urban-sniffle-p11zlpj.pages.github.io/reference/testing.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Report details of hypothesis rejections — graph_test_closure","text":"","code":"hypotheses <- c(0.5, 0.5, 0, 0) transitions <- rbind(   c(0, 0, 1, 0),   c(0, 0, 0, 1),   c(0, 1, 0, 0),   c(1, 0, 0, 0) )  g <- graph_create(hypotheses, transitions) p <- c(.01, .005, .015, .022)  test_corr <- list(NA, matrix(c(1, .5, .5, 1), nrow = 2, byrow = TRUE))  # The default is all Bonferroni with alpha = .025 graph_test_closure(g, p) #>  #> Test parameters ($inputs) ------------------------------------------------------ #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1  0  0  1  0 #>   H2  0  0  0  1 #>   H3  0  1  0  0 #>   H4  1  0  0  0 #>  #>   Alpha = 0.025 #>  #>                           H1    H2    H3    H4 #>   Unadjusted p-values: 0.010 0.005 0.015 0.022 #>  #>   Test types #>   bonferroni: (H1, H2, H3, H4) #>  #> Test summary ($outputs) -------------------------------------------------------- #>   Hypothesis Adj. P-value Reject #>           H1         0.02   TRUE #>           H2         0.01   TRUE #>           H3         0.03  FALSE #>           H4         0.03  FALSE #>  #>   Final updated graph after removing rejected hypotheses #>  #>   --- Hypothesis weights --- #>   H1:  NA #>   H2:  NA #>   H3: 0.5 #>   H4: 0.5 #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1 NA NA NA NA #>   H2 NA NA NA NA #>   H3 NA NA  0  1 #>   H4 NA NA  1  0 #>   # But tests can be specified at the hypothesis-level graph_test_closure(   graph = g,   p = p,   alpha = .025,   test_groups = list(1:2, 3:4),   test_types = c(\"bonferroni\", \"parametric\"),   test_corr = test_corr ) #>  #> Test parameters ($inputs) ------------------------------------------------------ #>   Initial graph #>  #>   --- Hypothesis weights --- #>   H1: 0.5 #>   H2: 0.5 #>   H3: 0.0 #>   H4: 0.0 #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1  0  0  1  0 #>   H2  0  0  0  1 #>   H3  0  1  0  0 #>   H4  1  0  0  0 #>  #>   Alpha = 0.025 #>  #>                           H1    H2    H3    H4 #>   Unadjusted p-values: 0.010 0.005 0.015 0.022 #>  #>   Correlation matrix:     H3  H4 #>                       H3 1.0 0.5 #>                       H4 0.5 1.0 #>  #>   Test types #>   bonferroni: (H1, H2) #>   parametric: (H3, H4) #>  #> Test summary ($outputs) -------------------------------------------------------- #>   Hypothesis Adj. P-value Reject #>           H1      0.02000   TRUE #>           H2      0.01000   TRUE #>           H3      0.02773  FALSE #>           H4      0.02773  FALSE #>  #>   Final updated graph after removing rejected hypotheses #>  #>   --- Hypothesis weights --- #>   H1:  NA #>   H2:  NA #>   H3: 0.5 #>   H4: 0.5 #>  #>   --- Transition weights --- #>      H1 H2 H3 H4 #>   H1 NA NA NA NA #>   H2 NA NA NA NA #>   H3 NA NA  0  1 #>   H4 NA NA  1  0 #>"}]