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001-experiments.html

@@ -467,7 +467,7 @@ <h3 data-number="1.1.2" class="anchored" data-anchor-id="the-problems-of-directi
 <section id="experiments-help-us-answer-causal-questions" class="level2 page-columns page-full" data-number="1.2">
 <h2 data-number="1.2" class="anchored" data-anchor-id="experiments-help-us-answer-causal-questions"><span class="header-section-number">1.2</span> Experiments help us answer causal questions</h2>
 <p>Imagine that you (a) created an exact replica of our world, (b) gave $1,000 to everybody in the replica world, and then (c) found a few years later that everyone in the replica world was happier than their matched self in the original world. This experiment would provide strong evidence that money makes people happier. Let’s think through why.</p>
-<p>Consider a particular person—if they are happier in the replica that in the original world, what could explain that difference? Since we have replicated the world exactly but made only one change—money—then that change is the only factor that could explain the difference in happiness. We can say that we <strong>held all variables constant</strong> except for money, which we <strong>manipulated</strong> experimentally, observing its effect on some <strong>measure</strong>—happiness. This idea—holding all variables constant except for the specific experimental manipulation—is the basic logic that underpins the experimental method <span class="citation" data-cites="mill1869">(as articulated by <a href="#ref-mill1869" role="doc-biblioref">Mill 1843</a>)</span>.<a href="#fn6" class="footnote-ref" id="fnref6" role="doc-noteref"><sup>6</sup></a> Let’s think back to our observational study of money and happiness. One big causal inference problem was the presence of “third-variable” confounds like having more friends. More friends could cause you to have more money and also cause you to be happier. The idea of an experiment is to hold everything else constant—including the number of friends that people have—so we can measure the effect of money on happiness. By holding the number of friends constant, we would be severing the causal links between friends and both money and happiness. This move is graphically conveyed in <a href="#fig-experiments-money3" class="quarto-xref">figure&nbsp;<span>1.3</span></a>, where we “snip away” the friend confound.</p>
+<p>Consider a particular person—if they are happier in the replica that in the original world, what could explain that difference? Since we have replicated the world exactly but made only one change—money—then that change is the only factor that could explain the difference in happiness. We can say that we <em>held all variables constant</em> except for money, which we <strong>manipulated</strong> experimentally, observing its effect on some <strong>measure</strong>—happiness. This idea—holding all variables constant except for the specific experimental manipulation—is the basic logic that underpins the experimental method <span class="citation" data-cites="mill1869">(as articulated by <a href="#ref-mill1869" role="doc-biblioref">Mill 1843</a>)</span>.<a href="#fn6" class="footnote-ref" id="fnref6" role="doc-noteref"><sup>6</sup></a> Let’s think back to our observational study of money and happiness. One big causal inference problem was the presence of “third-variable” confounds like having more friends. More friends could cause you to have more money and also cause you to be happier. The idea of an experiment is to hold everything else constant—including the number of friends that people have—so we can measure the effect of money on happiness. By holding the number of friends constant, we would be severing the causal links between friends and both money and happiness. This move is graphically conveyed in <a href="#fig-experiments-money3" class="quarto-xref">figure&nbsp;<span>1.3</span></a>, where we “snip away” the friend confound.</p>
 <div class="no-row-height column-margin column-container"><div id="fn6"><p><sup>6</sup>&nbsp;Another way to reason about why we can infer causality here follows the counterfactual logic we described in an earlier footnote. If the definition of causality is counterfactual (“What would have happened if the cause had been different?”), then this experiment fulfills that definition. In our impossible experiment, we can literally <em>see</em> the counterfactual: if the person had $1,000 more, here’s how much happier they would be!</p></div><div id="fig-experiments-money3" class="quarto-float quarto-figure quarto-figure-center anchored" alt="A DAG with scissored arrows from friends to money and to happiness, experimental intervention arrow from money to happiness.">
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