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1.2 Experiments help us answer causal questions

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.

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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 held all variables constant except for money, which we manipulated experimentally, observing its effect on some measure—happiness. This idea—holding all variables constant except for the specific experimental manipulation—is the basic logic that underpins the experimental method (as articulated by Mill 1843).6 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 figure 1.3, where we “snip away” the friend confound.

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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 held all variables constant except for money, which we manipulated experimentally, observing its effect on some measure—happiness. This idea—holding all variables constant except for the specific experimental manipulation—is the basic logic that underpins the experimental method (as articulated by Mill 1843).6 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 figure 1.3, where we “snip away” the friend confound.

6 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 see the counterfactual: if the person had $1,000 more, here’s how much happier they would be!

diff --git a/002-theories.html b/002-theories.html index f9ae6054..ac2c1c45 100644 --- a/002-theories.html +++ b/002-theories.html @@ -458,7 +458,7 @@

2.1.1 Psychological constructs

Constructs are the psychological variables that we want our theory to describe, like “money” and “happiness” in the example from last chapter. At first glance, it might seem odd that we need a specific name for these variables. But in probing the relationship between money and happiness, we will have to figure out a way to measure happiness. Let’s say we just ask people to answer the question “How happy are you?” by giving ratings on a 1 (miserable) to 10 (elated) scale.

Now say someone in the study reports they are an 8 on this scale. Is this really how happy they are? What if they weren’t concentrating very hard on the rating, or if they thought the researcher wanted them to be happy? What if they act much less happy in their interactions with family and friends?

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We resolve this dilemma by saying that the self-report ratings we collect are only a measure of a latent construct, happiness. The construct is latent because we can never see it directly, but we think it has a causal influence on the measure: happier people should, on average, provide higher ratings. But many other factors can lead to noise or bias in the measurement, so we shouldn’t mistake those ratings as actually being the construct.

+

We resolve this dilemma by saying that the self-report ratings we collect are only a measure of a latent construct, happiness. The construct is latent because we can never see it directly, but we think it has a causal influence on the measure: happier people should, on average, provide higher ratings. But many other factors can lead to noise or bias in the measurement, so we shouldn’t mistake those ratings as actually being the construct.

The particular question “How happy are you?” is one way of going from the general construct to a specific measure. The general process of going from construct to a specific instantiation that can be measured or manipulated is called operationalization. Happiness can be operationalized by self-report, but it can also be operationalized many other way—for example, through a measure like the use of positive language in a personal essay, or by ratings by friends, family, or a clinician. These decisions about how to operationalize a construct with a particular measure are tricky and consequential, and we discuss them extensively in chapter 8. Each different operationalization might be appropriate for a specific study, yet it would require some justification and argument to connect each one to the others.

Proposing a particular construct is a very important part of making a theory. For example, a researcher might worry that self-reported happiness is very different than someone’s well-being as observed by the people around them and assert that happiness is not a single construct but rather a group of distinct constructs. This researcher would then be surprised to know that self-reports of happiness relate very highly to others’ perceptions of a person’s well-being (Sandvik, Diener, and Seidlitz 1993).1

1 Sometimes positing the construct is the key part of a theory. g (general intelligence) is the classic psychological example of a single-construct theory. The idea behind g theory is that the best measure of general intelligence is the shared variance between a wide variety of different tests. The decision to theorize about and measure a single unified construct for intelligence—rather than say, many different separate kinds of intelligence—is itself a controversial move.

Even external, apparently nonpsychological variables like money don’t have direct effects on people but rather operate through psychological constructs. People studying money seriously as a part of psychological theories think about perceptions of money in different ways depending on the context. For example, researchers have written about the importance of how much money you have on hand based on when in the month your paycheck arrives (Ellwood-Lowe, Foushee, and Srinivasan 2022) but have also considered perceptions of long-term accumulation of wealth as a way of conceptualizing people’s understanding of the different resources available to White and Black families in the United States (Kraus et al. 2019).

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(Ungar 2002). Concretely, a therapist might be more likely to examine family, peer, and school environments when analyzing a child’s situation through the lens of EST.

There’s a continuum between precisely specified theories and broad frameworks. Some theories propose interconnected constructs but don’t specify the relationships between them, or don’t specify how those constructs should be operationalized. So when you read a paper that says it proposes a “theory,” it’s a good idea to to ask whether it describes specific relations between operationalized constructs. If it doesn’t, it may be more of a framework than a theory.

@@ -526,16 +526,16 @@

Our view of psychological theories is that they describe a set of relationships between different constructs. How can we test theories and decide which one is best? We’ll first describe falsificationism, a historical viewpoint on this issue that has been very influential in the past and that connects to ideas about statistical inference presented in chapter 6. We’ll then turn to a more modern viewpoint, holism, that recognizes the interconnections between theory and measurement.

2.2.1 Falsificationism

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One historical view that resonates with many scientists is the philosopher Karl Popper’s falsificationism. In particular, there is a simplistic version of falsificationism that is often repeated by working scientists, even though it’s much less nuanced than what Popper actually said! On this view, a scientific theory is a set of hypotheses about the world that instantiate claims like the connection between money and happiness.5 What makes a statement a scientific hypothesis is that it can be disproved (i.e., it is falsifiable) by an observation that contradicts it. For example, observing a lottery winner who immediately becomes depressed would falsify the hypothesis that receiving money makes you happier.

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5 Earlier we treated the claim that money caused happiness as a theory. It is one! It’s just a very simple theory that has only one hypothesized connection in it.

For the simplistic falsificationist, theories are never confirmed. The hypotheses that form parts of theories are universal statements. You can never prove them right; you can only fail to find falsifying evidence. Seeing hundreds of people get happier when they received money would not prove that the money-happiness hypothesis was universally true. There could always be a counter-example around the corner.

+

One historical view that resonates with many scientists is the philosopher Karl Popper’s falsificationism. In particular, there is a simplistic version of falsificationism that is often repeated by working scientists, even though it’s much less nuanced than what Popper actually said! On this view, a scientific theory is a set of hypotheses about the world that instantiate claims like the connection between money and happiness.5 What makes a statement a scientific hypothesis is that it can be disproved (i.e., it is falsifiable) by an observation that contradicts it. For example, observing a lottery winner who immediately becomes depressed would falsify the hypothesis that receiving money makes you happier.

+

5 Earlier we treated the claim that money caused happiness as a theory. It is one! It’s just a very simple theory that has only one hypothesized connection in it.

For the simplistic falsificationist, theories are never confirmed. The hypotheses that form parts of theories are universal statements. You can never prove them right; you can only fail to find falsifying evidence. Seeing hundreds of people get happier when they received money would not prove that the money-happiness hypothesis was universally true. There could always be a counter-example around the corner.

This theory doesn’t really describe how scientists work. For example, scientists like to say that their evidence “supports” or “confirms” their theory, and falsificationism rejects this kind of talk. A falsificationist says that confirmation is an illusion; that the theory is simply surviving to be tested another day. This strict falsificationist perspective is unpalatable to many scientists. After all, if we observe that hundreds of people get happier when they receive money, it seems like this should at least slightly increase our confidence that money causes happiness!6

6 An alternative perspective comes from the Bayesian tradition that we’ll learn more about in chapters 5 and 6. In a nutshell, Bayesians propose that our subjective belief in a particular hypothesis can be captured by a probability, and that our scientific reasoning can then be described by a process of normative probabilistic reasoning (Strevens 2006). The Bayesian scientist distributes probability across a wide range of alternative hypotheses; observations that are more consistent with a hypothesis increase the hypothesis’s probability (Sprenger and Hartmann 2019).

2.2.2 A holistic viewpoint on theory testing

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The key issue that leads us to reject strict falsificationism is the observation that no individual hypothesis (a part of a theory) can be falsified independently. Instead, a large series of what are called auxiliary assumptions (or auxilliary hypotheses) are usually necessary to link an observation to a theory (Lakatos 1976). For example, if giving some individual person money didn’t change their happiness, we wouldn’t immediately throw out our theory that money causes happiness. Instead, the fault might be in any one of our auxiliary assumptions, like our measurement of happiness, or our choice of how much money to give or when to give it. The idea that individual parts of a theory can’t be falsified independently is sometimes called holism.

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The key issue that leads us to reject strict falsificationism is the observation that no individual hypothesis (a part of a theory) can be falsified independently. Instead, a large series of what are called auxiliary assumptions (or auxilliary hypotheses) are usually necessary to link an observation to a theory (Lakatos 1976). For example, if giving some individual person money didn’t change their happiness, we wouldn’t immediately throw out our theory that money causes happiness. Instead, the fault might be in any one of our auxiliary assumptions, like our measurement of happiness, or our choice of how much money to give or when to give it. The idea that individual parts of a theory can’t be falsified independently is sometimes called holism.

One consequence of holism is that the relationship between data and theory isn’t always straightforward. An unexpected observation may not cause us to give up on a main hypothesis in our theory—but it will often cause us to question our auxiliary assumptions instead (e.g., how we operationalize our constructs). Thus, before abandoning our theory of money causing happiness, we might want to try several happiness questionnaires.

The broader idea of holism is supported by historical and sociological studies of how science progresses, especially in the work of Kuhn (1962). Examining historical evidence, Kuhn found that scientific revolutions didn’t seem to be caused by the falsification of a theoretical statement via an incontrovertible observation. Instead, Kuhn described scientists as mostly working within paradigms: sets of questions, assumptions, methods, phenomena, and explanatory hypotheses.

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Paradigms allow for activities Kuhn described as normal science—that is, testing questions within the paradigm, explaining new observations, or modifying theory to fit these paradigms. But normal science is punctuated by periods of crisis when scientists begin to question their theory and their methods. Crises don’t happen just because a single observation is inconsistent with the current theory. Rather, there will often be a holistic transition to a new paradigm, typically because of a striking explanatory or predictive success—often one that’s outside the scope of the current working theory entirely.

+

Paradigms allow for activities Kuhn described as normal science—that is, testing questions within the paradigm, explaining new observations, or modifying theory to fit these paradigms. But normal science is punctuated by periods of crisis when scientists begin to question their theory and their methods. Crises don’t happen just because a single observation is inconsistent with the current theory. Rather, there will often be a holistic transition to a new paradigm, typically because of a striking explanatory or predictive success—often one that’s outside the scope of the current working theory entirely.

In sum, the lesson of holism is that we can’t just put our theories in direct contact with evidence and think that they will be supported or overturned. Instead, we need to think about the scope of our theory (in terms of the phenomena and measures it is meant to explain), as well as the auxiliary hypotheses—operationalizations—that link it to specific observations.

diff --git a/003-replication.html b/003-replication.html index 3706d203..24622c9b 100644 --- a/003-replication.html +++ b/003-replication.html @@ -466,8 +466,8 @@

<

We’ll then discuss some of the main reasons why findings might not replicate—especially analytic flexibility and publication bias. We end by taking up the issue of how reproducibility and replicability relate to theory building in psychology, and the role of open science in this discussion. This discussion focuses on the key role of transparency (one of our major book themes) in supporting theory building.

3.1 Reproducibility

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Scientific papers are full of numbers: sample sizes, measurements, statistical results, and visualizations. For those numbers to have meaning, and for other scientists to be able to verify them, we need to know where they came from (their provenance). The chain of actions that scientists perform on the raw data, all the way through to reporting numbers in their papers, is sometimes called the analysis pipeline. For much of history, scientific papers have only provided a verbal description of the analysis pipeline, usually with little detail.2

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2 The situation is nicely summed up by a prescient quote from Buckheit and Donoho (1995, 5): “A scientific publication is not the scholarship itself, it is merely advertising of the scholarship. The actual scholarship is the complete software development environment and the complete set of instructions which generated the figures.”.

3 For many years, professional societies, like the American Psychological Association, have mandated data sharing (https://www.apa.org/ethics/code), but only for purposes of verification, and only “on request”—in other words, scientists could keep data hidden by default and it was their responsibility to share if another scientist requested access. In practice, this kind of policy does not work; data are rarely made available on request (Wicherts et al. 2006). We believe this situation is untenable. We provide a longer argument justifying data sharing in chapter 4 and discuss some of the practicalities of sharing in chapter 13.

Moreover, researchers typically do not share key research objects from this pipeline, such as the analysis scripts or the raw data (Hardwicke, Thibault, et al. 2021).3 Without code and data, the numbers reported in scientific papers are often not reproducible—an independent scientist cannot repeat all of the steps in the analysis pipeline and get the same results as the original scientists.

+

Scientific papers are full of numbers: sample sizes, measurements, statistical results, and visualizations. For those numbers to have meaning, and for other scientists to be able to verify them, we need to know where they came from (their provenance). The chain of actions that scientists perform on the raw data, all the way through to reporting numbers in their papers, is sometimes called the analysis pipeline. For much of history, scientific papers have only provided a verbal description of the analysis pipeline, usually with little detail.2

+

2 The situation is nicely summed up by a prescient quote from Buckheit and Donoho (1995, 5): “A scientific publication is not the scholarship itself, it is merely advertising of the scholarship. The actual scholarship is the complete software development environment and the complete set of instructions which generated the figures.”.

3 For many years, professional societies, like the American Psychological Association, have mandated data sharing (https://www.apa.org/ethics/code), but only for purposes of verification, and only “on request”—in other words, scientists could keep data hidden by default and it was their responsibility to share if another scientist requested access. In practice, this kind of policy does not work; data are rarely made available on request (Wicherts et al. 2006). We believe this situation is untenable. We provide a longer argument justifying data sharing in chapter 4 and discuss some of the practicalities of sharing in chapter 13.

Moreover, researchers typically do not share key research objects from this pipeline, such as the analysis scripts or the raw data (Hardwicke, Thibault, et al. 2021).3 Without code and data, the numbers reported in scientific papers are often not reproducible—an independent scientist cannot repeat all of the steps in the analysis pipeline and get the same results as the original scientists.

Reproducibility is desirable for a number of reasons. Without it:

  • Errors in calculation or reporting could lead to disparities between the reported result and the actual result
    • @@ -539,7 +539,7 @@

    7 A fascinating study by Baribault et al. (2018) proposes a method for empirically understanding psychological invariances. Treating a subliminal priming effect as their model system, they sampled thousands of “micro-experiments” in which small parameters of their experimental procedure were randomly sampled. These parameters allowed for measurement of their effect of interest, averaging across this irrelevant variation. In their case, it turned out that color did not matter.

    8 In some sense, the research program of some branches of the social sciences amounts to an understanding of invariances across human cognition.

    9 Presumably not very much if Dr. Toad gave the original instructions in English instead of in German—that’s another one of these pesky invariances that we are always worrying about!

A lot is at stake in this discussion. If Dr. Frog publishes a finding with US undergraduates and Dr. Toad then “replicates” the procedure in Germany, to what extent should we be perturbed if the effect is different in magnitude or absent?9 Meta-researchers have made a number of replication taxonomies to try and quantify the degree of methodological consistency between two experiments.

Some researchers have tried to distinguish “direct replications”10 and “conceptual replications.” Direct replications are those that attempt to reproduce all of the salient features of the prior study, up to whatever invariances the experimenters believe are present (e.g., color of the paint or gender of the experimenter). In contrast, conceptual replications are typically paradigms that attempt to test the same hypothesis via different operationalizations of the manipulation and/or the measure. We agree with Zwaan et al. (2018): labeling this second type of experiment as a “replication” is a little misleading. Rather, so-called conceptual replications are actually different tests of the same part of your theory. Such tests can be extremely valuable, but they serve a different goal than replication.

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10 These also get called exact replications sometimes. We think this term is misleading because similarity between two different experiments is always going to be on a gradient, and where you cut this continuum is always going to be a theory-laden decision. One person’s “exact” is another’s “inexact.”

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10 These also get called “exact replications” sometimes. We think this term is misleading because similarity between two different experiments is always going to be on a gradient, and where you cut this continuum is always going to be a theory-laden decision. One person’s “exact” is another’s “inexact.”

An example dataset from our experiment is shown in figure 5.3. Eventually, we’ll want to estimate the effect of milk-first preparation on quality ratings (our effect of interest). But for now, our goal will be to estimate the quality of the tea when it is milk-first (which some data suggest is actually the better way, at least for British tea drinkers; Kennedy 2003). More formally, we want to use our sample of nine milk-first tea judgments to estimate a number that we can’t directly observe, namely the true perceived quality of all possible milk-first cups. We’ll call this number a population parameter for reasons that will become clear in a moment.

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We’ll try to go easy on notation, but some amount will hopefully make things clearer. We will use \(\theta_{\textrm{M}}\) (“theta”) to denote the parameter we want to estimate (the population parameter) and \(\widehat{\theta}_{\textrm{M}}\), its sample estimate.3

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We’ll try to go easy on notation, but some amount will hopefully make things clearer. We will use \(\theta_{\textrm{M}}\) (“theta”) to denote the parameter we want to estimate (the population parameter) and \(\widehat{\theta}_{\textrm{M}}\), its sample estimate.3

3 Statisticians use “hats” like this to denote estimates from a specific sample. One way to remember this is that the “person in the hat” is wearing a hat to dress up as the actual quantity.

5.1.1 Maximum likelihood estimation

Okay, you are probably saying, if we want our estimate of milk-first quality, shouldn’t we just take the average rating across the nine cups of milk-first tea? The answer is yes. But let’s unpack that choice: taking the sample mean as our estimate \(\widehat{\theta}_{\textrm{M}}\) is an example of an estimation approach called maximum likelihood estimation. In general terms, maximum likelihood estimation is a two-step process.

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First, we assume a model for how the data were generated.4 This model is specified in terms of certain population parameters. In our example, the model is as simple as they come: we just assume there is some average level of tea quality and that the measurements vary around it. Let’s take a look at the data from the milk-first condition, shown in figure 5.4. Our observations are clustered around the mean, but they also show some variation. Some are higher and some are lower. Variation of this type is a feature of every data set. This variation can be summarized via a probability distribution, a mathematical entity that describes the properties of possible datasets.

+

First, we assume a model for how the data were generated.4 This model is specified in terms of certain population parameters. In our example, the model is as simple as they come: we just assume there is some average level of tea quality and that the measurements vary around it. Let’s take a look at the data from the milk-first condition, shown in figure 5.4. Our observations are clustered around the mean, but they also show some variation. Some are higher and some are lower. Variation of this type is a feature of every data set. This variation can be summarized via a probability distribution, a mathematical entity that describes the properties of possible datasets.

4 This sense of “model” is actually a formal instantiation of the type of causal model we discussed in chapter 1. As you get deeper into causal modeling, typically what you do is define a causal “story” for the statistical process that generated a dataset, using both DAGs and the kinds of probability distributions we define below.

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\(\theta_M\)) describes where its center goes, and the standard deviation describes how wide it is. Technically, the mean is the expected value for new samples from the distribution. Our best guess about the value of these new samples is that they are at the mean. We can write this more formally by introducing \(E[M]\) to denote the expectation of the variable \(M\).

-

The standard deviation \(\sigma_M\) is then a way of describing the expected variation in these samples. A bigger standard deviation means that we expect samples to be on average further from the mean. We can write this formally as \(\sigma_M = \sqrt{E[(M - \theta_M)^2]}\): the standard deviation is the expected absolute distance between individual samples and the mean, with the square and square root being necessary to compute distance.

+

The standard deviation \(\sigma_M\) is then a way of describing the expected variation in these samples. A bigger standard deviation means that we expect samples to be on average further from the mean. We can write this formally as \(\sigma_M = \sqrt{E[(M - \theta_M)^2]}\): the standard deviation is the expected absolute distance between individual samples and the mean, with the square and square root being necessary to compute distance.

Using a probability distribution to describe our dataset gives us a way of summarizing our observations through the parameters of the distribution and encoding an assumption about what future observations might look like. How do we fit a normal distribution to our data? We try to find the values of the population parameters that make our observed data as likely as possible. Let’s start with the mean.

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A comparison of the best-fitting normal distribution and a substantially worse curve.{#fig-estimation-ml2 .margin-caption width=60% fig-alt=“A plot with a”substantially worse curve” and “maximum likelihood curve”, each point has dashed line to curve.”}

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For example, if our sample mean is \(\widehat{\theta}_{\textrm{M}} = 4.5\), what underlying value of \(\theta_{\textrm{M}}\) would make these data most likely to occur? Well, suppose the underlying parameter were \(\theta_{\textrm{M}}=2.5\). Then it would be pretty unlikely that our sample mean would be so much bigger. So, \(\widehat{\theta}_{\textrm{M}}=2.5\) is a poor estimate of the population parameter based on these data (?fig-estimation-ml2). Conversely, if the parameter were \(\theta_{\textrm{M}}=6.5\), it would be a bit unlikely that our sample mean would be so much smaller. The value of \(\widehat{\theta}_{\textrm{M}}\) that makes these data most likely is just 4.5 itself: the sample mean! That is why the sample mean in this case is the maximum likelihood estimate.

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+A plot with a "substantially worse curve" and "maximum likelihood curve", each point has dashed line to curve. +
+
+Figure 5.5: A comparison of the best-fitting normal distribution and a substantially worse curve. +
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+
+

For example, if our sample mean is \(\widehat{\theta}_{\textrm{M}} = 4.5\), what underlying value of \(\theta_{\textrm{M}}\) would make these data most likely to occur? Well, suppose the underlying parameter were \(\theta_{\textrm{M}}=2.5\). Then it would be pretty unlikely that our sample mean would be so much bigger. So, \(\widehat{\theta}_{\textrm{M}}=2.5\) is a poor estimate of the population parameter based on these data (figure 5.5). Conversely, if the parameter were \(\theta_{\textrm{M}}=6.5\), it would be a bit unlikely that our sample mean would be so much smaller. The value of \(\widehat{\theta}_{\textrm{M}}\) that makes these data most likely is just 4.5 itself: the sample mean! That is why the sample mean in this case is the maximum likelihood estimate.

@@ -544,11 +553,11 @@

-Figure 5.5: Bayes’s rule, annotated. +Figure 5.6: Bayes’s rule, annotated.

How do we quantify this trade-off between our prior expectations and our current observations? We can do this via Bayesian estimation of \(\widehat{\theta}_{\textrm{M}}\). Bayesian estimation provides a principled framework for integrating prior beliefs and data. These estimation techniques can be very helpful in cases where data are sparse or prior beliefs are strong.

-

In Bayesian estimation, we observe some data \(d\), consisting of the set of responses in the experiment. Now we can use Bayes’s rule, a tool from basic probability theory, to estimate this number (figure 5.5). Each part of this equation has a name, and it’s worth becoming familiar with them. The thing we want to compute, \(p(\theta_{\textrm{M}} |\text{data})\), is called the posterior probability—it tells us what we should believe about the population parameter on tea quality, given the data we observed.5

+

In Bayesian estimation, we observe some data \(d\), consisting of the set of responses in the experiment. Now we can use Bayes’s rule, a tool from basic probability theory, to estimate this number (figure 5.6). Each part of this equation has a name, and it’s worth becoming familiar with them. The thing we want to compute, \(p(\theta_{\textrm{M}} |\text{data})\), is called the posterior probability—it tells us what we should believe about the population parameter on tea quality, given the data we observed.5

5 We’re making the posterior purple to indicate the combination of likelihood (red) and prior (blue).

6 Speaking informally, “likelihood” is just a synonym for probability, but in Bayesian estimation, “likelihood” is a technical term specifically referring to probability of the data given our hypothesis. This ambiguity can get a bit confusing.

The first part of the numerator is \(p(\text{data}|\theta_{\textrm{M}})\), the probability of the data we observed given our hypothesis about the participant’s ability. This part is called the likelihood.6 This term tells us about the relationship between our hypothesis and the data we observed—so, if we think the tea is of high quality (say \(\theta_{\textrm{M}} = 6.5\)), then the probability of observing a bunch of low-quality ratings will be fairly low.

@@ -557,7 +566,7 @@

-Figure 5.6: Bayesian inference about tea ratings with a strong prior on low values. +Figure 5.7: Bayesian inference about tea ratings with a strong prior on low values.

The second term in the numerator, \(p(\theta_{\textrm{M}} )\), is called the prior. This term encodes our beliefs about the likely distribution of tea quality. Intuitively, if we think that the tea is likely of high quality, we should require more evidence to convince us that it’s bad. In contrast, if we think it’s probably bad, a few examples of low ratings might serve to convince us.

@@ -568,10 +577,10 @@

-Figure 5.7: Bayesian inference about tea ratings with a weak prior on low values. +Figure 5.8: Bayesian inference about tea ratings with a weak prior on low values.
-

Figure 5.6 gives an example of the combination of prior and data. In this example, we look at what difference the prior makes after observing 9 ratings. If we go in assuming that the tea is likely to be bad, the posterior mean (purple line) will be pushed downward relative to the maximum likelihood estimate (red line). This prior is operating only over on ratings—estimates of tea quality. Later on when we talk about comparing milk-first and tea-first ratings to get an estimate of the experimental effect, we could consider putting a prior on tea discrimination (e.g., the experimental effect).

+

Figure 5.7 gives an example of the combination of prior and data. In this example, we look at what difference the prior makes after observing 9 ratings. If we go in assuming that the tea is likely to be bad, the posterior mean (purple line) will be pushed downward relative to the maximum likelihood estimate (red line). This prior is operating only over on ratings—estimates of tea quality. Later on when we talk about comparing milk-first and tea-first ratings to get an estimate of the experimental effect, we could consider putting a prior on tea discrimination (e.g., the experimental effect).

@@ -579,11 +588,11 @@

-Figure 5.8: Bayesian inference about tea ratings with a strong prior on low values and more data. +Figure 5.9: Bayesian inference about tea ratings with a strong prior on low values and more data.
-

Priors aren’t usually as strong as the one shown above. Figure 5.7 shows how the picture shifts when we have a weaker prior reflecting a flatter, more widely spread belief about the distribution of ratings. Now the posterior mean (purple) is closer to the maximum likelihood mean (red). This situation is more common—the prior encodes a weak assumption that ratings won’t cluster around the ends of the scale.

-

The effect of the prior is also decreased when you have more data. Take a look at figure 5.8. The prior is the same as in figure 5.6, but we have more data. As a result, the posterior distribution is much more peaked and also much closer to the data—the prior makes much less difference.

+

Priors aren’t usually as strong as the one shown above. Figure 5.8 shows how the picture shifts when we have a weaker prior reflecting a flatter, more widely spread belief about the distribution of ratings. Now the posterior mean (purple) is closer to the maximum likelihood mean (red). This situation is more common—the prior encodes a weak assumption that ratings won’t cluster around the ends of the scale.

+

The effect of the prior is also decreased when you have more data. Take a look at figure 5.9. The prior is the same as in figure 5.7, but we have more data. As a result, the posterior distribution is much more peaked and also much closer to the data—the prior makes much less difference.

Bayesian estimation is most important when you have strong beliefs and not a lot of data. That can be a case where you have just a few participants in your experiment, but it’s also good—and perhaps more common—to use Bayesian methods when you have a lot of data, but maybe not that much data about particular units that you care about. For example, you might have a large dataset about the effects of an educational intervention but not that much data about how it affects a particular subgroup. Bayesian estimates and maximum likelihood estimates will exactly coincide either under a flat prior (a prior that makes any value equally likely) or as the amount of data goes to infinity.

@@ -599,12 +608,12 @@

-Figure 5.9: Estimating the average treatment effect from the tea-tasting data. +Figure 5.10: Estimating the average treatment effect from the tea-tasting data.

9 For simplicity, we’re assuming that the standard deviations in each tea group are equal.

-

In the maximum likelihood framework, we could posit that ratings in each group (milk-first and tea-first) follow a normal distribution but that these normal distributions might have different means and standard deviations. Extending the notation introduced above, let’s term the parameters for the tea-first group \(\theta_{\textrm{T}}\) (the mean) and \(\sigma\) (the standard deviation). To estimate the treatment effect, we are positing a model in which the milk-first ratings are normally distributed with mean \(\theta_{\textrm{M}} = \theta_{\textrm{T}} + \beta\) and with standard deviation \(\sigma\).9 This equation says that milk-first ratings have the same distribution as tea-first ratings, except that their average is shifted by \(\beta\). Setting our model up this way then lets us compute \(\hat{\beta}\), our estimate of the treatment effect in our sample.

-

As in the one-sample case (i.e., estimating the mean of just the milk-first group), maximum likelihood estimation would then proceed by finding the value of \(\beta\) that makes the data most likely under the assumed model. As you’d probably expect, this estimate \(\widehat{\beta}\) turns out to be simply the difference in sample means, \(\widehat{\theta}_{\textrm{M}} - \widehat{\theta}_{\textrm{T}}\). You can see this difference pictured in figure 5.9.

+

In the maximum likelihood framework, we could posit that ratings in each group (milk-first and tea-first) follow a normal distribution but that these normal distributions might have different means and standard deviations. Extending the notation introduced above, let’s term the parameters for the tea-first group \(\theta_{\textrm{T}}\) (the mean) and \(\sigma\) (the standard deviation). To estimate the treatment effect, we are positing a model in which the milk-first ratings are normally distributed with mean \(\theta_{\textrm{M}} = \theta_{\textrm{T}} + \beta\) and with standard deviation \(\sigma\).9 This equation says that milk-first ratings have the same distribution as tea-first ratings, except that their average is shifted by \(\beta\). Setting our model up this way then lets us compute \(\hat{\beta}\), our estimate of the treatment effect in our sample.

+

As in the one-sample case (i.e., estimating the mean of just the milk-first group), maximum likelihood estimation would then proceed by finding the value of \(\beta\) that makes the data most likely under the assumed model. As you’d probably expect, this estimate \(\widehat{\beta}\) turns out to be simply the difference in sample means, \(\widehat{\theta}_{\textrm{M}} - \widehat{\theta}_{\textrm{T}}\). You can see this difference pictured in figure 5.10.

In the Bayesian framework, we would again specify a prior \(p(\beta)\) that encodes our prior beliefs about the size and direction of the treatment effect. If we have no prior beliefs at all, then we could specify a flat prior, \(p(\beta) \propto 1\).10. If we believe the treatment effect is likely to favor milk-first pouring (\(\beta>0\)), we could specify that the prior is a normal distribution centered at some positive value (e.g., \(\beta=0.5\)); the standard deviation of this prior would encode how certain we are about our prior beliefs. And, if we have no prior beliefs about the direction of the treatment effect, but we think it is unlikely to be very large, we could specify a normal prior centered at 0, which has the effect of “shrinking” the estimates closer to 0.11

10 This equation says that the probability of any value of \(\beta\) is “proportional to” 1, meaning that it’s constant (“flat”) regardless of what value \(\beta\) takes.

11 The measures of variability that we discuss here account for statistical uncertainty reflecting the fact that we have only a finite sample size. If the sample size were infinite, there would be no uncertainty of this kind. Statistical uncertainty is only one kind of uncertainty, though. A more holistic view of the overall credibility of an estimate should also account for other things outside of the model, like study design issues and bias.

As in our example above, maximum likelihood estimates and Bayesian estimates are going to be pretty similar if we have a lot of data or weak priors. They will only diverge when we have strong priors or relatively little data. The reason we are setting up these two different frameworks, however, is that they provide very different inferential tools, as we’ll see in the next chapter.

@@ -612,7 +621,7 @@

5.2.2 Measures of effect size

Once we have measured something, we need to make a decision about how to describe this effect to others. Sometimes we are working with fairly intuitive relationships that are easy to describe. A researcher might say, for example, that people who received milk-first tea drank the tea, on average, five minutes quicker than people who received tea-first tea (i.e., that \(\widehat{\beta} = 5\) minutes). Time is measured in units like minutes and seconds and so we all have a shared understanding of what five minutes means.

But what about our participants’ ratings of tea quality, which were provided on an arbitrary 7-point rating scale that we devised? What does it mean to that participants who drank milk-first tea rated it 1 point higher than participants who drank tea-first tea (i.e., that \(\widehat{\beta} = 1\) point)? And how is this difference comparable to, for instance, a 1-point change on a scale that has similar anchors (“terrible” and “delicious”) but uses a 100-point rating system?

-

To provide a common language for describing these relationships, some researchers use standardized effect sizes. A common standardized effect size is Cohen’s d, which provides a standardized estimate of the difference between two means. There are many different ways to calculate Cohen’s d (Lakens 2013), but all approaches are usually some variant of the following formula:

+

To provide a common language for describing these relationships, some researchers use standardized effect sizes. A common standardized effect size is Cohen’s d, which provides a standardized estimate of the difference between two means. There are many different ways to calculate Cohen’s d (Lakens 2013), but all approaches are usually some variant of the following formula:

\[ d = \frac{\theta_{\textrm{M}} - \theta_{\textrm{T}}}{\sigma_{\text{pooled}}} \]

@@ -623,13 +632,13 @@

-Figure 5.10: Schematic effect size computation. +Figure 5.11: Schematic effect size computation.
-

Let’s compute this measure for our tea-drinking study. We can just plug in the estimates we see in figure 5.9 and compute the standard deviation of our observed data:

-

\[\widehat{d} = \frac{\widehat{\theta}_{\textrm{M}} - \widehat{\theta}_{\textrm{T}}}{\widehat{\sigma}_{\text{pooled}}} = \frac{4.5- 3.5}{1.25} = \frac{1}{1.25} = 0.80\] In other words, the effect size of the difference between the two conditions is 0.8 standard deviations. This process is shown graphically in figure 5.10.12

-

12 Cohen’s \(d\), also referred to as a standardized mean difference (SMD), can be tricky to apply to more complex experimental designs, such as when you have within-participant designs and multiple measurements of each participant. For some guidance on this topic, see Lakens (2013).

We previously said that people who drank milk-first tea had quality ratings that were, on average, 1 point higher on a 7-point scale (\(\beta = 1\) point). Cohen’s d translates the arbitrary units of our rating scale into a unit-less effect size that is measured in terms of the variation in the data. You may find yourself wondering: “Why would I ever describe things in terms of standard deviations?” The key benefit is that it allows us to compare the size of the effect between studies that use different measures.

+

Let’s compute this measure for our tea-drinking study. We can just plug in the estimates we see in figure 5.10 and compute the standard deviation of our observed data:

+

\[\widehat{d} = \frac{\widehat{\theta}_{\textrm{M}} - \widehat{\theta}_{\textrm{T}}}{\widehat{\sigma}_{\text{pooled}}} = \frac{4.5- 3.5}{1.25} = \frac{1}{1.25} = 0.80\] In other words, the effect size of the difference between the two conditions is 0.8 standard deviations. This process is shown graphically in figure 5.11.12

+

12 Cohen’s \(d\), also referred to as a standardized mean difference (SMD), can be tricky to apply to more complex experimental designs, such as when you have within-participant designs and multiple measurements of each participant. For some guidance on this topic, see Lakens (2013).

We previously said that people who drank milk-first tea had quality ratings that were, on average, 1 point higher on a 7-point scale (\(\beta = 1\) point). Cohen’s d translates the arbitrary units of our rating scale into a unitless effect size that is measured in terms of the variation in the data. You may find yourself wondering: “Why would I ever describe things in terms of standard deviations?” The key benefit is that it allows us to compare the size of the effect between studies that use different measures.

Let’s say that we ran a replication of our tea study with two changes: (1) we studied patrons in a US cafe instead of a UK cafe, and (2) we used a 100-point quality rating scale instead of a 7-point scale. Imagine that, just as we found that participants in the UK rated the milk-first tea 1 point higher on a 7-point quality scale, US participants rated the milk-first tea 1 point higher on a 100-point quality scale. It seems clear that these effects are different because of the difference in scale. But how different?

It might at first seem reasonable just to normalize by the length of the scale. So, maybe the UK experimental participants showed a 1/7 rating effect and the US participants showed a 1/100 rating effect. The trouble with this move is that it presupposes that participants from two different populations are using two different scales in exactly the same way! For example, maybe US participants made very clumpy judgments that were mostly centered around 50 (perhaps because of a lack of milk tea experience). Standardized effect sizes get around this kind of issue by scaling according to the variability of the data.

Let’s compute the effect size for the cross-cultural replication. We’ll imagine that participants who drank milk-first tea gave an average rating of 50/100 and participants who drank tea-first tea rated it 49 on average. But if their variability was also relatively lower, perhaps the standard deviation of their ratings was only 5. Using the formula above, we find

@@ -659,20 +668,20 @@

-Figure 5.11: The relationship between age and milk-first tea rating. +Figure 5.12: The relationship between age and milk-first tea rating.
-

14 Remember, this is a correlational relationship, and there’s no causal inference possible here.

Our focus up until now has been in estimating individual effects, but sometimes we also want to estimate the relationship between two different variables. Extending our example, figure 5.11 shows the relationship between the age of the tea taster and their rating of milk-first tea. It seems that younger people overall like tea less than older people.14 How could we quantify this result?

+

14 Remember, this is a correlational relationship, and there’s no causal inference possible here.

Our focus up until now has been in estimating individual effects, but sometimes we also want to estimate the relationship between two different variables. Extending our example, figure 5.12 shows the relationship between the age of the tea taster and their rating of milk-first tea. It seems that younger people overall like tea less than older people.14 How could we quantify this result?

The first concept we need is covariance. Covariance captures the degree to which we expect two variables to deviate from their means in the same direction. We’re looking at milk-first tea ratings \(M\) and participant ages \(A\). We can write the covariance between these two variables as

\[Cov(M, A) = E[(M - \theta_M) (A - \theta_A)]\]

-

This formula expresses the expected product of how much each observation differs from its expectation (mean) along each variable. Figure 5.12 shows these differences, which are multiplied together for each point to get the covariation.15

+

This formula expresses the expected product of how much each observation differs from its expectation (mean) along each variable. Figure 5.13 shows these differences, which are multiplied together for each point to get the covariation.15

15 This looks a little tricky, but it’s actually very related to the basic concepts we’ve already seen. Remember when we introduced the standard deviation, we described it as the expected distance between new samples from a distribution and the mean of that distribution. The covariance is very related: the standard deviation is just \(\sqrt{Cov(X,X)}\), in other words, the square root of the covariance of a variable with itself.

A plot the same as 5.12 with each point having by segment to mean age (theta_A) and segment to mean rating (theta_M).
-Figure 5.12: Estimating the covariation between age and milk-first tea rating. +Figure 5.13: Estimating the covariation between age and milk-first tea rating.
diff --git a/006-inference.html b/006-inference.html index 81ba30b2..68ba7106 100644 --- a/006-inference.html +++ b/006-inference.html @@ -515,7 +515,7 @@

If NHST approaches have so many issues, what should replace them? Figure 6.1 shows one way of organizing different inferential approaches. There has been a recent move toward the use of Bayes Factors to quantify the evidence in support of different candidate hypotheses. Bayes Factors can help answer questions like (2). We introduce these tools and we believe that they have broader applicability than the NHST framework and should be known by students. On the other hand, Bayes Factors are not a panacea. They have many of the same problems as NHST when they are applied dichotomously.

-

Instead of dichotomous frequentist or Bayesian hypothesis testing, we follow our thematic emphasis on measurement precision and advocate for a measurement strategy, which is more suited toward questions (3) and (4) (Cumming 2014; Kruschke and Liddell 2018b). The goal of these strategies is to yield an accurate and precise estimate of the relationships underlying observed variation in the data.

+

Instead of dichotomous frequentist or Bayesian hypothesis testing, we follow our thematic emphasis on measurement precision and advocate for a measurement strategy, which is more suited toward questions (3) and (4) (Cumming 2014; Kruschke and Liddell 2018b). The goal of these strategies is to yield an accurate and precise estimate of the relationships underlying observed variation in the data.

This isn’t a statistics book, and we won’t attempt to teach the full array of important statistical concepts that will allow students to build good models of a broad array of datasets (Sorry!).2 But we do want you to be able to reason about inference and modeling. In this chapter, we’ll start by making some inferences about our tea-tasting example from the last chapter, using this example to build up intuitions about hypothesis testing and inference. Then, in chapter 7, we’ll start to look at more sophisticated models and how they can be fit to real datasets.

2 If you’re interested in going deeper, here are two books that have been really influential for us. The first is Gelman and Hill (2006) and its successor Gelman, Hill, and Vehtari (2020), which teach regression and multi-level modeling from the perspective of data description. The second is McElreath (2018), a course on building Bayesian models of the causal structure of your data. Honestly, neither is an easy book to sit down and read (unless you are the kind of person who reads statistics books on the subway for fun), but both really reward detailed study. We encourage you to get together a reading group and go through the exercises in one of these together. It’ll be well worthwhile in its impact on your statistical and scientific thinking.

6.1 Sampling variation

@@ -656,7 +656,7 @@

\(\beta\)) is zero.

Even with an actual treatment effect of zero, across repeated sampling, we should see some variation in \(\widehat{\beta}\), our estimate of the treatment effect. Sometimes we’ll get a small positive effect, sometimes a small negative one. Occasionally just by chance we’ll get a big effect. This is just sampling variation as we described above.

-

Fisher’s innovation was to quantify the probability of observing various values of \(\hat{\beta}\), given the null model. Then, if the observed data that were very low probability under the null model, we could declare that the null was rejected. How unlikely must the observed data be in order to reject the null? Fisher declared that it is “usual and convenient for experimenters to take 5 percent as a standard level of convenience,” establishing the 0.05 cutoff that has become gospel throughout the sciences.4

+

Fisher’s innovation was to quantify the probability of observing various values of \(\hat{\beta}\), given the null model. Then, if the observed data that were very low probability under the null model, we could declare that the null was rejected. How unlikely must the observed data be in order to reject the null? Fisher declared that it is “usual and convenient for experimenters to take 5% as a standard level of convenience,” establishing the 0.05 cutoff that has become gospel throughout the sciences.4

4 Actually, right after establishing 0.05 as a cutoff, Fisher then writes that “in the statistical sense, we thereby admit that no isolated experiment, however significant in itself, can suffice for the experimental demonstration of any natural phenomenon … in order to assert that a natural phenomenon is experimentally demonstrable we need, not an isolated record, but a reliable method of procedure. In relation to the test of significance, we may say that a phenomenon is experimentally demonstrable when we know how to conduct an experiment which will rarely fail to give us a statistically significant result.” In other words, Fisher was all for replication!

Let’s take a look at what the null model might look like. We already tried out repeating our tea-tasting experiment thousands of times in our discussion of sampling above. Now in figure 6.6, we do the same thing but we assume that the null hypothesis of no treatment effect is true. The plot shows the distribution of treatment effects \(\hat{\beta}\) we observe: some a little negative, some a little positive, and a few substantially positive or negative, but mostly zero.

Let’s apply the \(p<0.05\) standard. If our observation has less than a 5% probability under the null model, then the null model is likely wrong. The red dashed lines on figure 6.6 show the point below which only 2.5% of the data are found and the point above which only 2.5% of the data are found. These are called the tails of the distribution. Because we’d be equally willing to accept milk-first tea or tea-first tea being better, we consider both positive and negative observations as possible.5

5 Because we’re looking at both tails of the distribution, this is called a “two-tailed” test.

@@ -871,7 +871,7 @@

10 Two good ones beyond the McElreath book mentioned above are Gelman et al. (1995), which is a bit more statistical, and Kruschke (2014), which is a bit more focused on psychological data analysis. An in-prep web-book by Nicenboim et al. (https://vasishth.github.io/bayescogsci/book) also looks great.

6.3.2 \(p\)-values

-

Now let’s turn back to NHST and the \(p\)-value. We already have a working definition of what a \(p\)-value is from our discussion above: it’s the probability of the data (or any data that would be more extreme) under the null hypothesis. How is this quantity related to either our Bayesian estimate or the BF? Well, the first thing to notice is that the \(p\)-value is very close (but not identical) to the likelihood itself.11

+

Now let’s turn back to NHST and the \(p\)-value. We already have a working definition of what a \(p\)-value is from our discussion above: it’s the probability of the data (or any data that would be more extreme) under the null hypothesis. How is this quantity related to either our Bayesian estimate or the BF? Well, the first thing to notice is that the \(p\)-value is very close (but not identical) to the likelihood itself.11

11 The likelihood—for both Bayesians and frequentists—is the probability of the data, just like the \(p\)-value. But unlike the \(p\)-value, it doesn’t include the probability of more extreme data as well.

12 \(t\)-tests can also be used in cases where one sample is being compared to some baseline.

Next we can use a simple statistical test, a \(t\)-test, to compute \(p\)-values for our experiment. In case you haven’t encountered one, a \(t\)-test is a procedure for computing a \(p\)-value by comparing the distribution of two variables using the null hypothesis that there is no difference between them.12 The \(t\)-test uses the data to compute a test statistic whose distribution under the null hypothesis is known. Then the value of this statistic can be converted to \(p\)-values for making an inference.

diff --git a/007-models.html b/007-models.html index e1a4a6be..c609b0d0 100644 --- a/007-models.html +++ b/007-models.html @@ -502,7 +502,7 @@

7.1.1 Regression for estimating a simple treatment effect

Let’s start with one of these special cases, namely estimating a treatment effect, \(\beta\), in a two-group design. In chapter 5, we solved this exact challenge for the tea-tasting experiment. We applied a classical model in which the milk-first ratings are assumed to be normally distributed with mean \(\theta_{\text{M}} = \theta_{\text{T}} + \beta\) and standard deviation \(\sigma\).4

4 Here’s a quick reminder that “model” here is a way of saying “set of assumptions about the data generating procedure.” So saying that some equation is a “model” is the same as saying that we think this is where the data came from. We can “turn the crank”—that is, generate data through the process that’s specified in those equations, such as by pulling numbers from a normal distribution with mean \(\theta_{\text{T}} + \beta\) and standard deviation \(\sigma\). In essence, we’re committing to the idea that this process will give us data that are substantively similar to the ones we have already.

5 Using 0 and 1 is known as dummy coding and allows us to interpret the parameter as the difference of the treatment group (tea-first) from the baseline (milk-first). There are many other ways to code categorical variables, with other interpretations. As a practical tip, be careful to check how your variables are coded before reporting anything!

Let’s now write that model as a regression model—that is, as a model that predicts each participant’s tea rating, \(Y_i\), given that participant’s treatment assignment, \(X_i\). \(X_i=0\) represents the control (milk-first) group and \(X_i=1\) represents the treatment (tea-first) group.5 Here, \(Y_i\) is the dependent variable and \(X_i\) is the independent variable. The subscripts \(i\) index the participants. To make this concrete, you can see some sample tea-tasting data (the first three observations from each condition) below (table 7.1), with the index \(i\), the condition and its predictor \(X_i\), and the rating \(Y\).

-

Let’s write this model more formally as a linear regression of Y on X. Conventionally, regression models are written with \(\beta\) symbols for all parameters, so we’ll now use \(\beta_0 = \theta_M\) for the mean in the milk-first group and \(\beta_1 = \theta_T - \theta_M\) as the average difference between the tea-first and milk-first groups. This \(\beta\) is a generalization of the one we were using to denote the causal effect above and in the previous two chapters.

+

Let’s write this model more formally as a “linear regression of Y on X”. Conventionally, regression models are written with \(\beta\) symbols for all parameters, so we’ll now use \(\beta_0 = \theta_M\) for the mean in the milk-first group and \(\beta_1 = \theta_T - \theta_M\) as the average difference between the tea-first and milk-first groups. This \(\beta\) is a generalization of the one we were using to denote the causal effect above and in the previous two chapters.

\[ Y_i = \beta_0 + \beta_1 X_i + \epsilon_i \]

@@ -576,7 +576,7 @@

6 Formally, we’d write \(\epsilon_i \sim \mathcal{N}(0, \sigma^2)\). The tilde means “is distributed as,” and what follows is a normal distribution with mean 0 and variance \(\sigma^2\).

7 You may be wondering why so much effort was put into building boutique solutions for these special cases when a unified framework was available the whole time. A partial answer is that the classical infrastructure of statistics was developed before computers were widespread, and these special cases were chosen because they were easy to work with “analytically” (meaning to work out all the math by hand, using values from big numerical tables). Now that we have computers with more flexible algorithms, the model-based perspective is more practical and accessible than it used to be.

Now we have the model. But how do we estimate the regression coefficients \(\beta_0\) and \(\beta_1\)? The usual method is called ordinary least squares (OLS). Here’s the basic idea. For any given regression coefficient estimates \(\widehat{\beta}_0\) and \(\widehat{\beta}_1\), we would obtain different predicted values, \(\widehat{Y}_i = \widehat{\beta}_0 + \widehat{\beta}_1 X_i\), for each participant. Some regression coefficient estimates will yield better predictions than others. Ordinary least squares estimation is designed to find the values of the regression coefficients that optimize these predictions, meaning that the predictions are as close as possible to participants’ true outcomes, \(Y_i\).

-
+
A plot with lines from each rating point to horizontal line beta_0 for milk-first and beta_0 + beta_1 for tea-first. @@ -848,7 +848,7 @@

-
+
A plot with data points, means, and confidence intervals for proportion correct over age group, for experimental and control. @@ -965,10 +965,10 @@

Let’s estimate this model and see how it looks. We’ll focus here on interpretation of the so-called fixed effects (the main predictors), as opposed to the participant and item random effects.31 Table 7.2 shows the coefficients. Again, let’s walk through each:

31 Participant means are estimated to have a standard deviation of 0.23 (in log-odds), while items have a standard deviation of 0.27. These indicate that both of our random effects capture meaningful variation.

    -

  • The intercept (control condition estimate) is \(\hat{\beta} = 0.80\), 95% CI \([0.42, 1.18]\), \(z = 4.16\), \(p < 0.001\). This estimate reflects that the log-odds of a correct response for an average-age participant in the control condition is 0.8, which corresponds to a probability of 0.69. If we look at figure 7.4, that estimate makes sense: 0.69 seems close to the average for the control condition.

    • -

  • The age effect estimate is \(\hat{\beta} = 0.55\), 95% CI \([0.21, 0.88]\), \(z = 3.19\), \(p = 0.001\). This means there is a slight decrease in the log-odds of a correct response for older children in the control condition. Again, looking at figure 7.4, this estimate is interpretable: we see a small decline in the probability of a correct response for the oldest age group.

    • +

  • The *intercept** (control condition estimate) is \(\hat{\beta} = 0.80\), 95% CI \([0.42, 1.18]\), \(z = 4.16\), \(p < 0.001\). This estimate reflects that the log-odds of a correct response for an average-age participant in the control condition is 0.8, which corresponds to a probability of 0.69. If we look at figure 7.4, that estimate makes sense: 0.69 seems close to the average for the control condition.

    • +

  • The age effect estimate is \(\hat{\beta} = 0.55\), 95% CI \([0.21, 0.88]\), \(z = 3.19\), \(p = 0.001\). This means there is a slight decrease in the log-odds of a correct response for older children in the control condition. Again, looking at figure 7.4, this estimate is interpretable: we see a small decline in the probability of a correct response for the oldest age group.

  • The key experimental condition estimate then is \(\hat{\beta} = -2.26\), 95% CI \([-2.70, -1.82]\), \(z = -10.07\), \(p < 0.001\). This estimate means that the log-odds of a correct response for an average-age participant in the experimental condition is the sum of the estimates for the control (intercept) and the experimental conditions: 0.8 + -2.26, which corresponds to a probability of 0.19. Grounding our interpretation in figure 7.4, this estimate corresponds to the average value for the experimental condition.

    • -

  • Finally, the interaction of age and condition is \(\hat{\beta} = -0.92\), 95% CI \([-1.43, -0.42]\), \(z = -3.60\), \(p < 0.001\). This positive coefficient reflects that with every year of age, the difference between control and experimental conditions grows.

    • +

  • Finally, the interaction of age and condition is \(\hat{\beta} = -0.92\), 95% CI \([-1.43, -0.42]\), \(z = -3.60\), \(p < 0.001\). This positive coefficient reflects that with every year of age, the difference between control and experimental conditions grows.

In sum, this model suggests that there was a substantial difference in performance between experimental and control conditions, in turn supporting the hypothesis that children in the sampled age group can perform pragmatic inferences above chance.

This example illustrates the “default saturated model” framework that we recommend—the idea that a single regression model corresponding to the design of the experiment can yield an interpretable estimate of the causal effect of interest, even in the presence of other sources of variation.

@@ -1001,8 +1001,8 @@

Figure 7.5 shows the results of a simple simulation investigating the relationship between estimation error and the inclusion of covariates. Only when the correlation between covariate and outcome (e.g., age and tea rating) is greater than \(r=0.6\) to \(r=0.8\) does this adjustment really help.

That said, there are quite a few reasons not to include covariates. These motivate our recommendation to skip them in your default model unless you have very strong theory-based expectations for either (A) a correlation with the outcome or (B) a strong moderation relationship.

The first reason not to include covariates is simply because we don’t need to. Because randomization cuts causal links, our experimental estimate is an unbiased estimate of the causal effect of interest (at least for large samples). We are guaranteed that, in the limit of many different experiments, even though people with different ages will be in the different tea-tasting conditions, this source of variation will be averaged out. Actually, including unnecessary covariates into models (slightly) decreases the probability that the model can detect a true effect (that is, it decreases statistical precision and power). Just by chance, covariates can “soak up” variation in the outcome, leaving less to be accounted for by the true effect!

-

The second reason is that you can actually compromise your causal inference by including some covariates, particularly those that are collected after randomization. The logic of randomization is that you cut all causal links between features of the sample and the condition manipulation. But you can “uncut” these links by accident by adding variables into your model that are related to group status. This problem is generically called conditioning on post-treatment variables, and a full discussion of is out of the scope of this book, but it’s something to avoid (and read up on if you’re worried about it, see Montgomery, Nyhan, and Torres 2018).

-

Finally, one of the standard justifications for adding covariates—because your groups are unbalanced—is actually ill-founded as well. People often talk about “unhappy randomization”: you randomize to the different tea-tasting groups, for example, but then it turns out the mean age is a bit different between groups. Then you do a \(t\)-test or some other statistical test and find out that you actually have a significant age difference. This practice makes no sense! Because you randomized, you know that the difference in ages occurred by chance. Further, incidental demographic differences between groups are unlikely to be important unless that characteristic is highly correlated with the outcome (see above). Instead, if the sample size is small enough that meaningfully large incidental differences could arise in important confounders, then it is preferable to stratify on that confounder at the outset—we’ll have lot more to say about this issue in chapter 10.

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The second reason is that you can actually compromise your causal inference by including some covariates, particularly those that are collected after randomization. The logic of randomization is that you cut all causal links between features of the sample and the condition manipulation. But you can “uncut” these links by accident by adding variables into your model that are related to group status. This problem is generically called conditioning on post-treatment variables, and a full discussion of is out of the scope of this book, but it’s something to avoid (and read up on if you’re worried about it, see Montgomery, Nyhan, and Torres 2018).

+

Finally, one of the standard justifications for adding covariates—because your groups are unbalanced—is actually ill-founded as well. People often talk about “unhappy randomization”: you randomize to the different tea-tasting groups, for example, but then it turns out the mean age is a bit different between groups. Then you do a \(t\)-test or some other statistical test and find out that you actually have a significant age difference. This practice makes no sense! Because you randomized, you know that the difference in ages occurred by chance. Further, incidental demographic differences between groups are unlikely to be important unless that characteristic is highly correlated with the outcome (see above). Instead, if the sample size is small enough that meaningfully large incidental differences could arise in important confounders, then it is preferable to stratify on that confounder at the outset—we’ll have lot more to say about this issue in chapter 10.

So these are our options: if a covariate is known to be very strongly related to our outcome, we can include it in our default model. Otherwise, we avoid a lot of trouble by leaving covariates out.

diff --git a/008-measurement.html b/008-measurement.html index 9bea1eec..a8437a7d 100644 --- a/008-measurement.html +++ b/008-measurement.html @@ -454,7 +454,7 @@

<

In the previous section of the book, we described a set of measurement-focused statistical techniques for quantifying (and maximizing) our precision. In this next set of three chapters focusing on planning experiments, we will develop our toolkit for designing the measures (this chapter), design manipulations (chapter 9), and sampling (chapter 10) strategies that will allow us to create and evaluate experiments. These chapters form a core part of our approach to “experimentology”: a set of decisions to reduce bias, maximize measurement precision, and assess generalizability. Let’s begin with measurement.

Throughout the history of science, advances in measurement have gone hand in hand with advances in knowledge.1 Telescopes revolutionized astronomy, microscopes revolutionized biology, and patch clamping revolutionized physiology. But measurement isn’t easy. Even the humble thermometer, allowing reliable measurement of temperature, required centuries of painstaking effort to perfect (Chang 2004). Psychology and the behavioral sciences are no different—we need reliable instruments to measure the things we care about. In this next section of the book, we’re going to discuss the challenges of measurement in psychology, and the properties that distinguish good instruments from bad.

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1 As such, measurement is a perennially controversial topic in philosophy of science. For an overview of competing frameworks, see Tal (2020) or Maul, Irribarra, and Wilson (2016), which focuses specifically on measurement in psychology.

What does it mean to measure something? Intuitively, we know that a ruler measures the quantity of length and a scale measures the quantity of mass (Kisch 1965). As we discussed in chapter 2, those quantities are latent (unobserved). Individual measurements, in contrast, are manifest: they are observable to us. What does it mean to measure a psychological construct—a hypothesized theoretical quantity inside the head?

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1 As such, measurement is a perennially controversial topic in philosophy of science. For an overview of competing frameworks, see Tal (2020) or Maul, Irribarra, and Wilson (2016), which focuses specifically on measurement in psychology.

What does it mean to measure something? Intuitively, we know that a ruler measures the quantity of length and a scale measures the quantity of mass (Kisch 1965). As we discussed in chapter 2, those quantities are latent (unobserved). Individual measurements, in contrast, are manifest: they are observable to us. What does it mean to measure a psychological construct—a hypothesized theoretical quantity inside the head?

We first have to keep in mind that not every measure is equally precise. This point is obvious when you think about physical measurement instruments: a caliper will give you a much more precise estimate of thickness than a ruler will. One way to see that the measurement is more precise is by repeating it a bunch of times. The measurements from the caliper will likely be more similar to one another, reflecting the fact that the amount of error in each individual measurement is smaller. We can do the same thing with a psychological measurement—repeat and assess variation—though as we’ll see below it’s a little trickier. Measurement instruments that have less error are called more reliable instruments.2

2 Is reliability the same as precision? Yes, more or less. Confusingly, different fields call these concepts different things (there’s a helpful table of these names in Brandmaier et al. 2018). Here we’ll talk about reliability as a property of instruments specifically while using the term precision to talk about the measurements themselves.

3 We are also going to talk in chapter 9 about the validity of manipulations. The way you identify a causal effect on some measure is by operationalizing some construct as well. To identify causal effects, we must link a particular construct of interest to something we can concretely manipulate in an experiment, like the stimuli or instructions.

Second, psychological measurements do not directly reflect latent theoretical constructs of interest, quantities like happiness, intelligence, or language processing ability. And unlike quantities like length and mass, there is often disagreement in psychology about what the right theoretical quantities are. Thus, we have to measure an observable behavior—our operationalization of the construct—and then make an argument about how the measure relates to a proposed construct of interest (and sometimes whether the construct really exists at all). This argument is about the validity of our measurements.3

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but does not appear to be a face-valid measure of intelligence in that simply being fast doesn’t accord with our intuition about what it means to be intelligent!
  • Ecological validity: The measure relates to the context of people’s lives. For example, a rating of a child’s behavioral self-control in the classroom is a more ecologically valid measure of executive function than a reaction-time task administered in a lab context. Ecological validity arguments can be made on the basis of the experimental task, the stimuli, and the general setting of the experiment (Schmuckler 2001). Researchers differ in how much weight they assign to ecological validity based on their goals and their theoretical orientation.
  • Internal validity: Usually used negatively. A “challenge to internal validity” is a description of a case where the measure is administered in such a way as to weaken the relationship between measure and construct. For example, if later items on a math test showed lower performance due to test-taker’s fatigue rather than lower knowledge of the concepts, the test might have an internal validity issue.11
    • -
  • Convergent validity: The classic strategy for showing validity is to show that a measure relates (usually, correlates) with other putative measures of the same construct. When these relationships are measured concurrently, this is sometimes called concurrent validity. As we mentioned in chapter 2, self-reports of happiness relate to independent ratings by friends and family, suggesting that both measure the same underlying construct (Sandvik, Diener, and Seidlitz 1993).12
    • +
  • Convergent validity: The classic strategy for showing validity is to show that a measure relates (usually, correlates) with other putative measures of the same construct. When these relationships are measured concurrently, this is sometimes called concurrent validity . As we mentioned in chapter 2, self-reports of happiness relate to independent ratings by friends and family, suggesting that both measure the same underlying construct (Sandvik, Diener, and Seidlitz 1993).12
  • Predictive validity. If the measure predicts other later measures of the construct, or related outcomes that might be of broader significance. Predictive validity is often used in lifespan and developmental studies where it is particularly prized for a measure to be able to predict meaningful life outcomes such as educational success in the future. For example, classroom self-control ratings (among other measures) appear strongly predictive of later life health and wealth outcomes (Moffitt et al. 2011).
  • Divergent validity. If the measure can be shown to be distinct from measure(s) of a different construct, this evidence can help establish that the measure is specifically linked to the target construct. For example, measures of happiness (specifically, life satisfaction) can be distinguished from measures of optimism as well as both positive and negative affect, suggesting that these are distinct constructs (Lucas, Diener, and Suh 1996).
    • diff --git a/009-design.html b/009-design.html index e16be173..aa400b1e 100644 --- a/009-design.html +++ b/009-design.html @@ -443,13 +443,13 @@

  • Describe key elements of experimental design
  • Define randomization and counterbalancing strategies for removing confounds
    • -
  • Discuss strategies to design experiments that are appropriate to the populations of interest
    • +
  • Discuss strategies to design experiments that are appropriate to the populations of interest

    • The key thesis of our book is that experiments should be designed to yield precise and unbiased measurements of a causal effect. But the causal effect of what? The manipulation! In an experiment we manipulate (intervene on) some aspect of the world and measure the effects of that manipulation. We then compare that measurement to a situation where the intervention has not occurred.

    -

    We refer to different intervention states as conditions of the experiment. The most common experimental design is the comparison between a control condition, in which the intervention is not performed, and an experimental (sometimes called treatment) condition in which the intervention is performed.

    +

    We refer to different intervention states as conditions of the experiment. The most common experimental design is the comparison between a controlcondition, in which the intervention is not performed, and an experimental (sometimes called treatment) condition in which the intervention is performed.

    But many other experimental designs are possible. In more complex experiments, manipulations along different dimensions (sometimes called factors in this context) can be combined. In the first part of the chapter, we’ll introduce some common experimental designs and the vocabulary for describing them. Our focus here is in identifying designs that maximize measurement precision.

    A good experimental measure must be a valid measure of the construct of interest. The same is true for a manipulation—it must validly relate to the causal effect of interest. In the second part of the chapter, we’ll discuss issues of manipulation validity, including both issues of ecological validity and confounding. We’ll talk about how practices like randomization and counterbalancing can help remove nuisance confounds, an important part of bias reduction for experimental designs.1

    1 This section will draw on our introduction to causal inference in chapter 1, so if you haven’t read that, now’s the time.

    To preview our general take-home points from this chapter: we think that your default experiment should manipulate one or two factors—usually not more—and should manipulate those factors continuously and within participants. Although such designs are not always possible, they are typically the most likely to yield precise estimates of a particular effect that can be used to constrain future theorizing. We’ll start by considering a case study in which a subtle confound led to difficulties interpreting an experimental result.

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    11 Our advice is both to use within-participants designs and to get multiple measurements from each participant.

    11 We’re talking about multiple trials with the same measure, not multiple distinct measures. As we discussed in chapter 8, we tend to be against measuring lots of different things in a single experiment—in part because of the concerns that we’re articulating in this chapter: if you have time, it’s better to make more precise measures of what you care about most. Measuring one thing well is hard enough. Much better to measure one thing well than many things badly.

    Why? In the last subsection, we described how variability in our estimates in a between-participants design depends on three components: (1) true condition differences; (2) sampling variation between conditions; and (3) measurement error.

    Within-participants designs are good because they don’t include (2). Repeated measures reduce (3): the more times you measure, the lower your measurement error, leading to greater measure reliability!

    -

    There are problems with repeating the same measure many times, however. Some measures can’t be repeated without altering the response. To take an obvious example, we can’t give the exact same math problem twice and get two useful measurements of mathematical ability! The typical solution to this problem is to create multiple items. In the case of a math assessment, you create multiple problems that you believe test the same concept but have different numbers or other superficial characteristics.

    +

    There are problems with repeating the same measure many times, however. Some measures can’t be repeated without altering the response. To take an obvious example, we can’t give the exact same math problem twice and get two useful measurements of mathematical ability! The typical solution to this problem is to create multiple items. In the case of a math assessment, you create multiple problems that you believe test the same concept but have different numbers or other superficial characteristics.

    Using multiple items for measurement is good for two reasons. First, it reduces measurement error by allowing responses to be combined across items. But second, it increases the generalizability of the measurement. An effect that is consistent across many different items is more likely to be an effect that can be generalized to a whole class of stimuli—in precisely the same way that the use of multiple participants can license generalizations across a population of people (Clark 1973).

    diff --git a/010-sampling.html b/010-sampling.html index 6af69bee..1b42d9ea 100644 --- a/010-sampling.html +++ b/010-sampling.html @@ -484,12 +484,12 @@

    Discuss sampling theory and stratified sampling
  • Reason about the limitations of different samples, especially convenience samples
  • Consider sampling biases and how they affect your inferences
    • -
  • Learn how to choose and justify an appropriate sample size for your experiment
    • +
  • Learn how to choose and justify an appropriate sample size for your experiment

    • -

    As we keep reminding you, experiments are designed to yield measurements of a causal effect. But a causal effect of what, and for whom? These are questions that are often given surprisingly little air time in our papers. Titles in our top journals read “Daxy thinking promotes fribbles,” “Doing fonzy improves smoodling,” or “Blicket practice produces more foozles than smonkers.”1 Each of these uses generic language to state a claim that is implied to be generally true (DeJesus et al. 2019),2 but for each of these, we could reasonably ask, “For whom?” Is it everyone? Or a particular set of people? These are questions about our key theme, generalizability.

    +

    As we keep reminding you, experiments are designed to yield measurements of a causal effect. But a causal effect of what, and for whom? These are questions that are often given surprisingly little air time in our papers. Titles in our top journals read “Daxy thinking promotes fribbles,” “Doing fonzy improves smoodling,” or “Blicket practice produces more foozles than smonkers.”1 Each of these uses generic language to state a claim that is implied to be generally true (DeJesus et al. 2019),2 but for each of these, we could reasonably ask, “For whom?” Is it everyone? Or a particular set of people? These are questions about our key theme, generalizability.

    1 Titles changed to protect the original authors. These researchers might very well have said more specific things in the text of their paper.

    2 Generic language is a fascinating linguistic phenomenon. When we say things like “Mosquitoes transmit malaria,” we don’t mean that all mosquitoes do it, only something like “It’s a valid and diagnostic generalization about mosquitoes in contrast to other relevant insects or other creatures that they are spreaders of malaria” (see Tessler and Goodman 2019).

    Let’s focus on smoodling. We wouldn’t let the authors get away with a fully universal version of their claim: “Doing [any] fonzy improves smoodling [for everyone].” The nongeneric version states a generalization that goes way beyond the evidence we actually have. But it seems that we are often okay with authors implying (with generic language) that their findings generalize broadly. Imagine for a second what the completely specific version of one of these titles might look like: “Reading one particular selection of fonzy for fifteen minutes in the lab improved 36 college students’ smoodling scores on a questionnaire.” This paper sounds pretty narrow in its applicability!

    We’ve already run into generalizability in our treatment of statistical estimation and inference. When we estimated a particular quantity (say, the effect of fonzy), we did so in our own sample. But we then used inferential tools to reason about how the estimate in this sample related to the parameter in the population as a whole. How do we link up these statistical tools for generalization to the scientific questions we have about the generalizability of our findings? That’s the question of this chapter.

    A key set of decisions in experiment planning is what population to sample from and how to sample. We’ll start by talking about the basics of sampling theory: different ways of sampling and the generalizations they do and don’t license. The second section of the chapter will then deal with sampling biases that can compromise our effect estimates. A final set of key decisions is about sample size planning. In the third part of the chapter we’ll address this issue, starting with classic power analysis but then introducing several other ways that an experimenter can plan and justify their sample size.

    @@ -741,14 +741,14 @@

    -

    The same dynamic holds with sample size: the same effect size will be easier to detect with a larger sample size than with a small one. Figure 10.8 shows how this relationship works. A large sample size creates a tighter null distribution (right side) by reducing sampling error. A tighter null distribution means you can reject the null more of the time based on the variation in a true effect. If your sample size is too small to detect your effect much of the time, we call this being under-powered.10

    -

    10 You can also refer to a design as over-powered, though we object slightly to this characterization, since the value of large datasets is typically not just to reject the null but also to measure an effect with high precision and to investigate how it is moderated by other characteristics of the sample.

    11 Our focus here is on giving you a conceptual introduction to power analysis, but we refer you to Cohen (1992) for a more detailed introduction.

    Classical power analysis involves computing the sample size \(N\) that’s necessary in order to achieve some level of power, given \(\alpha\) and a known effect size.11 The mathematics of the relationship between \(\alpha\), \(\beta\), \(N\), and effect size have been worked out for a variety of different statistical tests (Cohen 2013) and codified in software like G*Power (Faul et al. 2007) and the pwr package for R (Champely 2020). For other cases (including mixed effects models), you may have to conduct a simulation in which you generate many simulated experimental runs under known assumptions and compute how many of these lead to a significant effect; luckily, R packages exist for this purpose as well, including the simr package (Green and MacLeod 2016).

    +

    The same dynamic holds with sample size: the same effect size will be easier to detect with a larger sample size than with a small one. Figure 10.8 shows how this relationship works. A large sample size creates a tighter null distribution (right side) by reducing sampling error. A tighter null distribution means you can reject the null more of the time based on the variation in a true effect. If your sample size is too small to detect your effect much of the time, we call this being underpowered.10

    +

    10 You can also refer to a design as overpowered, though we object slightly to this characterization, since the value of large datasets is typically not just to reject the null but also to measure an effect with high precision and to investigate how it is moderated by other characteristics of the sample.

    11 Our focus here is on giving you a conceptual introduction to power analysis, but we refer you to Cohen (1992) for a more detailed introduction.

    Classical power analysis involves computing the sample size \(N\) that’s necessary in order to achieve some level of power, given \(\alpha\) and a known effect size.11 The mathematics of the relationship between \(\alpha\), \(\beta\), \(N\), and effect size have been worked out for a variety of different statistical tests (Cohen 2013) and codified in software like G*Power (Faul et al. 2007) and the pwr package for R (Champely 2020). For other cases (including mixed effects models), you may have to conduct a simulation in which you generate many simulated experimental runs under known assumptions and compute how many of these lead to a significant effect; luckily, R packages exist for this purpose as well, including the simr package (Green and MacLeod 2016).

    10.4.2 Power analysis in practice

    Let’s do a power analysis for our hypothetical money and happiness experiment. Imagine the experiment is a simple two-group design in which participants from a convenience population are randomly assigned either to receive $1,000 and some advice on saving money (experimental condition) vs just receiving the advice and no money (control condition). We then follow up a month later and collect self-reported happiness ratings. How many people should we have in our study in order to be able to reject the null? The answer to this question depends on our desired values of \(\alpha\) and \(\beta\) as well as our expected effect size for the intervention.

    For \(\alpha\) we will just set a conventional significance threshold of \(\alpha = 0.05\). But what should be our desired level of power? The usual standard in the social sciences is to aim for power above 80% (e.g., \(\beta < 0.20\)); this gives you four out of five chances to observe a significant effect. But just like \(\alpha = 0.05\), this is a conventional value that is perhaps a little bit too loose for modern standards—a strong test of a particular effect should probably have 90% or 95% power.12

    -

    12 Really, researchers interested in using power analysis in their work should give some thought to what sort of chance of a false negative they are willing to accept. In exploratory research, perhaps a higher chance of missing an effect is reasonable; in contrast, in confirmatory research it might make sense to aim for a higher level of power.

    These choices are relatively easy, compared to the fundamental issue: our power analysis requires some expectation about our effect size. This is the first fundamental problem of power analysis: if you knew the effect size, you might not need to do the experiment!

    +

    12 Really, researchers interested in using power analysis in their work should give some thought to what sort of chance of a false negative they are willing to accept. In exploratory research, perhaps a higher chance of missing an effect is reasonable; in contrast, in confirmatory research it might make sense to aim for a higher level of power.

    These choices are relatively easy, compared to the fundamental issue: our power analysis requires some expectation about our effect size. This is the first fundamental problem of power analysis: if you knew the effect size, you might not need to do the experiment!

    So how are you supposed to get an estimate of effect size? Here are a few possibilities:

    • Meta-analysis. If there is a good meta-analysis of the effect that you are trying to measure (or something closely related), then you are in luck. A strong meta-analysis will have not only a precise effect size estimate but also some diagnostics detecting and correcting potential publication bias in the literature (see chapter 16). While these diagnostics are imperfect, they still can give you a sense for whether you can use the meta-analytic effect size estimate as the basis for a power analysis.

      • @@ -781,7 +781,7 @@

      chapter 6, the null hypothesis significance testing framework is just not set up to let you accept the null hypothesis. If you are interested in a bidirectional approach to hypothesis testing in which you can accept and reject the null, you may need to consider Bayes Factor or equivalence testing approaches (Lakens, Scheel, and Isager 2018), which don’t fit the assumptions of classical power analysis.

      +

      There’s a second issue, however. The second fundamental problem of power analysis is that the real effect size for an experiment may be zero. And in that case, no sample size will let you correctly reject the null. Going back to our discussion in chapter 6, the null hypothesis significance testing framework is just not set up to let you accept the null hypothesis. If you are interested in a bidirectional approach to hypothesis testing in which you can accept and reject the null, you may need to consider Bayes Factor or equivalence testing approaches (Lakens, Scheel, and Isager 2018), which don’t fit the assumptions of classical power analysis.

    10.4.3 Alternative approaches to sample size planning

    diff --git a/011-prereg.html b/011-prereg.html index c6b1c91d..ece09088 100644 --- a/011-prereg.html +++ b/011-prereg.html @@ -574,8 +574,17 @@

    11.1.2 Hypothesizing after results are known

    In addition to degrees of freedom in experimental design and analysis, there is additional flexibility in how researchers interpret research results. As we discussed in chapter 2, theories can accommodate even conflicting results in many different ways—for example, by positing auxiliary hypotheses that explain why a particular datapoint is special.

    -

    The practice of selecting or developing your hypothesis after observing the data has been called “hypothesizing after the results are known,” or “HARKing” (Kerr 1998). HARKing is potentially problematic because it expands the garden of forking paths and helps to justify the use of various additional design and analysis decisions (?fig-grid). For example, you may come up with an explanation for why an intervention is effective in men but not in women in order to justify a post hoc subgroup analysis based on sex (see the Case study box). The extent to which HARKing is problematic is contested (for discussion see Hardwicke and Wagenmakers 2023). But at the very least, it’s important to be honest about whether hypotheses were developed before or after observing the data.

    -

    A grid of individual research results. The horizontal axis provides a simplified illustration of the many justifiable design and analysis choices that a scientist can use to generate the evidence. The vertical axis illustrates that there are often several potential hypotheses that could be constructed or selected when interpreting the evidence. An unconstrained scientist can simultaneously fit evidence to hypotheses and fit hypotheses to evidence in order to obtain their preferred overall result.{#fig-grid .margin-caption width=70% fig-alt=“A diagram of a grid with axes degrees of freedom to fit”evidence to hypotheses” and “hypotheses to evidence”.”}

    +

    The practice of selecting or developing your hypothesis after observing the data has been called “hypothesizing after the results are known,” or “HARKing” (Kerr 1998). HARKing is potentially problematic because it expands the garden of forking paths and helps to justify the use of various additional design and analysis decisions (figure 11.6). For example, you may come up with an explanation for why an intervention is effective in men but not in women in order to justify a post hoc subgroup analysis based on sex (see the Case study box). The extent to which HARKing is problematic is contested (for discussion see Hardwicke and Wagenmakers 2023). But at the very least, it’s important to be honest about whether hypotheses were developed before or after observing the data.

    +
    +
    +
    +A diagram of a grid with axes degrees of freedom to fit "evidence to hypotheses" and "hypotheses to evidence". +
    +
    +Figure 11.6: A grid of individual research results. The horizontal axis provides a simplified illustration of the many justifiable design and analysis choices that a scientist can use to generate the evidence. The vertical axis illustrates that there are often several potential hypotheses that could be constructed or selected when interpreting the evidence. An unconstrained scientist can simultaneously fit evidence to hypotheses and fit hypotheses to evidence in order to obtain their preferred overall result. +
    +
    +

    But hang on a minute! Isn’t it a good thing to seek out interesting results if they are there in the data? Shouldn’t we “let the data speak”? The answer is yes! But it’s crucial to understand the distinction between exploratory and confirmatory modes of research.3 Confirmation involves making research decisions before you’ve seen the data whereas exploration involves making research decisions after you’ve seen data.

    3 In practice, an individual study may contain both exploratory and confirmatory aspects, which is why we describe them as different “modes.”

    The key things to remember about exploratory research are that you need to (1) be aware of the increased risk of bias arising from data-dependent decision-making and calibrate your confidence in the results accordingly; and (2) be honest with other researchers about your analysis strategy so they are also aware of the risk of bias and can calibrate their confidence in the outcomes accordingly. In the next section, we will learn about how preregistration helps us to make this important distinction between exploratory and confirmation research.

    @@ -590,11 +599,11 @@

    -Figure 11.6: Preregistration clarifies where research activities fall on the continuum of prespecification. When the preregistration provides little constraint over researcher degrees of freedom (i.e., more exploratory research), decisions are more likely to be data-dependent, and consequently there is a higher risk of bias. When preregistration provides strong constraint over researcher degrees of freedom (i.e., more confirmatory research), decisions are less likely to be data dependent, and consequently there is a lower risk of bias. Exploratory research activities are more sensitive to serendipitous discovery but also have a higher risk of bias relative to confirmatory research activities. Preregistration transparently communicates where particular results are located along the continuum, helping readers to appropriately calibrate their confidence. +Figure 11.7: Preregistration clarifies where research activities fall on the continuum of prespecification. When the preregistration provides little constraint over researcher degrees of freedom (i.e., more exploratory research), decisions are more likely to be data-dependent, and consequently there is a higher risk of bias. When preregistration provides strong constraint over researcher degrees of freedom (i.e., more confirmatory research), decisions are less likely to be data dependent, and consequently there is a lower risk of bias. Exploratory research activities are more sensitive to serendipitous discovery but also have a higher risk of bias relative to confirmatory research activities. Preregistration transparently communicates where particular results are located along the continuum, helping readers to appropriately calibrate their confidence.
    -

    Preregistration does not require that you specify all research decisions in advance, only that you are transparent about what was planned, and what was not planned. This transparency helps to make a distinction between which aspects of the research were exploratory and which were confirmatory (figure 11.6). All else being equal, we should have more confidence in confirmatory results, because there is a lower risk of bias. Exploratory results have a higher risk of bias, but they are also more sensitive to serendipitous (unexpected) discoveries. So the confirmatory mode is best suited to testing hypotheses, and the exploratory mode is best suited to generating them. Therefore, exploratory and confirmatory research are both valuable activities—it is just important to differentiate them (Tukey 1980)! Preregistration offers the best of both worlds by clearly separating one from the other.

    +

    Preregistration does not require that you specify all research decisions in advance, only that you are transparent about what was planned, and what was not planned. This transparency helps to make a distinction between which aspects of the research were exploratory and which were confirmatory (figure 11.7). All else being equal, we should have more confidence in confirmatory results, because there is a lower risk of bias. Exploratory results have a higher risk of bias, but they are also more sensitive to serendipitous (unexpected) discoveries. So the confirmatory mode is best suited to testing hypotheses, and the exploratory mode is best suited to generating them. Therefore, exploratory and confirmatory research are both valuable activities—it is just important to differentiate them (Tukey 1980)! Preregistration offers the best of both worlds by clearly separating one from the other.

    In addition to the benefits described above, preregistration may improve the quality of research by encouraging closer attention to study planning. We’ve found that the process of writing a preregistration really helps facilitate communication between collaborators, and can catch addressable problems before time and resources are wasted on a poorly designed study. Detailed advanced planning can also create opportunities for useful community feedback, particularly in the context of registered reports (see the Depth box below), where dedicated peer reviewers will evaluate your study before it has even begun.

    -Figure 11.7: Registered reports (from https://www.cos.io/initiatives/registered-reports, licensed under CC BY 4.0). +Figure 11.8: Registered reports (from https://www.cos.io/initiatives/registered-reports, licensed under CC BY 4.0).
    -

    Registered reports. Registered reports (figure 11.7) are a type of article format that embeds preregistration directly into the publication pipeline (Chambers and Tzavella 2020). The idea is that you submit your preregistered protocol to a journal and it is peer reviewed before you’ve even started your study. If the study is approved, the journal agrees to publish it, regardless of the results. This is a radical departure from traditional publication models where peer reviewers and journals evaluate your study after its been completed and the results are known. Because the study is accepted for publication independently of the results, registered reports can offer the benefits of preregistration with additional protection against publication bias. They also provide a great opportunity to obtain feedback on your study design while you can still change it!

    +

    Registered reports. Registered reports (figure 11.8) are a type of article format that embeds preregistration directly into the publication pipeline (Chambers and Tzavella 2020). The idea is that you submit your preregistered protocol to a journal and it is peer reviewed before you’ve even started your study. If the study is approved, the journal agrees to publish it, regardless of the results. This is a radical departure from traditional publication models where peer reviewers and journals evaluate your study after its been completed and the results are known. Because the study is accepted for publication independently of the results, registered reports can offer the benefits of preregistration with additional protection against publication bias. They also provide a great opportunity to obtain feedback on your study design while you can still change it!

    diff --git a/012-collection.html b/012-collection.html index 226434dd..50054ce9 100644 --- a/012-collection.html +++ b/012-collection.html @@ -665,8 +665,8 @@

    14

    -

    14 We mean especially when deploying a new experimental paradigm or when collecting data from a new population. Once you have run many studies with a similar procedure and similar sample, extensive piloting is less important. Any time you change something, it’s always good to run one or two pilots, though, just to check that you didn’t inadvertently mess up your experiment.

    15 In a pinch you can even run yourself through the experiment a bunch of times (though this isn’t preferable because you’re likely to miss a lot of aspects of the experience that you are habituated to, especially if you’ve been debugging the experiment already).

    The first pilot, which we call your non-naive participant pilot, can make use of participants who know the goals of the experiment and understand the experimental manipulation—this could be a friend, collaborator, colleague, or family member.15 The goal of this pilot study is to ensure that your experiment is comprehensible, that participants can complete it, and that the data are logged appropriately. You must analyze the data from the non-naive pilot, at least to the point of checking that the relevant data about each trial is logged.

    -

    The second pilot, your naive participant pilot, should consist of a test of a small set of participants recruited via the channel you plan to use for your main study. The number of participants you should pilot depends on the cost of the experiment in time, money, and opportunity as well as its novelty. A brand new paradigm is likely more prone to error than a tried and tested paradigm. For a short online survey-style experiment, a pilot of 10–20 people is reasonable. A more time-consuming laboratory study might require piloting just two or three people.16

    +

    14 We mean especially when deploying a new experimental paradigm or when collecting data from a new population. Once you have run many studies with a similar procedure and similar sample, extensive piloting is less important. Any time you change something, it’s always good to run one or two pilots, though, just to check that you didn’t inadvertently mess up your experiment.

    15 In a pinch you can even run yourself through the experiment a bunch of times (though this isn’t preferable because you’re likely to miss a lot of aspects of the experience that you are habituated to, especially if you’ve been debugging the experiment already).

    The first pilot, which we call your non-naive participant pilot, can make use of participants who know the goals of the experiment and understand the experimental manipulation—this could be a friend, collaborator, colleague, or family member.15 The goal of this pilot study is to ensure that your experiment is comprehensible, that participants can complete it, and that the data are logged appropriately. You must analyze the data from the non-naive pilot, at least to the point of checking that the relevant data about each trial is logged.

    +

    The second pilot, your naive participant pilot, should consist of a test of a small set of participants recruited via the channel you plan to use for your main study. The number of participants you should pilot depends on the cost of the experiment in time, money, and opportunity as well as its novelty. A brand new paradigm is likely more prone to error than a tried and tested paradigm. For a short online survey-style experiment, a pilot of 10–20 people is reasonable. A more time-consuming laboratory study might require piloting just two or three people.16

    16 In the case of especially expensive experiments, it can be a dilemma whether to run a larger pilot to identify difficulties since such a pilot will be costly. In these cases, one possibility is to plan to include the pilot participants in the main dataset if no major procedural changes are required. In this case, it is helpful to preregister a contingent testing strategy to avoid introducing data-dependent bias (see chapter 11). For example, in a planned sample of 100 participants, you could preregister running 20 as a pilot sample with the stipulation that you will look only at their dropout rate—and not at any condition differences. Then the preregistration can state that, if the dropout rate is lower than 25%, you will collect the next 80 participants and analyze the whole dataset, including the initial pilot, but if dropout rate is higher than 25%, you will discard the pilot sample and make changes. This kind of strategy can help you split the difference between cautious piloting and conservation of rare or costly data.

    The goal of the naive pilot study is to understand properties of the participant experience. Were participants confused? Did they withdraw before the study finished? Even a small number of pilots can tell you that your dropout rate is likely too high: for example, if five of ten pilot participants withdraw, you likely need to reconsider aspects of your design. It’s critical for your naive participant pilot that you debrief more extensively with your participants. This debriefing often takes the form of an interview questionnaire after the study is over. “What did you think the study was about?” and “Is there any way we could improve the experience of being in the study?” can be helpful questions. Often this debriefing is more effective if it is interactive, so even if you are running an online study you may want to find some way to chat with your participants.

    Piloting—especially piloting with naive participants to optimize the participant experience—is typically an iterative process. We frequently launch an experiment for a naive pilot, then recognize from the data or from participant feedback that the experience can be improved. We make tweaks and pilot again. Be careful not to over-fit to small differences in pilot data, however. Piloting should be more like workshopping a manuscript to remove typos than doing statistical analysis. If someone has trouble understanding a particular sentence—whether in your manuscript or in your experiment instructions—you should edit to make it clearer!

    diff --git a/013-management.html b/013-management.html index 171139a6..f1e995e5 100644 --- a/013-management.html +++ b/013-management.html @@ -505,8 +505,17 @@

    13.1.2 Versioning

    Probably everyone who has ever collaborated electronically has experienced the frustration of editing a document, only to find out that you are editing the wrong version—perhaps some of the problems you are working on have already been corrected, or perhaps the section you are adding has already been written by someone else. A second common source of frustration comes when you take a wrong turn in a project, perhaps by reorganizing a manuscript in a way that doesn’t work or refactoring code in a way that turns out to be short-sighted.

    These two problems are solved by modern version control systems. Here we focus on the use of Git, which is the most widely used version control system. Git is a great general solution for version control, but many people—including several of us—don’t love it for collaborative manuscript writing. We’ll introduce Git and its principles here, while noting that online collaboration tools like Google Docs and Overleaf7 can be easier for writing prose (as opposed to code); we cover this topic in a bit more depth in chapter 14.

    -

    7 Overleaf is actually supported by Git on the backend!

    A visualization of Git version control showing a series of commits (circles) on three different branches: the main branch (green) and two others (blue and red). Branches can be created and then merged back into the main branch.{#fig-management-git .column-margin fig-alt=“A diagram of connected circles where”your work” and “someone else’s work” branch off of “main branch” then merge back in.”}

    -

    Git is a tool for creating and managing projects, which are called repositories. A Git repository is a directory whose revision history is tracked via a series of commits—snapshots of the state of the project. These commits can form a tree with different branches, as when two contributors to the project are working on two different parts simultaneously (?fig-management-git). These branches can later be merged either automatically or via manual intervention in the case of conflicting changes.

    +

    7 Overleaf is actually supported by Git on the backend!

    +
    +
    +A diagram of connected circles where "your work" and "someone else's work" branch off of "main branch" then merge back in. +
    +
    +Figure 13.4: A visualization of Git version control showing a series of commits (circles) on three different branches: the main branch (green) and two others (blue and red). Branches can be created and then merged back into the main branch. +
    +
    +
    +

    Git is a tool for creating and managing projects, which are called repositories. A Git repository is a directory whose revision history is tracked via a series of commits—snapshots of the state of the project. These commits can form a tree with different branches, as when two contributors to the project are working on two different parts simultaneously (figure 13.4). These branches can later be merged either automatically or via manual intervention in the case of conflicting changes.

    Commonly, Git repositories are hosted by an online service like GitHub to facilitate collaboration. With this workflow, a user makes changes to a local version of the repository on their own computer and pushes those changes to the online repository. Another user can then pull those changes from the online repository to their own local version. The online “origin” copy is always the definitive copy of the project, and a record is kept of all changes. Chapter B provides a practical introduction to Git and GitHub, and there are a variety of good tutorials available online and in print (Blischak, Davenport, and Wilson 2016).

    Collaboration using version control tools is designed to solve many of the problems we’ve been discussing:

      @@ -516,7 +525,7 @@

      A project’s commit history is labeled with each commit’s author and date, facilitating record-keeping and collaboration.
    • Automatic merging can allow synchronous editing of different parts of a manuscript or codebase.9

    -

    8 In 48BC, Julius Caesar accidentally burned down part of the Great Library of Alexandria where the sole copies of many valuable ancient works were stored. To this day, many scientists have apparently retained the habit of storing single copies of important information in vulnerable locations. Even in the age of cloud computing, hard drive failure is a surprisingly common source of problems!

    9 Version control isn’t magic, and if you and a collaborator edit the same paragraph or function, you will likely have to merge your changes by hand. But Git will at least show you where the conflict is!

    Organizing a project repository for collaboration and hosting on a remote platform is an important first step toward sharing! Many of our projects (like this book) are actually born open: we do all of our work on a publicly hosted repository for everyone to see (Rouder 2015). This philosophy of “working in the open” encourages good organization practices from the beginning. It can feel uncomfortable at first, but this discomfort soon vanishes as you realize that basically no one is looking at your in-progress project.

    +

    8 In 48BC, Julius Caesar accidentally burned down part of the Great Library of Alexandria where the sole copies of many valuable ancient works were stored. To this day, many scientists have apparently retained the habit of storing single copies of important information in vulnerable locations. Even in the age of cloud computing, hard drive failure is a surprisingly common source of problems!

    9 Version control isn’t magic, and if you and a collaborator edit the same paragraph or function, you will likely have to merge your changes by hand. But Git will at least show you where the conflict is!

    Organizing a project repository for collaboration and hosting on a remote platform is an important first step toward sharing! Many of our projects (like this book) are actually born open: we do all of our work on a publicly hosted repository for everyone to see (Rouder 2015). This philosophy of “working in the open” encourages good organization practices from the beginning. It can feel uncomfortable at first, but this discomfort soon vanishes as you realize that basically no one is looking at your in-progress project.

    One concern that many people raise about sharing in-progress research openly is the possibility of “scooping”—that is, other researchers getting an idea or even data from the repository and writing a paper before you do. We have two responses to this concern. First, the empirical frequency of this sort of scooping is difficult to determine but likely very low—we don’t know of any documented cases. Mostly, the problem is getting people to care about your experiment at all, not people caring so much that they would publish using your data or materials! In Gary King’s words (King and Shieber 2013), “The thing that matters the least is being scooped. The thing that matters the most is being ignored.” On the other hand, if you are in an area of research that you perceive to be competitive, or where there is some significant risk of this kind of shenanigans, it’s very easy to keep part, or all, of a repository private among your collaborators until you are ready to share more widely. All of the benefits we described still accrue. For an appropriately organized and hosted project, often the only steps required to share materials, data, and code are (1) to make the hosted repository public and (2) to link it to an archival storage platform like the Open Science Framework.

    @@ -554,7 +563,7 @@

    13.2.3 Organize your data for later analysis: Spreadsheets

    Data come in many forms, but chances are that at some point during your project you will end up with a spreadsheet full of information. Well-organized spreadsheets can mean the difference between project success and failure! A wonderful article by Broman and Woo (2018) lays out principles of good spreadsheet design. We highlight some of their principles here (with our own, opinionated ordering):

      -
    1. Make it a rectangle.18 Nearly all data analysis software, like SPSS, Stata, Jamovi, and JASP (and many R packages), require data to be in a tabular format.19 If you are used to analyzing data exclusively in a spreadsheet, this kind of tabular data isn’t quite as readable, but readable formatting gets in the way of almost any analysis you want to do. Figure 13.4 gives some examples of nonrectangular spreadsheets. All of these will cause any analytic package to choke because of inconsistencies in how rows and columns are used!
      2. +
    2. Make it a rectangle.18 Nearly all data analysis software, like SPSS, Stata, Jamovi, and JASP (and many R packages), require data to be in a tabular format.19 If you are used to analyzing data exclusively in a spreadsheet, this kind of tabular data isn’t quite as readable, but readable formatting gets in the way of almost any analysis you want to do. Figure 13.5 gives some examples of nonrectangular spreadsheets. All of these will cause any analytic package to choke because of inconsistencies in how rows and columns are used!

    18 Think of your data like a well-ordered plate of sushi, neatly packed together without any gaps.

    19 Tabular data is a precursor to “tidy” data, which we describe in more detail in appendix D.

    @@ -562,7 +571,7 @@

    -Figure 13.4: Examples of non-rectangular spreadsheet formats that are likely to cause problems in analysis. Adapted from Broman and Woo (2018). +Figure 13.5: Examples of non-rectangular spreadsheet formats that are likely to cause problems in analysis. Adapted from Broman and Woo (2018).
    @@ -645,14 +654,14 @@

    13.2.5 Document the format of your data

    -

    Even the best-organized tabular data are not always easy to understand by other researchers, or even yourself, especially after some time has passed. For that reason, you should make a codebook (also known as a data dictionary) that explicitly documents what each variable is. Figure 13.6 shows an example codebook for the trial-level data in the bottom of figure 13.5. Each row represents one variable in the associated dataset. Codebooks often describe what type of variable a column is (e.g., numeric, string), and what values can appear in that column. A human-readable explanation is often given as well, providing units (e.g., “seconds”) and a translation of numeric codes (e.g., “test condition is coded as 1”) where relevant.

    +

    Even the best-organized tabular data are not always easy to understand by other researchers, or even yourself, especially after some time has passed. For that reason, you should make a codebook (also known as a data dictionary) that explicitly documents what each variable is. Figure 13.7 shows an example codebook for the trial-level data in the bottom of figure 13.6. Each row represents one variable in the associated dataset. Codebooks often describe what type of variable a column is (e.g., numeric, string), and what values can appear in that column. A human-readable explanation is often given as well, providing units (e.g., “seconds”) and a translation of numeric codes (e.g., “test condition is coded as 1”) where relevant.

    2 spreadsheets: each row has one participant with ID, age, etc.; each row has one trial with trial number, looking time, etc.
    -Figure 13.5: Example participant (top) and trial (bottom) level data from the ManyBabies (2020) case study. +Figure 13.6: Example participant (top) and trial (bottom) level data from the ManyBabies (2020) case study.
    @@ -662,7 +671,7 @@

    -Figure 13.6: Codebook for trial-level data (see above) from the ManyBabies (2020) case study. +Figure 13.7: Codebook for trial-level data (see above) from the ManyBabies (2020) case study.
    @@ -678,7 +687,7 @@

    (Ross, Iguchi, and Panicker 2018) and needs to be balanced against the ethical imperatives to share (see chapter 4).23

    23 Meyer (2018) gives an excellent overview of how to navigate various legal and ethical issues around data sharing in the US context.

    Furthermore, there are legal regulations that protect participants’ data, though these vary from country to country. In the US, the relevant regulation is HIPAA, the Health Insurance Portability and Accountability Act, which limits disclosures of private health information (PHI). In the European Union, the relevant regulation is the European GDPR (General Data Protection Regulation). It’s beyond the scope of this book to give a full treatment of these regulatory frameworks; you should consult with your local ethics board regarding compliance, but here is the way we have navigated this situation while still sharing data.

    Under both frameworks, anonymization (or equivalently de-identification) of data is a key concept, such that data sharing is generally just fine if the data meet the relevant standard. Under US guidelines, researchers can follow the “safe harbor” standard24 under which data are considered to be anonymized if they do not contain identifiers like names, telephone numbers, email addresses, social security numbers, dates of birth, faces, and others. Thus, data that only contain participant IDs and nothing from this list can typically be shared without participant consent without a problem.25

    -

    25 US IRBs are a very decentralized bunch, and their interpretations often vary considerably. For reasons of liability or ethics, they may not allow data sharing even though it is permitted by US law. If you feel like arguing with an IRB that takes this kind of stand, you could mention that the DHHS rule actually doesn’t consider de-identified data to be “human subjects” data at all, and thus the IRB may not have regulatory authority over it. We’re not lawyers, and we’re not sure if you’ll succeed, but it could be worth a try.

    The EU’s GDPR also allows fully anonymized data sharing, with one big complication. Putting anonymous identifiers in a data file and removing identifiable fields does not itself suffice for GDPR anonymization if the data are still in principle reidentifiable because you have maintained documentation linking IDs to identifiable data like names or email addresses. Only when the key linking identifiers to data has been destroyed are the data truly de-identified according to this standard.

    +

    25 US IRBs are a very decentralized bunch, and their interpretations often vary considerably. For reasons of liability or ethics, they may not allow data sharing even though it is permitted by US law. If you feel like arguing with an IRB that takes this kind of stand, you could mention that the DHHS rule actually doesn’t consider de-identified data to be “human subjects” data at all, and thus the IRB may not have regulatory authority over it. We’re not lawyers, and we’re not sure if you’ll succeed, but it could be worth a try.

    The EU’s GDPR also allows fully anonymized data sharing, with one big complication. Putting anonymous identifiers in a data file and removing identifiable fields does not itself suffice for GDPR anonymization if the data are still in principle reidentifiable because you have maintained documentation linking IDs to identifiable data like names or email addresses. Only when the key linking identifiers to data has been destroyed are the data truly de-identified according to this standard.

    -

    De-identification is not always enough. As datasets get richer, statistical reidentification risks go up substantially such that, with a little bit of outside information, data can be matched with a unique individual. These risks are especially high with linguistic, physiological, and geospatial data, but they can be present even for simple behavioral experiments. In one influential demonstration, knowing a person’s location on two occasions was often enough to identify their data uniquely in a huge database of credit card transactions (De Montjoye et al. 2015).26 Thus, simply removing fields from the data is a good starting point—but if you are collecting richer data about participants’ behavior you may need to consult an expert.

    -

    26 For an example closer to home, many of the contributing labs in the ManyBabies project logged the date of test for each participant. This useful and seemingly innocuous piece of information is unlikely to identify any particular participant—but alongside a social media post about a lab visit or a dataset about travel records, it could easily reveal a particular participant’s identity.

    Privacy issues are ubiquitous in data sharing, and almost every experimental research project will need to solve them before sharing data. For simple projects, often these are the only issues that preclude data sharing. However, in more complex projects, other concerns can arise. Funders may have specific mandates regarding where your data should be shared. Data use agreements or collaborator preferences may restrict where and when you can share. And certain data types require much more sensitivity since they are more consequential than, say, the reaction times on a Stroop task. We include here a set of questions to walk through to plan your sharing (figure 13.7). When in doubt, it’s often a good idea to consult with the relevant local authority—for example, your ethics board for ethical issues or your research management office for regulatory issues.

    +

    De-identification is not always enough. As datasets get richer, statistical reidentification risks go up substantially such that, with a little bit of outside information, data can be matched with a unique individual. These risks are especially high with linguistic, physiological, and geospatial data, but they can be present even for simple behavioral experiments. In one influential demonstration, knowing a person’s location on two occasions was often enough to identify their data uniquely in a huge database of credit card transactions (De Montjoye et al. 2015).26 Thus, simply removing fields from the data is a good starting point—but if you are collecting richer data about participants’ behavior you may need to consult an expert.

    +

    26 For an example closer to home, many of the contributing labs in the ManyBabies project logged the date of test for each participant. This useful and seemingly innocuous piece of information is unlikely to identify any particular participant—but alongside a social media post about a lab visit or a dataset about travel records, it could easily reveal a particular participant’s identity.

    Privacy issues are ubiquitous in data sharing, and almost every experimental research project will need to solve them before sharing data. For simple projects, often these are the only issues that preclude data sharing. However, in more complex projects, other concerns can arise. Funders may have specific mandates regarding where your data should be shared. Data use agreements or collaborator preferences may restrict where and when you can share. And certain data types require much more sensitivity since they are more consequential than, say, the reaction times on a Stroop task. We include here a set of questions to walk through to plan your sharing (figure 13.8). When in doubt, it’s often a good idea to consult with the relevant local authority—for example, your ethics board for ethical issues or your research management office for regulatory issues.

    A flowchart asking can/must you share; what to share; when to share; how to share.
    -Figure 13.7: A decision chart for thinking about sharing research products. Adapted from Klein et al. (2018). +Figure 13.8: A decision chart for thinking about sharing research products. Adapted from Klein et al. (2018).
    diff --git a/014-writing.html b/014-writing.html index 52a8230e..1fa89e33 100644 --- a/014-writing.html +++ b/014-writing.html @@ -528,7 +528,7 @@

    6 to final product—should be present in the project repository. By consulting the repository documentation, a reader should be able to follow the steps to go from the raw data to the final manuscript, including tables and figures.

  • Script everything. Try to ensure that each step of the analysis pipeline is executed by computer code rather than manual actions such as copying and pasting or directly editing spreadsheets. This practice ensures that every step is documented via executable code rather than ambiguous description, ensuring that it can be reproduced. Imagine, for example, that you decided to recode a variable in your dataset. You could use the “find and replace” function in Excel, but this action would not be documented—you might even forget that you did it! A better option would be to write an R script. While a scripted pipeline can be a pain to set up the first time, by the third time you rerun it, it will save you time.

  • Use literate programming. The meaning of a chunk of computer code is not always obvious to another user, especially if they’re not an expert. Indeed, we frequently look at our own code and scratch our heads, wondering what on earth it’s doing. To avoid this problem, try to structure your code around plain language comments that explain what it should be doing, a technique known as “literate programming” (Knuth 1992).

    • -

  • Use defensive programming. Errors can still occur in scripted analyses. Defensive programming is a series of strategies to help anticipate, detect, and avoid errors in advance. A typical defensive programming tool is the inclusion of tests in your code, snippets that check if the code or data meet some assumptions. For example, you might test if a variable storing reaction times has taken on values below zero (which should be impossible). If the test passes, the analysis pipeline continues; if the test fails, the pipeline halts and an error message appears to alert you to the problem.

    • +

  • Use defensive programming. Errors can still occur in scripted analyses. Defensive programming is a series of strategies to help anticipate, detect, and avoid errors in advance. A typical defensive programming tool is the inclusion of tests in your code, snippets that check if the code or data meet some assumptions. For example, you might test if a variable storing reaction times has taken on values below zero (which should be impossible). If the test passes, the analysis pipeline continues; if the test fails, the pipeline halts and an error message appears to alert you to the problem.

  • Use free/open-source software and programming languages. If possible, avoid using commercial software, like SPSS or Matlab, and instead use free, open-source software and programming languages, like JASP, Jamovi, R, or Python. This practice will make it easier for others to access, reuse, and verify your work—including yourself!7

  • Use version control. In chapter 13, we introduced the benefits of version control—a great way to save your analysis pipeline incrementally as you build it, allowing you to roll back to a previous version if you accidentally introduce errors.

  • Preserve the computational environment. Even if your analysis pipeline is entirely reproducible on your own computer, you still need to consider whether it will run on somebody else’s computer, or even your own computer after software updates. You can address this issue by documenting and preserving the computational environment in which the analysis pipeline runs successfully. Various tools are available to help with this, including Docker, Code Ocean, renv (for R), and pip (for Python).8

    • @@ -538,8 +538,8 @@

    14.3.3 The reproducibility-collaboration trade-off

    We would love to leave it there and watch you walk off into the sunset with a spring in your step and a reproducible report under your arm. Unfortunately, we have to admit that writing reproducibly can create a few practical difficulties when it comes to collaboration.

    A major aspect of collaboration is exchanging comments and inline text edits with coauthors. You can do this exchange with R Markdown files and Git, but these tools are not as user-friendly as, say, Word or Google Docs, and some collaborators will be completely unfamiliar with them. Most journals also expect articles to be submitted as Word documents. Outputting R Markdown files to Word can often introduce formatting issues, especially for moderately complex tables. So, until more user-friendly tools are introduced, some compromise between reproducibility and collaboration may be necessary. Here are two workflow styles for you to consider.

    -

    First, the maximal reproducibility approach. If your collaborators are familiar with R Markdown and you don’t mind exchanging comments and edits via Git—or if they don’t mind giving you lists of comments and changes that you implement in the R Markdown document—then you can maintain a fully reproducible workflow for your project. The journal submission and publication process may still introduce some issues, such as incorporating changes made by the copy editor, but at least your submitted manuscript (and the preprint you have hopefully posted) will be fully reproducible.

    -

    Second, the two worlds approach. This workflow is a bit clunky, but it facilitates collaboration and maintains reproducibility. First, write your results section in R Markdown and generate a Word document (see appendix C). Then, write the remainder of the manuscript in Word, including incorporating comments and changes from collaborators. When you have a final version, copy and paste the abstract, introduction, methods, and discussion into the R Markdown document.9 Integrating any changes made to the results section back into the R Markdown requires a bit more effort, either using manual checking or Word’s “compare documents” feature.10 The advantage of this approach is that you have a reproducible document and your collaborators have not had to deviate from their preferred workflow. Unfortunately, it requires more effort from you and is slightly more error-prone than the maximal reproducibility approach.

    +

    First, the maximal reproducibility approach. If your collaborators are familiar with R Markdown and you don’t mind exchanging comments and edits via Git—or if they don’t mind giving you lists of comments and changes that you implement in the R Markdown document—then you can maintain a fully reproducible workflow for your project. The journal submission and publication process may still introduce some issues, such as incorporating changes made by the copy editor, but at least your submitted manuscript (and the preprint you have hopefully posted) will be fully reproducible.

    +

    Second, the two worlds approach. This workflow is a bit clunky, but it facilitates collaboration and maintains reproducibility. First, write your results section in R Markdown and generate a Word document (see appendix C). Then, write the remainder of the manuscript in Word, including incorporating comments and changes from collaborators. When you have a final version, copy and paste the abstract, introduction, methods, and discussion into the R Markdown document.9 Integrating any changes made to the results section back into the R Markdown requires a bit more effort, either using manual checking or Word’s “compare documents” feature.10 The advantage of this approach is that you have a reproducible document and your collaborators have not had to deviate from their preferred workflow. Unfortunately, it requires more effort from you and is slightly more error-prone than the maximal reproducibility approach.

    9 You can also incorporate Google Docs into this workflow—we find that cloud platforms like Docs are especially useful when gathering comments from multiple collaborators on the same document. Unfortunately, you cannot generate a Google Doc from R Markdown, so you will need to import and convert or else copy and paste.

    10 Packages such as trackdown (Kothe et al. 2021) could help as well (https://claudiozandonella.github.io/trackdown).

    @@ -560,7 +560,7 @@

    12 Should you just make your claims more modest and avoid writing about your study’s limitations? The balance between claims and limitations is tricky. One way to navigate this issue is to ask yourself, “Is it OK to say X in the abstract of my article, if I later go on to say state a limitation relevant to that claim, or will the reader feel tricked?”

    14.4.2 Responsible handling of errors

    -

    It is not your responsibility to never make mistakes. But it is your responsibility to respond to errors in a timely, transparent, and professional manner (Bishop 2018).13 Regardless of how the error was identified (e.g., by yourself or by a reader), we recommend contacting the journal and requesting that they publish a correction statement (sometimes called an erratum). Several of us have corrected papers in the past. If the error is serious and cannot be fixed, you should consider retracting the article.

    +

    It is not your responsibility to never make mistakes. But it is your responsibility to respond to errors in a timely, transparent, and professional manner (Bishop 2018).13 Regardless of how the error was identified (e.g., by yourself or by a reader), we recommend contacting the journal and requesting that they publish a correction statement (sometimes called an erratum). Several of us have corrected papers in the past. If the error is serious and cannot be fixed, you should consider retracting the article.

    13 As jazz musician Miles Davis once said, “If you hit a wrong note, it’s the next note that you play that determines if it’s good or bad.”

    A correction/retraction statement should include the following information:

    1. Acknowledge the error. Be clear that an error has occurred.
      2. diff --git a/015-viz.html b/015-viz.html index cdca5c5e..55ff1b07 100644 --- a/015-viz.html +++ b/015-viz.html @@ -552,7 +552,7 @@

      Mapping a pa

      15.1 Basic principles of (confirmatory) visualization

      In this section, we begin by introducing a few simple guidelines to keep in mind when making informative visualizations in the context of experimental psychology.1 Remember that our needs may be distinct from other fields, such as journalism or public policy. You may have seen beautiful and engaging full-page graphics with small print and a wealth of information. The art of designing and producing these graphics is typically known as infoviz and should be distinguished from what we call statistical visualization (Gelman and Unwin 2013).

      -

      1 For the purposes of understanding the examples in this chapter, it should be sufficient to work through the tutorials on data manipulation and visualization in appendix D and appendix E.

      Roughly, infoviz aims to construct rich and immersive worlds to visually explore: a reader can spend hours pouring over the most intricate graphics and continue to find new and intriguing patterns. Statistical visualization, on the other hand, aims to crisply convey the logic of a specific inference at a glance. These visualizations are the production-ready figures that anchor the results section of a paper and accompany the key, preregistered analyses of interest. In this section, we review several basic principles of making statistical visualizations. We then return below to the role of visualization in more exploratory analyses.

      +

      1 For the purposes of understanding the examples in this chapter, it should be sufficient to work through the tutorials on data manipulation and visualization in appendix D and appendix E.

      Roughly, infoviz aims to construct rich and immersive worlds to visually explore: a reader can spend hours pouring over the most intricate graphics and continue to find new and intriguing patterns. Statistical visualization, on the other hand, aims to crisply convey the logic of a specific inference at a glance. These visualizations are the production-ready figures that anchor the results section of a paper and accompany the key, preregistered analyses of interest. In this section, we review several basic principles of making statistical visualizations. We then return below to the role of visualization in more exploratory analyses.

      15.1.1 Principle 1: Show the design

      There are so many different kinds of graphs (bar graphs, line graphs, scatter plots, and pie charts) and so many different possible attributes of those graphs (colors, sizes, line types). How do we begin to decide how to navigate these decisions? The first principle guiding good statistical visualizations is to show the design of your experiment.

      diff --git a/016-meta.html b/016-meta.html index 240b50ac..34b55061 100644 --- a/016-meta.html +++ b/016-meta.html @@ -591,7 +591,7 @@

      figure 16.1,5 the larger squares correspond to more precise studies; notice how much narrower their confidence intervals are than the confidence intervals of less precise studies.

      5 You can ignore for now the final line, “RE Model”; we will return to this later.

      -
      +
      A forest plot with estimated standardized mean difference points and 95% confidence interval error bars for each study. @@ -774,7 +774,7 @@

      10 Evidence is mixed regarding whether including gray literature actually reduces across-study biases in meta-analysis (Tsuji et al. 2020; Mathur and VanderWeele 2021), but it is still common practice to try to include this literature.

      There are also statistical methods to help assess how robust the results may be to across-study biases. Among the most popular tools to assess and correct for publication bias is the funnel plot (Duval and Tweedie 2000; Egger et al. 1997). A funnel plot shows the relationship between studies’ effect estimates and their precision (usually their standard error). These plots are called “funnel plots” because, if there is no publication bias, then as precision increases, the effects “funnel” toward the meta-analytic estimate. As the precision is smaller, they spread out more because of greater measurement error. Figure 16.4] is an example of one type of funnel plot (Mathur and VanderWeele 2020b) for a simulated meta-analysis of 100 studies with no publication bias.

      -
      +
      A plot showing a funnel-shaped distribution of observed effect sizes against their corresponding standard errors. @@ -808,7 +808,7 @@

      -
      +
      A plot showing an inverted funnel-shaped distribution of observed effect sizes against their corresponding standard errors. @@ -821,10 +821,10 @@

      11

      -

      11 Classic funnel plots look more like figure 16.5). Our version is different in a couple of ways. Most prominently, we don’t have the vertical axis reversed (which we think is confusing). We also don’t have the left boundary highlighted, because we think folks don’t typically select for negative studies.

      Not all funnel plots are symmetric! figure 16.6 is what happens to our hypothetical meta-analysis if all studies with \(p<0.05\) and positive estimates are published, but only 10% of studies with \(p \ge 0.05\) or with negative estimates are published. The introduction of publication bias dramatically inflates the pooled estimate from 0.34 to 1.15. Also, there appears to be a correlation between studies’ estimates and their standard errors, such that smaller studies tend to have larger estimates than do larger studies. This correlation is often called funnel plot asymmetry because the funnel plot starts to look like a right triangle rather than a funnel. Funnel plot asymmetry can be a diagnostic for publication bias, though it isn’t always a perfect indicator, as we’ll see in the next subsection.

      +

      11 Classic funnel plots look more like figure 16.5). Our version is different in a couple of ways. Most prominently, we don’t have the vertical axis reversed (which we think is confusing). We also don’t have the left boundary highlighted, because we think folks don’t typically select for negative studies.

      Not all funnel plots are symmetric! figure 16.6 is what happens to our hypothetical meta-analysis if all studies with \(p<0.05\) and positive estimates are published, but only 10% of studies with \(p \ge 0.05\) or with negative estimates are published. The introduction of publication bias dramatically inflates the pooled estimate from 0.34 to 1.15. Also, there appears to be a correlation between studies’ estimates and their standard errors, such that smaller studies tend to have larger estimates than do larger studies. This correlation is often called funnel plot asymmetry because the funnel plot starts to look like a right triangle rather than a funnel. Funnel plot asymmetry can be a diagnostic for publication bias, though it isn’t always a perfect indicator, as we’ll see in the next subsection.

      -
      +
      A plot showing a positively sloped linear distribution of observed effect sizes against their corresponding standard errors. diff --git a/017-conclusion.html b/017-conclusion.html index 103c2922..c3d231b0 100644 --- a/017-conclusion.html +++ b/017-conclusion.html @@ -445,7 +445,6 @@

      17.3 Final thoughts

      Doing experiments is a craft, one that requires practice and attention. The first experiment you run will have limitations and issues. So will the 100th. But as you refine your skills, the quality of the studies you design will get better. Further, your own ability to judge others’ experiments will improve as well, making you a more discerning consumer of empirical results. We hope you enjoy this journey!

      -

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