[citation needed]

There’s a nice post on Science-Based Medicine by Harriet Hall pushing back (kind of) against the increasingly popular idea that antidepressants don’t work. For context, there have been a couple of large recent meta-analyses that used comprehensive FDA data on clinical trials of antidepressants (rather than only published studies, which are biased towards larger, statistically significant, effects) to argue that antidepressants are of little or no use in mild or moderately-depressed people, and achieve a clinically meaningful benefit only in the severely depressed.

Hall points out that whether you think antidepressants have a clinically meaningful benefit or not depends on how you define clinically meaningful (okay, this sounds vacuous, but bear with me). Most meta-analyses of antidepressant efficacy reveal an effect size of somewhere between 0.3 and 0.5 standard deviations. Historically, psychologists consider effect sizes of 0.2, 0.5, and 0.8 standard deviations to be small, medium, and large, respectively. But as Hall points out:

The psychologist who proposed these landmarks [Jacob Cohen] admitted that he had picked them arbitrarily and that they had “no more reliable a basis than my own intuition.” Later, without providing any justification, the UK’s National Institute for Health and Clinical Excellence (NICE) decided to turn the 0.5 landmark (why not the 0.2 or the 0.8 value?) into a one-size-fits-all cut-off for clinical significance.

She goes on to explain why this ultimately leaves the efficacy of antidepressants open to interpretation:

In an editorial published in the British Medical Journal (BMJ), Turner explains with an elegant metaphor: journal articles had sold us a glass of juice advertised to contain 0.41 liters (0.41 being the effect size Turner, et al. derived from the journal articles); but the truth was that the “glass” of efficacy contained only 0.31 liters. Because these amounts were lower than the (arbitrary) 0.5 liter cut-off, NICE standards (and Kirsch) consider the glass to be empty. Turner correctly concludes that the glass is far from full, but it is also far from empty. He also points out that patients’ responses are not all-or-none and that partial responses can be meaningful.

I think this pretty much hits the nail on the head; no one really doubts that antidepressants work at this point; the question is whether they work well enough to justify their side effects and the social and economic costs they impose. I don’t have much to add to Hall’s argument, except that I think she doesn’t sufficiently emphasize how big a role scale plays when trying to evaluate the utility of antidepressants (or any other treatment). At the level of a single individual, a change of one-third of a standard deviation may not seem very big (then again, if you’re currently depressed, it might!). But on a societal scale, even canonically ‘small’ effects can have very large effects in the aggregate.

The example I’m most fond of here is Robert Rosenthal’s famous illustration of the effects of aspirin on heart attack. The correlation between taking aspirin daily and decreased risk of heart attack is, at best, .03 (I say at best because the estimate is based on a large 1988 study, but my understanding is that more recent studies have moderated even this small effect). In most domains of psychology, a correlation of .03 is so small as to be completely uninteresting. Most psychologists would never seriously contemplate running a study to try to detect an effect of that size. And yet, at a population level, even an r of .03 can have serious implications. Cast in a different light, what this effect means is that 3% of people who would be expected to have a heart attack without aspirin would be saved from that heart attack given a daily aspirin regimen. Needless to say, this isn’t trivial. It amounts to a potentially life-saving intervention for 30 out of every 1,000 people. At a public policy level, you’d be crazy to ignore something like that (which is why, for a long time, many doctors recommended that people take an aspirin a day). And yet, by the standards of experimental psychology, this is a tiny, tiny effect that probably isn’t worth getting out of bed for.

The point of course is that when you consider how many people are currently on antidepressants (millions), even small effects–and certainly an effect of one-third of a standard deviation–are going to be compounded many times over. Given that antidepressants demonstrably reduce the risk of suicide (according to Hall, by about 20%), there’s little doubt that tens of thousands of lives have been saved by antidepressants. That doesn’t necessarily justify their routine use, of course, because the side effects and costs also scale up to the societal level (just imagine how many millions of bouts of nausea could be prevented by eliminating antidepressants from the market!). The point is that just that, if you think the benefits of antidepressants outweigh their costs even slightly at the level of the average depressed individual, you’re probably committing yourself to thinking that they have a hugely beneficial impact at a societal level–and that holds true irrespective of whether the effects are ‘clinically meaningful’ by conventional standards.

Tangentially related to the last post, Games With Words has a post up soliciting opinions about the merit of effect sizes. The impetus is a discussion we had in the comments on his last post about Jonah Lehrer’s New Yorker article. It started with an obnoxious comment (mine, of course) and then rapidly devolved into a  murderous duel civil debate about the importance (or lack thereof) of effect sizes in psychology. What I argued is that consideration of effect sizes is absolutely central to most everything psychologists do, even if that consideration is usually implicit rather than explicit. GWW thinks effect sizes aren’t that important, or at least, don’t have to be.

The basic observation in support of thinking in terms of effect sizes rather than (or in addition to) p values is simply that the null hypothesis is nearly always false. (I think I said “always” in the comments, but I can live with “nearly always”). There are exceedingly few testable associations between two or more variables that could plausibly have an effect size of exactly zero. Which means that if all you care about is rejecting the null hypothesis by reaching p < .05, all you really need to do is keep collecting data–you will get there eventually.

I don’t think this is a controversial point, and my sense is that it’s the received wisdom among (most) statisticians. That doesn’t mean that the hypothesis testing framework isn’t useful, just that it’s fundamentally rooted in ideas that turn out to be kind of silly upon examination. (For the record, I use significance tests all the time in my own work, and do all sorts of other things I know on some level to be silly, so I’m not saying that we should abandon hypothesis testing wholesale).

Anyway, GWW’s argument is that, at least in some areas of psychology, people don’t really care about effect sizes, and simply want to know if there’s a real effect or not. I disagree for at least two reasons. First, when people say they don’t care about effect sizes, I think what they really mean is that they don’t feel a need to explicitly think about effect sizes, because they can just rely on a decision criterion of p < .05 to determine whether or not an effect is ‘real’. The problem is that, since the null hypothesis is always false (i.e., effects are never exactly zero in the population), if we just keep collecting data, eventually all effects become statistically significant, rendering the decision criterion completely useless. At that point, we’d presumably have to rely on effect sizes to decide what’s important. So it may look like you can get away without considering effect sizes, but that’s only because, for the kind of sample sizes we usually work with, p values basically end up being (poor) proxies for effect sizes.

Second, I think it’s simply not true that we care about any effect at all. GWW makes a seemingly reasonable suggestion that even if it’s not sensible to care about a null of exactly zero, it’s quite sensible to care about nothing but the direction of an effect. But I don’t think that really works either. The problem is that, in practice, we don’t really just care about the direction of the effect; we also want to know that it’s meaningfully large (where ‘meaningfully’ is intentionally vague, and can vary from person to person or question to question). GWW gives a priming example: if a theoretical model predicts the presence of a priming effect, isn’t it enough just to demonstrate a statistically significant priming effect in the predicted direction? Does it really matter how big the effect is?

Yes. To see this, suppose that I go out and collect priming data online from 100,000 subjects, and happily reject the null at p < .05 based on a priming effect of a quarter of a millisecond (where the mean response time is, say, on the order of a second). Does that result really provide any useful support for my theory, just because I was able to reject the null? Surely not. For one thing, a quarter of a millisecond is so tiny that any reviewer worth his or her salt is going to point out that any number of confounding factors could be responsible for that tiny association. An effect that small is essentially uninterpretable. But there is, presumably, some minimum size for every putative effect which would lead us to say: “okay, that’s interesting. It’s a pretty small effect, but I can’t just dismiss it out of hand, because it’s big enough that it can’t be attributed to utterly trivial confounds.” So yes, we do care about effect sizes.

The problem, of course, is that what constitutes a ‘meaningful’ effect is largely subjective. No doubt that’s why null hypothesis testing holds such an appeal for most of us (myself included)–it may be silly, but it’s at least objectively silly. It doesn’t require you to put your subjective beliefs down on paper. Still, at the end of the day, that apprehensiveness we feel about it doesn’t change the fact that you can’t get away from consideration of effect sizes, whether explicitly or implicitly. Saying that you don’t care about effect sizes doesn’t actually make it so; it just means that you’re implicitly saying that you literally care about any effect that isn’t exactly zero–which is, on its face, absurd. Had you picked any other null to test against (e.g., a range of standardized effect sizes between -0.1 and 0.1), you wouldn’t have that problem.

To reiterate, I’m emphatically not saying that anyone who doesn’t explicitly report, or even think about, effect sizes when running a study should be lined up against a wall and fired upon at will is doing something terribly wrong. I think it’s a very good idea to (a) run power calculations before starting a study, (b) frequently pause to reflect on what kinds of effects one considers big enough to be worth pursuing; and (c) report effect size measures and confidence intervals for all key tests in one’s papers. But I’m certainly not suggesting that if you don’t do these things, you’re a bad person, or even a bad researcher. All I’m saying is that the importance of effect sizes doesn’t go away just because you’re not thinking about them. A decision about what constitutes a meaningful effect size is made every single time you test your data against the null hypothesis; so you may as well be the one making that decision explicitly, instead of having it done for you implicitly in a silly way. No one really cares about anything-but-zero.

Over the last four or five years, there’s been a growing awareness in the scientific community that science is an imperfect process. Not that everyone used to think science was a crystal ball with a direct line to the universe or anything, but there does seem to be a growing recognition that scientists are human beings with human flaws, and are susceptible to common biases that can make it more difficult to fully trust any single finding reported in the literature. For instance, scientists like interesting results more than boring results; we’d rather keep our jobs than lose them; and we have a tendency to see what we want to see, even when it’s only sort-of-kind-of there, and sometimes not there at all. All of these things contrive to produce systematic biases in the kinds of findings that get reported.

The single biggest contributor to the zeitgeist shift nudge is undoubtedly John Ioannidis (recently profiled in an excellent Atlantic article), whose work I can’t say enough good things about (though I’ve tried). But lots of other people have had a hand in popularizing the same or similar ideas–many of which actually go back several decades. I’ve written a bit about these issues myself in a number of papers (1, 2, 3) and blog posts (1, 2, 3, 4, 5), so I’m partial to such concerns. Still, important as the role of the various selection and publication biases is in charting the course of science, virtually all of the discussions of these issues have had a relatively limited audience. Even Ioannidis’ work, influential as it’s been, has probably been read by no more than a few thousand scientists.

Last week, the debate hit the mainstream when the New Yorker (circulation: ~ 1 million) published an article by Jonah Lehrer suggesting–or at least strongly raising the possibility–that something might be wrong with the scientific method. The full article is behind a paywall, but I can helpfully tell you that some people seem to have un-paywalled it against the New Yorker’s wishes, so if you search for it online, you will find it.

The crux of Lehrer’s argument is that many, and perhaps most, scientific findings fall prey to something called the “decline effect”: initial positive reports of relatively large effects are subsequently followed by gradually decreasing effect sizes, in some cases culminating in a complete absence of an effect in the largest, most recent studies. Lehrer gives a number of colorful anecdotes illustrating this process, and ends on a decidedly skeptical (and frankly, terribly misleading) note:

The decline effect is troubling because it reminds us how difficult it is to prove anything. We like to pretend that our experiments define the truth for us. But that’s often not the case. Just because an idea is true doesn’t mean it can be proved. And just because an idea can be proved doesn’t mean it’s true. When the experiments are done, we still have to choose what to believe.

While Lehrer’s article received pretty positive reviews from many non-scientist bloggers (many of whom, dismayingly, seemed to think the take-home message was that since scientists always change their minds, we shouldn’t trust anything they say), science bloggers were generally not very happy with it. Within days, angry mobs of Scientopians and Nature Networkers started murdering unicorns; by the end of the week, the New Yorker offices were reduced to rubble, and the scientists and statisticians who’d given Lehrer quotes were all rumored to be in hiding.

Okay, none of that happened. I’m just trying to keep things interesting. Anyway, because I’ve been characteristically lazy slow on the uptake, by the time I got around to writing this post you’re now reading, about eighty hundred and sixty thousand bloggers had already weighed in on Lehrer’s article. That’s good, because it means I can just direct you to other people’s blogs instead of having to do any thinking myself. So here you go: good posts by Games With Words (whose post tipped me off to the article), Jerry Coyne, Steven Novella, Charlie Petit, and Andrew Gelman, among many others.

Since I’ve blogged about these issues before, and agree with most of what’s been said elsewhere, I’ll only make one point about the article. Which is that about half of the examples Lehrer talks about don’t actually seem to me to qualify as instances of the decline effect–at least as Lehrer defines it. The best example of this comes when Lehrer discusses Jonathan Schooler’s attempt to demonstrate the existence of the decline effect by running a series of ESP experiments:

In 2004, Schooler embarked on an ironic imitation of Rhine’s research: he tried to replicate this failure to replicate. In homage to Rhirie’s interests, he decided to test for a parapsychological phenomenon known as precognition. The experiment itself was straightforward: he flashed a set of images to a subject and asked him or her to identify each one. Most of the time, the response was negative—-the images were displayed too quickly to register. Then Schooler randomly selected half of the images to be shown again. What he wanted to know was whether the images that got a second showing were more likely to have been identified the first time around. Could subsequent exposure have somehow influenced the initial results? Could the effect become the cause?

The craziness of the hypothesis was the point: Schooler knows that precognition lacks a scientific explanation. But he wasn’t testing extrasensory powers; he was testing the decline effect. “At first, the data looked amazing, just as we’d expected,” Schooler says. “I couldn’t believe the amount of precognition we were finding. But then, as we kept on running subjects, the effect size”–a standard statistical measure–“kept on getting smaller and smaller.” The scientists eventually tested more than two thousand undergraduates. “In the end, our results looked just like Rhinos,” Schooler said. “We found this strong paranormal effect, but it disappeared on us.”

This is a pretty bad way to describe what’s going on, because it makes it sound like it’s a general principle of data collection that effects systematically get smaller. It isn’t. The variance around the point estimate of effect size certainly gets smaller as samples get larger, but the likelihood of an effect increasing is just as high as the likelihood of it decreasing. The absolutely critical point Lehrer left out is that you would only get the decline effect to show up if you intervened in the data collection or reporting process based on the results you were getting. Instead, most of Lehrer’s article presents the decline effect as if it’s some sort of mystery, rather than the well-understood process that it is. It’s as though Lehrer believes that scientific data has the magical property of telling you less about the world the more of it you have. Which isn’t true, of course; the problem isn’t that science is malfunctioning, it’s that scientists are still (kind of!) human, and are susceptible to typical human biases. The unfortunate net effect is that Lehrer’s article, while tremendously entertaining, achieves exactly the opposite of what good science journalism should do: it sows confusion about the scientific process and makes it easier for people to dismiss the results of good scientific work, instead of helping people develop a critical appreciation for the amazing power science has to tell us about the world.

There’s a beautiful paper in Nature this week by Adrian Owen and colleagues that provides what’s probably as close to definitive evidence as you can get in any single study that “brain training” programs don’t work. Or at least, to the extent that they do work, the effects are so weak they’re probably not worth caring about.

Owen et al used a very clever approach to demonstrate their point. Rather than spending their time running small-sample studies that require people to come into the lab over multiple sessions (an expensive and very time-intensive effort that’s ultimately still usually underpowered), they teamed up with the BBC program ‘Bang Goes The Theory‘. Participants were recruited via the tv show, and were directed to an experimental website where they created accounts, engaged in “pre-training” cognitive testing, and then could repeatedly log on over the course of six weeks to perform a series of cognitive tasks supposedly capable of training executive abilities. After the training period, participants again performed the same battery of cognitive tests, enabling the researchers to compare performance pre- and post-training.

Of course, you expect robust practice effects with this kind of thing (i.e., participants would almost certainly do better on the post-training battery than on the pre-training battery solely because they’d been exposed to the tasks and had some practice). So Owen et al randomly assigned participants logging on to the website to two different training programs (involving different types of training tasks) or to a control condition in which participants answered obscure trivia questions rather than doing any sort of intensive cognitive training per se. The beauty of doing this all online was that the authors were able to obtain gargantuan sample sizes (several thousand in each condition), ensuring that statistical power wasn’t going to be an issue. Indeed, Owen et al focus almost explicitly on effect sizes rather than p values, because, as they point out, once you have several thousand participants in each group, almost everything is going to be statistically significant, so it’s really the effect sizes that matter.

The critical comparison was whether the experimental groups showed greater improvements in performance post-training than the control group did. And the answer, generally speaking, was no. Across four different tasks, the differences in training-related gains in the experimental group relative to the control group were always either very small (no larger than about a fifth of a standard deviation), or even nonexistent (to the extent that for some comparisons, the control group improved more than the experimental groups!). So the upshot is that if there is any benefit of cognitive training (and it’s not at all clear that there is, based on the data), it’s so small that it’s probably not worth caring about. Here’s the key figure:

You could argue that the fact the y-axis spans the full range of possible values (rather than fitting the range of observed variation) is a bit misleading, since it’s only going to make any effects seem even smaller. But even so, it’s pretty clear these are not exactly large effects (and note that the key comparison is not the difference between light and dark bars, but the relative change from light to dark across the different groups).

Now, people who are invested (either intellectually or financially) in the efficacy of cognitive training programs might disagree, arguing that an effect of one-fifth of a standard deviation isn’t actually a tiny effect, and that there are arguably many situations in which that would be a meaningful boost in performance. But that’s the best possible estimate, and probably overstates the actual benefit. And there’s also the opportunity cost to consider: the average participant completed 20 – 30 training sessions, which, even at just 20 minutes a session (an estimate based on the description of the length of each of the training tasks), would take about 8 – 10 hours to complete (and some participants no doubt spent many more hours in training).  That’s a lot of time that could have been invested in other much more pleasant things, some of which might also conceivably improve cognitive ability (e.g., doing Sudoku puzzles, which many people actually seem to enjoy). Owen et al put it nicely:

To illustrate the size of the transfer effects observed in this study, consider the following representative example from the data. The increase in the number of digits that could be remembered following training on tests designed, at least in part, to improve memory (for example, in experimental group 2) was three-hundredth of a digit. Assuming a linear relationship between time spent training and improvement, it would take almost four years of training to remember one extra digit. Moreover, the control group improved by two-tenths of a digit, with no formal memory training at all.

If someone asked you if you wanted to spend six weeks doing a “brain training” program that would provide those kinds of returns, you’d probably politely (or impolitely) refuse. Especially since it’s not like most of us spend much of our time doing digit span tasks anyway; odds are that the kinds of real-world problems we’d like to perform a little better at (say, something trivial like figuring out what to buy or not to buy at the grocery store) are even further removed from the tasks Owen et al (and other groups) have used to test for transfer, so any observable benefits in the real world would presumably be even smaller.

Of course, no study is perfect, and there are three potential concerns I can see. The first is that it’s possible that there are subgroups within the tested population who do benefit much more from the cognitive training. That is, the miniscule overall effect could be masking heterogeneity within the sample, such that some people (say, maybe men above 60 with poor diets who don’t like intellectual activities) benefit much more. The trouble with this line of reasoning, though, is that the overall effects in the entire sample are so small that you’re pretty much forced to conclude that either (a) any group that benefits substantially from the training is a very small proportion of the total sample, or (b) that there are actually some people who suffer as a result of cognitive training, effectively balancing out the gains seen by other people. Neither of these possibilities seem particularly attractive.

The second concern is that it’s conceivable that the control group isn’t perfectly matched to the experimental group, because, by the authors’ own admission, the retention rate was much lower in the control group. Participants were randomly assigned to the three groups, but only about two-thirds as many control participants completed the study. The higher drop-out rate was apparently due to the fact that the obscure trivia questions used as a control task were pretty boring. The reason that’s a potential problem is that attrition wasn’t random, so there may be a systematic difference between participants in the experimental conditions and those in the control conditions. In particular, it’s possible that the remaining control participants had a higher tolerance for boredom and/or were somewhat smarter or more intellectual on average (answering obscure trivia questions clearly isn’t everyone’s cup of tea). If that were true, the lack of any difference between experimental and control conditions might be due to participant differences rather than an absence of a true training effect. Unfortunately, it’s hard to determine whether this might be true, because (as far as I can tell) Owen et al don’t provide the raw mean performance scores on the pre- and post-training testing for each group, but only report the changes in performance. What you’d want to know is that the control participants didn’t do substantially better or worse on the pre-training testing than the experimental participants (due to selective attrition of low-performing subjects), which might make changes in performance difficult to interpret. But at face value, it doesn’t seem very plausible that this would be a serious issue.

Lastly, Owen et al do report a small positive correlation between number of training sessions performed (which was under participants’ control) and gains in performance on the post-training test. Now, this effect was, as the authors note, very small (a maximal Spearman’s rho of .06), so that it’s also not really likely to have practical implications. Still, it does suggest that performance increases as a function of practice. So if we’re being pedantic, we should say that intensive cognitive training may improve cognitive performance in a generalized way, but that the effect is really minuscule and probably not worth the time and effort required to do the training in the first place. Which isn’t exactly the type of careful and measured claim that the people who sell brain training programs are generally interested in making.

At any rate, setting aside the debate over whether cognitive training works or not, one thing that’s perplexed me for a long time about the training literature is why people focus to such an extent on cognitive training rather than other training regimens that produce demonstrably larger transfer effects. I’m thinking in particular of aerobic exercise, which produces much more robust and replicable effects on cognitive performance. There’s a nice meta-analysis by Colcombe and colleagues that found effect sizes on the order of half a standard deviation and up for physical exercise in older adults–and effects were particularly large for the most heavily g-loaded tasks. Now, even if you allow for publication bias and other manifestations of the fudge factor, it’s almost certain that the true effect of physical exercise on cognitive performance is substantially larger than the (very small) effects of cognitive training as reported by Owen et al and others.

The bottom line is that, based on everything we know at the moment, the evidence seems to pretty strongly suggest that if your goal is to improve cognitive function, you’re more likely to see meaningful results by jogging or swimming regularly than by doing crossword puzzles or N-back tasks–particularly if you’re older. And of course, a pleasant side effect is that exercise also improves your health and (for at least some people) mood, which I don’t think N-back tasks do. Actually, many of the participants I’ve tested will tell you that doing the N-back is a distinctly dysphoric experience.

On a completely unrelated note, it’s kind of neat to see a journal like Nature publish what is essentially a null result. It goes to show that people do care about replication failures in some cases–namely, in those cases when the replication failure contradicts a relatively large existing literature, and is sufficiently highly powered to actually say something interesting about the likely effect sizes in question.

Owen AM, Hampshire A, Grahn JA, Stenton R, Dajani S, Burns AS, Howard RJ, & Ballard CG (2010). Putting brain training to the test. Nature PMID: 20407435

Andrew Gelman posted a link on his blog today to a paper by John Ioannidis I hadn’t seen before. In many respects, it’s basically the same paper I wrote earlier this year as a commentary on the Vul et al “voodoo correlations” paper (the commentary was itself based largely on an earlier chapter I wrote with my PhD advisor, Todd Braver). Well, except that the Ioannidis paper came out a year earlier than mine, and is also much better in just about every respect (more on this below).

What really surprises me is that I never came across Ioannidis’ paper when I was doing a lit search for my commentary. The basic point I made in the commentary–which can be summarized as the observation that low power coupled with selection bias almost invariably inflates significant effect sizes–is a pretty straightforward statistical point, so I figured that many people, and probably most statisticians, were well aware of it. But no amount of Google Scholar-ing helped me find an authoritative article that made the same point succinctly; I just kept coming across articles that made the point tangentially, in an off-hand “but of course we all know we shouldn’t trust these effect sizes, because…” kind of way. So I chalked it down as one of those statistical factoids (of which there are surprisingly many) that live in the unhappy land of too-obvious-for-statisticians-to-write-an-article-about-but-not-obvious-enough-for-most-psychologists-to-know-about. And so I just went ahead and wrote the commentary in a non-technical way that I hoped would get the point across intuitively.

Anyway, after the commentary was accepted, I sent a copy to Andrew Gelman, who had written several posts about the Vul et al controversy. He promptly send me back a link to this paper of his, which basically makes the same point about sampling error, but with much more detail and much better examples than I did. His paper also cites an earlier article in American Scientist by Wainer, which I also recommend, and again expresses very similar ideas. So then I felt a bit like a fool for not stumbling across either Gelman’s paper or Wainer’s earlier. And now that I’ve read Ioannidis’ paper, I feel even dumber, seeing as I could have saved myself a lot of trouble by writing two or three paragraphs and then essentially pointing to Ioannidis’ work. Oh well.

That all said, it wasn’t a complete loss; I still think the basic point is important enough that it’s worth repeating loudly and often, no matter how many times it’s been said before. And I’m skeptical that many fMRI researchers would have appreciated the point otherwise, given that none of the papers I’ve mentioned were published in venues fMRI researchers are likely to read regularly (which is presumably part of the reason I never came across them!). Of course, I don’t think that many people who do fMRI research actually bothered to read my commentary, so it’s questionable whether it had much impact anyway.

At any rate, the Ioannidis paper makes a number of points that my paper didn’t, so I figured I’d talk about them a bit. I’ll start by revisiting what I said in my commentary, and then I’ll tell you why you should read Ioannidis’ paper instead of mine.

The basic intuition can be captured as follows. Suppose you’re interested in the following question: Do clowns suffer depression at a higher rate than us non-comical folk do? You might think this is a contrived (to put it delicately) question, but I can assure you it has all sorts of important real-world implications. For instance, you wouldn’t be so quick to book a clown for your child’s next birthday party if you knew that The Great Mancini was going to be out in the parking lot half an hour later drinking cheap gin out of a top hat. If that example makes you feel guilty, congratulations: you’ve just discovered the translational value of basic science.

Anyway, back to the question, and how we’re going to answer it. You can’t just throw a bunch of clowns and non-clowns in a room and give them a depression measure. There’s nothing comical about that. What you need to do, if you’re rigorous about it, is give them multiple measures of depression, because we all know how finicky individual questionnaires can be. So the clowns and non-clowns each get to fill out the Beck Depression Inventory (BDI), the Center for Epidemiologic Studies Depression Scale, the Depression Adjective Checklist, the Zung Self-Rating Depression Scale (ZSRDS), and, let’s say, six other measures. Ten measures in all. And let’s say we have 20 individuals in each group, because that’s all I personally a cash-strapped but enthusiastic investigator can afford. After collecting the data, we score the questionnaires and run a bunch of t-tests to determine whether clowns and non-clowns have different levels of depression. Being scrupulous researchers who care a lot about multiple comparisons correction, we decide to divide our critical p-value by 10 (the dreaded Bonferroni correction, for 10 tests in this case) and test at p < .005. That’s a conservative analysis, of course; but better safe than sorry!

So we run our tests and get what look like mixed results. Meaning, we get statistically significant positive correlations between clown-dom status and depression for 2 measures–the BDI and Zung inventories–but not for the other 8 measures. So that’s admittedly not great; it would have been better if all 10 had come out right. Still, it at least partially supports our hypothesis: Clowns are fucking miserable! And because we’re already thinking ahead to how we’re going to present these results when they (inevitably) get published in Psychological Science, we go ahead and compute the effect sizes for the two significant correlations, because, after all, it’s important to know not only that there is a “real” effect, but also how big that effect is. When we do that, it turns out that the point-biserial correlation is huge! It’s .75 for the BDI and .68 for the ZSRDS. In other words, about half of the variance in clowndom can be explained by depression levels. And of course, because we’re well aware that correlation does not imply causation, we get to interpret the correlation both ways! So we quickly issue a press release claiming that we’ve discovered that it’s possible to conclusively diagnose depression just by knowing whether or not someone’s a clown! (We’re not going to worry about silly little things like base rates in a press release.)

Now, this may all seem great. And it’s probably not an unrealistic depiction of how much of psychology works (well, minus the colorful scarves, big hair, and face paint). That is, very often people report interesting findings that were selectively reported from amongst a larger pool of potential findings on the basis of the fact that the former but not the latter surpassed some predetermined criterion for statistical significance. For example, in our hypothetical in press clown paper, we don’t bother to report results for the correlation between clownhood and the Center for Epidemiologic Studies Depression Scale (r = .12, p > .1). Why should we? It’d be silly to report a whole pile of additional correlations only to turn around and say “null effect, null effect, null effect, null effect, null effect, null effect, null effect, and null effect” (see how boring it was to read that?). Nobody cares about variables that don’t predict other variables; we care about variables that do predict other variables. And we’re not really doing anything wrong, we think; it’s not like the act of selective reporting is inflating our Type I error (i.e., the false positive rate), because we’ve already taken care of that up front by deliberately being overconservative in our analyses.

Unfortunately, while it’s true that our Type I error doesn’t suffer, the act of choosing which findings to report based on the results of a statistical test does have another unwelcome consequence. Specifically, there’s a very good chance that the effect sizes we end up reporting for statistically significant results will be artificially inflated–perhaps dramatically so.

Why would this happen? It’s actually entailed by the selection procedure. To see this, let’s take the classical measurement model, under which the variance in any measured variable reflects the sum of two components: the “true” scores (i.e., the scores we would get if our measurements were always completely accurate) and some random error. The error term can in turn be broken down into many more specific sources of error; but we’ll ignore that and just focus on one source of error–namely, sampling error. Sampling error refers to the fact that we can never select a perfectly representative group of subjects when we collect a sample; there’s always some (ideally small) way in which the sample characteristics differ from the population. This error term can artificially inflate an effect or artificially deflate it, and it can inflate or deflate it more or less, but it’s going to have an effect one way or the other. You can take that to the bank as sure as my name’s Bozo the Clown.

To put this in context, let’s go back to our BDI scores. Recall that what we observed is that clowns have higher BDI scores than non-clowns. But what we’re now saying is that that difference in scores is going to be affected by sampling error. That is, just by chance, we may have selected a group of clowns that are particularly depressed, or a group of non-clowns who are particularly jolly. Maybe if we could measure depression in all clowns and all non-clowns, we would actually find no difference between groups.

Now, if we allow that sampling error really is random, and that we’re not actively trying to pre-determine the outcome of our study by going out of our way to recruit The Great Depressed Mancini and his extended dysthymic clown family, then in theory we have no reason to think that sampling error is going to introduce any particular bias into our results. It’s true that the observed correlations in our sample may not be perfectly representative of the true correlations in the population; but that’s not a big deal so long as there’s no systematic bias (i.e., that we have no reason to think that our sample will systematically inflate correlations or deflate them). But here’s the problem: the act of choosing to report some correlations but not others on the basis of their statistical significance (or lack thereof) introduces precisely such a bias. The reason is that, when you go looking for correlations that are of a certain size or greater, you’re inevitably going to be more likely to select those correlations that happen to have been helped by chance than hurt by it.

Here’s a series of figures that should make the point even clearer. Let’s pretend for a moment that the truth of the matter is that there is in fact a positive correlation between clown status and all 10 depression measures. Except, we’ll make it 100 measures, because it’ll be easier to illustrate the point that way. Moreover, let’s suppose that the correlation is exactly the same for all 100 measures, at .3. Here’s what that would look like if we just plotted the correlations for all 100 measures, 1 through 100:

It’s just a horizontal red line, because all the individual correlations have the same value (0.3). So that’s not very exciting. But remember, these are the population correlations. They’re not what we’re going to observe in our sample of 20 clowns and 20 non-clowns, because depression scores in our sample aren’t a perfect representation of the population. There’s also error to worry about. And error–or at least, sampling error–is going to be greater for smaller samples than for bigger ones. (The reason for this can be expressed intuitively: other things being equal, the more observations you have, the more representative your sample must be of the population as a whole, because deviations in any given direction will tend to cancel each other out the more data you collect. And if you keep collecting, at the limit, your sample will constitute the whole population, and must therefore by definition be perfectly representative). With only 20 subjects in each group, our estimates of each group’s depression level are not going to be terrifically stable. You can see this in the following figure, which shows the results of a simulation on 100 different variables, assuming that all have an identical underlying correlation of .3:

Notice how much variability there is in the correlations! The weakest correlation is actually negative, at -.18; the strongest is much larger than .3, at .63. (Caveat for more technical readers: this assumes that the above variables are completely independent, which in practice is unlikely to be true when dealing with 100 measures of the same construct.) So even though the true correlation is .3 in all cases, the magic of sampling will necessarily produce some values that are below .3, and some that are above .3. In some cases, the deviations will be substantial.

By now you can probably see where this is going. Here we have a distribution of effect sizes that to some extent may reflect underlying variability in population effect sizes, but is also almost certainly influenced by sampling error. And now we come along and decide that, hey, it doesn’t really make sense to report all 100 of these correlations in a paper; that’s too messy. Really, for the sake of brevity and clarity, we should only report those correlations that are in some sense more important and “real”. And we do that by calculating p-values and only reporting the results of tests that are significant at some predetermined level (in our case, p < .005). Well, here’s what that would look like:

This is exactly the same figure as the previous one, except we’ve now grayed out all the non-significant correlations. And in the process, we’ve made Bozo the Clown cry:

Why? Because unfortunately, the criterion that we’ve chosen is an extremely conservative one. In order to detect a significant difference in means between two groups of 20 subjects at p < .005, the observed correlation (depicted as the horizontal black line above) needs to be .42 or greater! That’s substantially larger than the actual population effect size of .3. Effects of this magnitude don’t occur very frequently in our sample; in fact, they only occur 16 times. As a result, we’re going to end up failing to detect 84 of 100 correlations, and will walk away thinking they’re null results–even though the truth is that, in the population, they’re actually all pretty strong, at .3. This quantity–the proportion of “real” effects that we’re likely to end up calling statistically significant given the constraints of our sample–is formally called statistical power. If you do a power analysis for a two-sample t-test on a correlation of .3 at p < .005, it turns out that power is only .17 (which is essentially what we see above; the slight discrepancy is due to chance). In other words, even when there are real and relatively strong associations between depression and clownhood, our sample would only identify those associations 17% of the time, on average.

That’s not good, obviously, but there’s more. Now the other shoe drops, because not only have we systematically missed out on most of the effects we’re interested in (in virtue of using small samples and overly conservative statistical thresholds), but notice what we’ve also done to the effect sizes of those correlations that we do end up identifying. What is in reality a .3 correlation spuriously appears, on average, as  a .51 correlation in the 16 tests that surpass our threshold. So, through the combined magic of low power and selection bias, we’ve turned what may in reality be a relatively diffuse association between two variables (say, clownhood and depression) into a seemingly selective and extremely strong association. After all the excitement about getting a high-profile publication, it might ultimately turn out that clowns aren’t really so depressed after all–it’s all an illusion induced by the sampling apparatus. So you might say that the clowns get the last laugh. Or that the joke’s on us. Or maybe just that this whole clown example is no longer funny and it’s now time for it to go bury itself in a hole somewhere.

Anyway, that, in a nutshell, was the point my commentary on the Vul et al paper made, and it’s the same point the Gelman and Wainer papers make too, in one way or another. While it’s a very general point that really applies in any domain where (a) power is less than 100% (which is just about always) and (b) there is some selection bias (which is also just about always), there were some considerations that were particularly applicable to fMRI research. The basic issue is that, in fMRI research, we often want to conduct analyses that span the entire brain, which means we’re usually faced with conducting many more statistical comparisons than researchers in other domains generally deal with (though not, say, molecular geneticists conducting genome-wide association studies). As a result, there is a very strong emphasis in imaging research on controlling Type I error rates by using very conservative statistical thresholds. You can agree or disagree with this general advice (for the record, I personally think there’s much too great an emphasis in imaging on Type I error, and not nearly enough emphasis on Type II error), but there’s no avoiding the fact that following it will tend to produce highly inflated significant effect sizes, because in the act of reducing p-value thresholds, we’re also driving down power dramatically, and making the selection bias more powerful.

While it’d be nice if there was an easy fix for this problem, there really isn’t one. In behavioral domains, there’s often a relatively simple prescription: report all effect sizes, both significant and non-significant. This doesn’t entirely solve the problem, because people are still likely to overemphasize statistically significant results relative to non-significant ones; but at least at that point you can say you’ve done what you can. In the fMRI literature, this course of action isn’t really available, because most journal editors are not going to be very happy with you when you send them a 25-page table that reports effect sizes and p-values for each of the 100,000 voxels you tested. So we’re forced adopt other strategies. The one I’ve argued for most strongly is to increase sample size (which increases power and decreases the uncertainty of resulting estimates). But that’s understandably difficult in a field where scanning each additional subject can cost $1,000 or more. There are a number of other things you can do, but I won’t talk about them much here, partly because this is already much too long a post, but mostly because I’m currently working on a paper that discusses this problem, and potential solutions, in much more detail.

So now finally I get to the Ioannidis article. As I said, the basic point is the same one made in my paper and Gelman’s and others, and the one I’ve described above in excruciating clownish detail. But there are a number of things about the Ioannidis that are particularly nice. One is that Ioannidis considers not only inflation due to selection of statistically significant results coupled with low power, but also inflation due to the use of flexible analyses (or, as he puts it, “vibration” of effects–also known as massaging the data). Another is that he considers cultural aspects of the phenomenon, e.g., the fact that investigators tend to be rewarded for reporting large effects, even if they subsequently fail to replicate. He also discusses conditions under which you might actually get deflation of effect sizes–something I didn’t touch on in my commentary, and hadn’t really thought about. Finally, he makes some interesting recommendations for minimizing effect size inflation. Whereas my commentary focused primarily on concrete steps researchers could take in individual studies to encourage clearer evaluation of results (e.g., reporting confidence intervals, including power calculations, etc.), Ioannidis focuses on longer-term solutions and the possibility that we’ll need to dramatically change the way we do science (at least in some fields).

Anyway, this whole issue of inflated effect sizes is a critical one to appreciate if you do any kind of social or biomedical science research, because it almost certainly affects your findings on a regular basis, and has all sorts of implications for what kind of research we conduct and how we interpret our findings. (To give just one trivial example, if you’ve ever been tempted to attribute your failure to replicate a previous finding to some minute experimental difference between studies, you should seriously consider the possibility that the original effect size may have been grossly inflated, and that your own study consequently has insufficient power to replicate the effect.) If you only have time to read one article that deals with this issue, read the Ioannidis paper. And remember it when you write your next Discussion section. Bozo the Clown will thank you for it.

Ioannidis, J. (2008). Why Most Discovered True Associations Are Inflated Epidemiology, 19 (5), 640-648 DOI: 10.1097/EDE.0b013e31818131e7

Yarkoni, T. (2009). Big Correlations in Little Studies: Inflated fMRI Correlations Reflect Low Statistical Power-Commentary on Vul et al. (2009) Perspectives on Psychological Science, 4 (3), 294-298 DOI: 10.1111/j.1745-6924.2009.01127.x