In June 2016, the United Kingdom carried out a little study to test the hypothesis that it is the “will of the people” that the country should leave the European Union. The result favoured the Leave hypothesis, albeit with a really small effect size (1.89%). This finding came as a surprise to many but as so often it is the most surprising results that have the most impact.
Accusations of p-hacking soon emerged. Not only was there a clear sampling bias but data thugs suggested that the results might have even been obtained by fraud. Nevertheless, the original publication was never retracted. What’s wrong with inflating the results a bit? Massaging data to fit a theory is not the worst sin! The history of science is rich with errors. Such studies can be of value if they offer new clarity in looking at phenomena.
In fact, the 2016 study did offer a lot of new ways to look at the situation. There was a fair amount of HARKing about what the result of the 2016 study actually meant. Prior to conducting the study, at conferences and in seminars the proponents of the Leave hypotheses kept talking about the UK having a relationship with the EU like Norway and Switzerland. Yet somehow in the eventual publication of the 2016 findings, the authors had changed their tune. Now they professed that their hypothesis was obviously always that the UK should leave the EU without any deal whatsoever.
Sceptics of the Leave hypothesis pointed out various problems with this idea. For one thing, leaving the EU without a deal wasn’t a very plausible hypothesis. There were thousands of little factors to be considered and it seemed unlikely that this was really the will of the people. And of course, the nitpickers also said that “barely more than half” could never be considered the “will of the people”.
Almost immediately, there were calls for a replication to confirm that the “will of the people” really was what believers in the Leave-without-a-deal hypothesis claimed. At first, these voices came only from a ragtag band of second stringers – but as time went on and more and more people realised just how implausible the Leave hypothesis really was, their numbers grew.
Leavers however firmly disagreed. To them, a direct replication was meaningless. That was odd for some of them had openly admitted they wanted to p-hack the hell out of this thing until they got the result they wanted. But now they claimed that there had by now been several conceptual replications of the 2016 results, first in the United States and then later also Brazil, and some might argue even in Italy, Hungary, and Poland. Also in several other European countries similar results were found, albeit not statistically significant. Based on all this evidence, a meta-analysis surely supported the general hypothesis?
But the replicators weren’t dissuaded. The more radical among these methodological terrorists posited that any study in which the experimental design isn’t clearly defined and preregistered prior to data collection is inherently exploratory, and cannot be used to test any hypotheses. They instead called for a preregistered replication, ideally a Registered Report where the methods are peer-reviewed and the manuscript is in principle accepted for publication before data collection even commences. The fact that the 2016 study didn’t do this was just one of its many problems. But people still cite it simply because of its novelty. The replicators also pointed to other research fields, like Switzerland and Ireland, where this approach has long been used very successfully.
As an added twist, it turns out that nobody actually read the background literature. The 2016 study was already a replication attempt of previous findings from 1975. Sure, some people had vaguely heard about this earlier study. Everybody who has ever been to a conference knows that there is always one white-haired emeritus professor in the audience who will shout out “But I already did this four decades ago!”. But nobody really bothered to read this original study until now. It found an enormous result in the opposite direction, 17.23% in favour of remaining in Europe. As some commentators suggested, the population at large may have changed over the past four decades or that there may have been subtle but important differences in the methodology. What if leaving Europe then meant something different to what it means now? But if that were the case, couldn’t leaving Europe in 2016 also have meant something different than in 2019?
But the Leave proponents wouldn’t have any of that. They had already invested too much money and effort and spent all this time giving TED talks about their shiny little theory to give up now. They were in fact desperately afraid of a direct replication because they knew that as with most replications it would probably end in a null result and their beautiful theoretical construct would collapse like a house of cards. Deep inside, most of these people already knew they were chasing a phantom but they couldn’t ever admit it. People like Professor BoJo, Dr Moggy, and Micky “The Class Clown” Gove had built their whole careers on this Leave idea and so they defended the “will of the people” with religious zeal. The last straw they clutched to was to warn that all these failures to replicate would cause irreparable damage to the public’s faith in science.
Only Nigel Farage, unaffiliated garden gnome and self-styled “irreverent citizen scientist”, relented somewhat. Naturally, he claimed he would be doing all that just for science and the pursuit of the truth and that the result of this replication would be even clearer than the 2016 finding. But in truth, he smelled the danger on the wind. He knew that should the Leave hypothesis be finally accepted by consensus, he would be reduced to a complete irrelevance. What was more, he would not get that hefty paycheck.
As of today, the situation remains unresolved. The preregistered replication attempt is still stuck in editorial triage and hasn’t even been sent out for peer review yet. But meanwhile, people in the corridors of power in Westminster and Brussels and Tokyo and wherever else are already basing their decisions on the really weak and poor and quite possibly fraudulent data from the flawed 2016 study. But then, it’s all about the flair, isn’t it?
I have stayed out of the Wansink saga for the most part. If you don’t know what this is about, I suggest reading about this case on Retraction Watch. I had a few private conversations about this with Nick Brown, who has been one of the people instrumental in bringing about a whole series of retractions of Wansink’s publications. I have a marginal interest in some of Wansink’s famous research, specifically whether the size of plates can influence how much a person eats, because I have a broader interest in the interplay between perception and behaviour.
But none of that is particularly important. The short story is that considerable irregularities have been discovered in a string of Wansink’s publications, many of which has since been retracted. The whole affair first kicked off with a fundamental own-goal of a blog post (now removed, so posting Gelman’s coverage instead) he wrote in which he essentially seemed to promote p-hacking. Since then the problems that came to light ranged from irregularities (or impossibility) of some of the data he reported, evidence of questionable research practices in terms of cherry-picking or excluding data, to widespread self-plagiarism. Arguably, not all of these issues are equally damning and for some the evidence is more tenuous than for others – but the sheer quantity of problems is egregious. The resulting retractions seem entirely justified.
Today I read an article on Times Higher Education entitled “Massaging data to fit a theory is not the worst research sin” by Martin Cohen, which discusses Wansink’s research sins in a broader context of the philosophy of science. The argument is pretty muddled to me so I am not entirely sure what the author’s point is – but the effective gist seems to shrug off concerns about questionable research practices and that Wansink’s research is still a meaningful contribution to science. In my mind, Cohen’s article reflects a fundamental misunderstanding of how science works and in places sounds positively post-Truthian. In the following, I will discuss some of the more curious claims made by this article.
“Massaging data to fit a theory is not the worst research sin”
I don’t know about the “worst” sin. I don’t even know if science can have “sins” although this view has been popularised by Chris Chamber’s book and Neuroskeptic’s Circles of Scientific Hell. Note that “inventing data”, a.k.a. going Full-Stapel, is considered the worst affront to the scientific method in the latter worldview. “Massaging data” is perhaps not the same as outright making it up, but on the spectrum of data fabrication it is certainly trending in that direction.
Science is about seeking the truth. In Cohen’s words, “science should above all be about explanation”. It is about finding regularities, relationships, links, and eventually – if we’re lucky – laws of nature that help us make sense of a chaotic, complex world. Altering, cherry-picking, or “shoe-horning” data to fit your favourite interpretation is the exact opposite of that.
Now, the truth is that p-hacking, the garden of forking paths, flexible outcome-contingent analyses fall under this category. Such QRPs are extremely widespread and to some degree pervade most of the scientific literature. But just because it is common, doesn’t mean that this isn’t bad. Massaging data inevitably produces a scientific literature of skewed results. The only robust way to minimise these biases is through preregistration of experimental designs and confirmatory replications. We are working towards that becoming more commonplace – but in the absence of that it is still possible to do good and honest science.
In contrast, prolifically engaging in such dubious practices, as Wansink appears to have done, fundamentally undermines the validity of scientific research. It is not a minor misdemeanour.
“We forget too easily that the history of science is rich with errors”
I sympathise with the notion that science has always made errors. One of my favourite quotes about the scientific method is that it is about “finding better ways of being wrong.” But we need to be careful not to conflate some very different things here.
First of all, a better way of being wrong is an acknowledgement that science is never a done deal. We don’t just figure out the truth but constantly seek to home in on it. Our hypotheses and theories are constantly refined, hopefully by gradually becoming more correct, but there will also be occasional missteps down a blind alley.
But these “errors” are not at all the same thing as the practices Wansink appears to have engaged in. These were not mere mistakes. While the problems with many QRPs (like optional stopping) have long been underappreciated by many, a lot of the problems in Wansink’s retracted articles are quite deliberate distortions of scientific facts. For most, he could have and should have known better. This isn’t the same as simply getting things wrong.
The examples Cohen offers for the “rich errors” in past research are also not applicable. Miscalculating the age of the earth or presenting an incorrect equation are genuine mistakes. They might be based on incomplete or distorted knowledge. Publishing an incorrect hypothesis (e.g., that DNA is a triple helix) is not the same as mining data to confirm a hypothesis. It is perfectly valid to derive new hypotheses, even if they turn out to be completely false. For example, I might posit that gremlins cause the outdoor socket on my deck to fail. Sooner or later, a thorough empirical investigation will disprove this hypothesis and the evidence will support an alternative, such as that the wiring is faulty. The gremlin hypothesis may be false – and it is also highly implausible – but nothing stops me from formulating it. Wansink’s problem wasn’t with his hypotheses (some of which may indeed turn out to be true) but with the irregularities in the data he used to support them.
“Underlying it all is a suspicion that he was in the habit of forming hypotheses and then searching for data to support them”
Ahm, no. Forming hypotheses before collecting data is how it’s supposed to work. Using Cohen’s “generous perspective”, this is indeed how hypothetico-deductive research works. In how far this relates to Wansink’s “research sin” depends on what exactly is meant here by “searching for data to support” your hypotheses. If this implies you are deliberately looking for data that confirms your prior belief while ignoring or rejecting observations that contradict it, then that is not merely a questionable research practice, but antithetical to the whole scientific endeavour itself. It is also a perfect definition of confirmation bias, something that afflicts all human beings to some extent, scientists included. Scientists must find protections from fooling themselves in this way and that entails constant vigilance and scepticism of our own pet theories. In stark contrast, engaging in this behaviour actively and deliberately is not science but pure story-telling.
The critics are not merely “indulging themselves in a myth of neutral observers uncovering ‘facts'”. Quite to the contrary, I think Wansink’s critics are well aware of the human fallibility of scientists. People are rarely perfectly neutral when it pertains to hypotheses. Even when you are not emotionally invested in which one of multiple explanations for a phenomenon might be correct, they are frequently not equal in terms of how exciting it might be to confirm them. Finding gremlins under my deck would certainly be more interesting (and scary?) than evidence of faulty wiring.
But in the end, facts are facts. There are no “alternative facts”. Results are results. We can differ on how to interpret them but that doesn’t change the underlying data. Of course, some data are plainly wrong because they come from incorrect measurements, artifacts, or statistical flukes. These results are wrong. They aren’t facts even if we think of them as facts at the moment. Sooner or later, they will be refuted. That’s normal. But this is a long shot from deliberately misreporting or distorting facts.
“…studies like Wansink’s can be of value if they offer new clarity in looking at phenomena…”
This seems to be the crux of Cohen’s argument. Somehow, despite all the dubious and possibly fraudulent nature of his research, Wansink still makes a useful contribution to science. How exactly? What “new clarity” do we gain from cherry-picked results?
I can see though that Wansink may “stimulate ideas for future investigations”. There is no denying that he is a charismatic presenter and that some of his ideas were ingenuous. I like the concept of self-filling soup bowls. I do think we must ask some critical questions about this experimental design, such as whether people can be truly unaware that the soup level doesn’t go down as they spoon it up. But the idea is neat and there is certainly scope for future research.
But don’t present this as some kind of virtue. By all means, give credit to him for developing a particular idea or a new experimental method. But please, let’s not pretend that this excuses the dubious and deliberate distortion of the scientific record. It does not justify the amount of money that has quite possibly been wasted on changing how people eat, the advice given to schools based on false research. Deliberately telling untruths is not an error, it is called a lie.
TL-DR: No, men are not “better at science” than women.
Clickbaity enough for you? I cannot honestly say I have read a lot of OpEds in the Irish Times so the evidence for my titular claim is admittedly rather limited. But it is still more solidly grounded in actual data than this article published yesterday in the Irish Times. At least I have one data point.
The article in question, a prime example of Betteridge’s Law, is entitled “Are men just better at science than women?“. I don’t need to explain why such a title might be considered sensationalist and controversial. The article itself is an “Opinion” piece, thus allowing the publication to disavow any responsibility for its authorship whilst allowing it to rake in the views from this blatant clickbait. In it, the author discusses some new research reporting gender differences in systemising vs empathising behaviour and puts this in the context of some new government initiative to specifically hire female professors because apparently there is some irony here. He goes on a bit about something called “neurosexism” (is that a real word?) and talks about “hard-wired” brains*.
I cannot quite discern if the author thought he was being funny or if he is simply scientifically illiterate but that doesn’t really matter. I don’t usually spend much time commenting on stuff like that. I have no doubt that the Irish Times, and this author in particular, will be overloaded with outrage and complaints – or, to use the author’s own words, “beaten up” on Twitter. There are many egregious misrepresentations of scientific findings in the mainstream media (and often enough, scientists and/or the university press releases are the source of this). But this example of butchery is just so bad and infuriating in its abuse of scientific evidence that I cannot let it slip past.
The whole argument, if this is what the author attempted, is just riddled with logical fallacies and deliberate exaggerations. I have no time or desire to go through them all. Conveniently, the author already addresses a major point himself by admitting that the study in question does not actually speak to male brains being “hard-wired” for science, but that any gender differences could be arising due to cultural or environmental factors. Not only that, he also acknowledges that the study in question is about autism, not about who makes good professors. So I won’t dwell on these rather obvious points any further. There are much more fundamental problems with the illogical leaps and mental gymnastics in this OpEd:
What makes you “good at science”?
There is a long answer to this question. It most certainly depends somewhat on your field of research and the nature of your work. Some areas require more manual dexterity, whilst others may require programming skills, and others yet call for a talent for high-level maths. As far as we can generalise, in my view necessary traits of a good researcher are: intelligence, creativity, patience, meticulousness, and a dedication to seek the truth rather than confirming theories. That last one probably goes hand-in-hand with some scepticism, including a healthy dose of self-doubt.
There is also a short answer to this question. A good scientist is not measured by their Systemising Quotient (SQ), a self-report measure that quantifies “the drive to analyze or build a rule-based system”. Academia is obsessed with metrics like the h-index (see my previous post) but even pencil pushers and bean counters** in hiring or grant committees haven’t yet proposed to use SQ to evaluate candidates***.
I suspect it is true that many scientists score high on the SQ and also the related Autism-spectrum Quotient (AQ) which, among other things, quantifies a person’s self-reported attention to detail. Anecdotally, I can confirm that a lot of my colleagues score higher than the population average on AQ. More on this in the next section.
However, none of this implies that you need to have a high SQ or AQ to be “good at science”, whatever that means. That assertion is a logical fallacy called affirming the consequent. We may agree that “systemising” characterises a lot of the activities a typical scientist engages in, but there is no evidence that this is sufficient to being a good scientist. It could mean that systemising people are attracted to science and engineering jobs. It certainly does not mean that a non-systemising person cannot be a good scientist.
Small effect sizes
I know I rant a lot about relative effect sizes such as Cohen’s d, where the mean difference is normalised by the variability. I feel that in a lot of research contexts these are given undue weight because the variability itself isn’t sufficiently controlled. But for studies like this we can actually be fairly confident that they are meaningful. The scientific study had a pretty whopping sample size of 671,606 (although that includes all their groups) and also used validation data. The residual physiologist inside me retains his scepticism about self-report questionnaire type measures, but even I have come to admit that a lot of questionnaires can be pretty effective. I think it is safe to say that the Big 5 Personality Factors or the AQ tap into some meaningful real factors. Further, whatever latent variance there may be on these measures, that is probably outweighed by collecting such a massive sample. So the Cohen’s d this study reports is probably quite informative.
What does this say? Well, the difference in SQ between males and females was 0.31. In other words, the distributions of SQ between sexes overlap quite considerably but the distribution for males is somewhat shifted towards higher values. Thus, while the average man has a subtly higher SQ than the average woman, a rather considerable number of women will have higher SQs than the average man. The study helpfully plots these distributions in Figure 1****:
The relevant curves here are the controls in cyan and magenta. Sorry, colour vision deficient people, the authors clearly don’t care about you (perhaps they are retinasexists?). You’ll notice that the modes of the female and male distributions are really not all that far apart. More noticeable is the skew of all these distributions with a long tail to the right: Low SQs are most common in all groups (including autism) but values across the sample are spread across the full range. So by picking out a random man and a random woman from a crowd, you can be fairly confident that their SQs are both on the lower end but I wouldn’t make any strong guesses about whether the man has a higher SQ than the woman.
However, it gets even tastier because the authors of the study actually also conducted an analysis splitting their data from controls into people in Science, Technology, Engineering, or Maths (STEM) professions compared to controls who were not in STEM. The results (yes, I know the colour code is now weirdly inverted – not how I would have done it…) show that people in STEM, whether male or female, tend to have larger SQs than people outside of STEM. But again, the average difference here is actually small and most of it plays out in the rightward tail of the distributions. The difference between males and females in STEM is also much less distinct than for people outside STEM.
So, as already discussed in the previous section, it seems to be the case that people in STEM professions tend to “systemise” a bit more. It also suggests that men systemise more then women but that difference probably decreases for people in STEM. None of this tells us anything about whether people’s brains are “hard-wired” for systemising, if it is about cultural and environmental differences between men and women, or indeed if being trained in a STEM profession might make people more systemising. It definitely does not tell you who is “good at science”.
What if it were true?
So far so bad for those who might want to make that interpretive leap. But let’s give them the benefit of the doubt and ignore everything I said up until now. What if it were true that systemisers are in fact better scientists? Would that invalidate government or funders initiatives to hire more female scientists? Would that be bad for science?
No. Even if there were a vast difference in systemising between men and women, and between STEM and non-STEM professions, respectively, all such a hiring policy will achieve is to increase the number of good female scientists – exactly what this policy is intended to do. Let me try an analogy.
Basketball players in the NBA tend to be pretty damn tall. Presumably it is easier to dunk when you measure 2 meters than when you’re Tyrion Lannister. Even if all other necessarily skills here are equal there is a clear selection pressure for tall people to get into top basketball teams. Now let’s imagine a team decided they want to hire more shorter players. They declare they will hire 10 players who cannot be taller than 1.70m. The team will have try-outs and still seek to get the best players out of their pool of applicants. If they apply an objective criterion for what makes a good player, such as the ability to score consistently, they will only hire short players with excellent aim or who can jump really high. In fact, these shorties will be on average better at aiming and/or jumping than the giants they already have on their team. The team selects for the ability to score. Shorties and Tallies get there via different means but they both get there.
In this analogy, being a top scorer is being a systemiser, which in turn makes you a good scientist. Giants tend to score high because they find it easy to reach the basket. Shorties score high because they have other skills that compensate for their lack of height. Women can be good systemisers despite not being men.
The only scenario in which such a specific hiring policy could be counterproductive is if two conditions are met: 1) The difference between groups in the critical trait (i.e., systemising) is vast and 2) the policy mandates hiring from a particular group without any objective criteria. We have already established that the former condition isn’t fulfilled here – the difference in systemising between men and women is modest at best. The latter condition is really a moot point because this is simply not how hiring works in the real world. Hiring committees don’t usually just offer jobs to the relatively best person out of the pool but also consider the candidates’ objective abilities and achievements. This is even more pertinent here because all candidates in this case will already be eligible for a professorial position anyway. So all that will in fact happen is that we end up with more female professors who will also happen to be high in systemising.
Bad science reporting
Again, this previous section is based on the entirely imaginary and untenable assumption that systemisers are better scientists. I am not aware of any evidence of that – in part because we cannot actually quantify very well what makes a good scientist. The metrics academics actually (and sadly) use for hiring and funding decisions probably do not quantify that either but I am not even aware of any link between systemising and those metrics. Is there a correlation between h-indeces (relative to career age) and SQ? I doubt it.
What we have here is a case of awful science reporting. Bad science journalism and the abuse of scientific data for nefarious political purposes are hardly a new phenomenon – and this won’t just disappear. But the price of freedom (to practice science) is eternal vigilance. I believe as scientists we have a responsibility to debunk such blatant misapprehensions by journalists who I suspect have never even set foot in an actual lab or spoken to any actual scientists.
Some people assert that improving the transparency and reliability of research will hurt the public’s faith in science. Far from it, I believe those things can show people how science really works. The true damage to how the public perceives science is done by garbage articles in the mainstream media like this one – even if it is merely offered as an “opinion”.
*) Brains are not actually hard-wired to do anything. Leaving the old Hebbian analogy aside, brains aren’t wired at all, period. They are soft, squishy, wet sponges containing lots of neuronal and glial tissue plus blood vessels. Neurons connect via synapses between axons and dendrites and this connectivity is constantly regulated and new connections grown while others are pruned. This adaptability is one of the main reasons why we even have brains, and lies at the heart of the intelligence, ingenuity, and versatility of our species.
**) I suspect a lot of the pencil pushers and bean counters behind metrics like impact factors or the h-index might well be Systemisers.
***) I hope none of them read this post. We don’t want to give these people any further ideas…
****) Isn’t open access under Creative Commons license great?
Today’s post is inspired by another nonsensical proposal that made the rounds and that reminded me why I invented the Devil’s Neuroscientist back in the day (Don’t worry, that old demon won’t make a comeback…). So apparently RetractionWatch created a database allowing you to search for an author’s name to list any retractions or corrections of their publications*. Something called the Ochsner Journal then declared they would use this to scan “every submitting author’s name to ensure that no author published in the Journal has had a paper retracted.” I don’t want to dwell on this abject nonsense – you can read about this in this Twitter thread. Instead I want to talk about the wider mentality that I believe underlies such ideas.
In my view, using retractions as a stigma to effectively excommunicate any researcher from “science” forever is just another manifestation of a rather pervasive and counter-productive tendency of trying to reduce everything in academia to simple metrics and heuristics. Good science should be trustworthy, robust, careful, transparent, and objective. You cannot measure these things with a number.
Perhaps it is unsurprising that quantitative scientists want to find ways to quantify such things. After all, science is the endeavour to reveal regularities in our observations to explain the variance of the natural world and thus reduce the complexity in our understanding of it. There is nothing wrong with meta-science and trying to derive models of how science – and scientists – work. But please don’t pretend that these models are anywhere near good enough to actually govern all of academia.
Few people you meet still believe that the Impact Factor of a journal tells you much about the quality of a given publication in it. Neither does an h-index or citation count tell us anything about the importance or “impact” of somebody’s research, certainly not without looking at this relative to the specific field of science they operate in. The rate with which someone’s findings replicate doesn’t tell you anything about how great a scientist they are. And you certainly won’t learn anything about the integrity and ability of a researcher – and their right to publish in your journal – when all you have to go on is that they were an author on one retracted study.
Reducing people’s careers and scientific knowledge to a few stats is lazy at best. But it is also downright dangerous. As long as such metrics are used to make consequential real-life decisions, people are incentivised to game them. Nowhere can this be seen better than with the dubious tricks some journals use to inflate their Impact Factor or the occasional dodgy self-citation scandals. Yes, in the most severe cases these are questionable, possibly even fraudulent, practices – but there is a much greater grey area here. What do you think would happen, if we adopted the policy that only researchers with high replicability ratings get put up for promotion? Do you honestly think this would encourage scientists to do better science rather than merely safer, boring science?
This argument is sometimes used as a defence of the status quo and a reason why we shouldn’t change the way science is done. Don’t be fooled by that. We should reward good and careful science. We totally should give credit to people who preregister their experiments, self-replicate their findings, test the robustness of their methods, and go the extra mile to ensure their studies are solid. We should appreciate hypotheses based on clever, creative, and/or unique insights. We should also give credit to people for admitting when they are wrong – otherwise why should anyone seek the truth?
The point is, you cannot do any of that with a simple number in your CV. Neither can you do that by looking at retractions or failures to replicate as a plague mark on someone’s career. I’m sorry to break it to you, but the only way to assess the quality of some piece of research, or to understand anything about the scientist behind it, is to read their work closely and interpret it in the appropriate context. That takes time and effort. Often it also necessitates talking to them because no matter how clever you think you are, you will not understand everything they wrote, just as not everybody will comprehend the gibberish you write. If you believe a method is inadequate, by all means criticise it. Look at the raw data and the analysis code. Challenge interpretations you disagree with. Take nobody’s word for granted and all that…
But you can shove your metrics where the sun don’t shine.
TL-DR: If the title of this blog post is unsurprising to you, I suggest you go play outside.
Many discussions in my science social media bubble circle around p-values (what an exciting life I lead…). Just a few days ago, there was a big kerfuffle about p-curving and whether p-values just below 0.05 are a sign of whatever. One of the main concepts behind p-curves is that under the assumption that the null hypothesis (H0) of no effect/difference is true, p-values should be uniformly distributed (at least as long as the test assumptions are met reasonably). This once again supported my suspicions that most people don’t actually know what p-values mean. Reports of people defining p-values incorrectly abound, sometimes even in stats textbooks. It also seems to me that people find p-values rather unintuitive. And I get the impression a lot of people vastly overestimate how widely known things like p-curve actually are.
A few weeks ago I got embroiled in a Facebook discussion. A friend of mine was running a permutation analysis to test something about his experiment and found something very odd: the distribution of p-values was skewed severely to the left – there were very few low p-values but the proportion was steadily increasing with most p-values being just below 1. He expected this distribution to be uniform because under the random permutations H0 should be true. A lot of commenters on his post seemed rather surprised and/or confused by the whole idea that p-values should be distributed randomly when H0 is true. “Surely,” so the common intuition goes, “when there is actually no difference, most p-values should be high and close to 1?”
No, and the reason why not is the p-value itself. A p-value can be calculated/estimated in many different ways. Most people use parametric tests but essentially they all share one philosophy. If you have no underlying effect and randomly sample data ad infinitum you end up with a distribution of test statistics. In my example, I draw two variables each with n=100 from a normal distribution and calculate the Pearson correlation between them – and I repeat this 20,000 times. This produces a distribution of correlation coefficients like this:
There is no correlation between two random variables (H0 is true) and so the distribution is centred on zero. The spread of the distribution depends on the sample size. Larger samples will produce narrower distributions. Critically, we can use this distribution to get a p-value. If we had observed a correlation of r=0.3 in our experiment, we could calculate the proportion of correlation coefficients in this distribution that are equal or greater than 0.3. This would give us a one-tailed p-value. If you ignore the sign of the correlation, you get a two-tailed p-value.
In the plot above, I coloured the 5% most extreme correlation coefficients in blue (2.5% to the left and to the right, respectively). These regions are abutted by vertical red lines at just below +/-0.2 in this case. This reflects the critical effect size needed to get p<0.05 – only 5% of the correlations coefficients in this distribution are +/-0.19ish or even more extreme.
Now compare this to the region coloured in red. This region also makes up 5% of the whole distribution. However, the red region surrounds zero, that is, those correlation coefficients that are really close to the true correlation value. Random chance makes the distribution spread out (and that becomes more severe when your sample size is low) but most of the correlations will nevertheless be close to the true value of zero. Therefore, the range of values in this red region is much narrower because the values are much denser here.
But of course these nigh-zero correlation coefficients will have the largest p-values. Consider again what a p-value reflects. If your observed correlation is 0.006 and you again ignore the sign of the effects, almost all correlations in this null distribution would be equal or greater than 0.006. So this proportion, the p-value, is almost 1. Put in other words, 5% of low p-values below 0.05 are from the long, thin tails of the null distribution, while 5% of really high p-values above 0.95 are from a really narrow slither of the null distribution near zero:
Visualised the same way, you have the blue region with p<0.05 on the left. Here correlations are large (greater than 0.19ish). On the right, you have the red region with p>0.95. Here correlations are really close to zero.
In other words, you can directly read off the p-value from the x-axis of this distribution of p-values. This is a direct consequence of what p-values represent. They are the proportion of values in the null distribution where correlations are equal or more extreme than the observed correlation.
Of course, if the null hypothesis is false and there actually is a correlation between the two variables this distribution must become skewed. There should now be many more tests with low p-values than with large ones. This is exactly what happens and this is the pattern that analyses like p-curve seek to detect:
Now, my friend’s p-distribution looked essentially like the mirror image of this. I still haven’t learned what could have possibly caused this. It would mean that more effect sizes were close to zero than there should be under H0. This could suggest some assumptions not being met but none of my own feeble simulations managed to reproduce the pattern he found. His analyses sounded quite complex so there is a possibility that there were some complex errors in it.
TL;DR: You may get a very large relative effect size (like Cohen’s d), if the main source of the variability in your sample is the reliability of each observation and the measurement was made as exact as is feasible. Such a large d is not trivial, but in this case talking about d is missing the point.
In discussions of scientific findings you will often hear talk about relative effect sizes, like the ubiquitous Cohen’s d. Essentially, such effect sizes quantify the mean difference between groups/treatments/conditions relative to the variability across subjects/observations. The situation is actually a lot more complicated because even for a seemingly simple results like the difference between conditions you will find that there are several ways of calculating the effect size. You can read a nice summary by Jake Westfall here. There are also other effect sizes, such as correlation coefficients, and what I write here applies to that, too. I will however stick to the difference-type effect size because it is arguably the most common.
One thing that has irked me about those discussions for some years is that this ignores a very substantial issue: the between-subject variance of your sample depends on the within-subject variance. The more unreliable the measurement of each subject, the greater is the variability of your sample. Thus the reliability of individual measurements limits the relative effect size you can possibly achieve in your experiment given a particular experimental design. In most of science – especially biological and psychological sciences – the reliability of individual observations is strongly limited by the measurement error and/or the quality of your experiment.
There are some standard examples that are sometimes used to illustrate what a given effect size means. I stole a common one from this blog post about the average height difference between men and women, which apparently was d=1.482 in 1980 Spain. I have no idea if this is true exactly but that figure should be in the right ballpark. I assume most people will agree that men are on average taller than women but that there is nevertheless substantial overlap in the distributions – so that relatively frequently you will find a woman who is taller than many men. That is an effect size we might consider strong.
The height difference between men and women is a good reference for an effect size because it is largely limited by the between-subject variance, the variability in actual heights across the population. Obviously, the reliability of each observation also plays a role. There will definitely be a degree of measurement error. However, I suspect that this error is small, probably on the order of a few millimeters. Even if you’re quite bad at this measurement I doubt you will typically err by more than 1-2 cm and you can probably still replicate this effect in a fairly small sample. However, in psychology experiments your measurement rarely is that accurate.
Now, in some experiments you can increase the reliability of your individual measurement by increasing the number of trials (at this point I’d like to again refer to Richard Morey’s related post on this topic). In psychophysics, collecting hundreds or thousands of trials on one individual subject is not at all uncommon. Let’s take a very simple case. Contour integration refers to the ability of the visual system to detect “snake” contours better than “ladder” contours or those defined by other orientations (we like to call those “ropes”):
In the left image you should hopefully see a circle defined by 16 grating patches embedded in a background or randomly oriented gratings. This “snake” contour pops out from the background because the visual system readily groups orientations along a collinear (or cocircular) path into a coherent object. In contrast, when the contour is defined by patches of other orientations, for example the “rope” contour in the right image which is defined by patches at 45 degrees relative to the path, then it is much harder to detect the presence of this contour. This isn’t a vision science post so I won’t go into any debates on what this means. The take-home message here is that if healthy subjects with normal vision are asked to determine the presence or absence of a contour like this, especially with limited viewing time, they will perform very well for the “snake” contours but only barely above chance levels for the “rope” contours.
This is a very robust effect and I’d argue this is quite representative of many psychophysical findings. A psychophysicist probably wouldn’t simply measure the accuracy but conduct a broader study of how this depends on particular stimulus parameters – but that’s not really important here. It is still pretty representative.
What is the size of this effect?
If I study this in a group of subjects, the relative effect size at the group level will depend on how accurately I measure the performance in each individual. If I have 50 subjects (which is between 10-25 larger than your typical psychophysics study…) and each performs just one trial, then the sample variance will be much larger compared to if each of them does 100 trials or if they each do 1000 trials. As a result, the Cohen’s d of the group will be considerably different. A d>10 should be entirely feasible if we collect enough trials per person.
People will sometimes say that large effects (d>>2 perhaps) are trivial. But there is nothing trivial about this. In this particular example you may see the difference quite easily for yourself (so you are a single-subject and single-trial replication). But we might want to know just how much better we are at detecting the snake than the rope contours. Or, as I already mentioned, a psychophysicist might measure the sensitivity of subjects to various stimulus parameters in this experiment (e.g., the distance between patches, the amount of noise in the orientations we can tolerate, etc) and this could tell us something about how vision works. The Cohen’s d would be pretty large for all of these. That does not make it trivial but in my view it makes it useless:
Depending on my design choices the estimated effect size may be a very poor reflection of the true effect size. As mentioned earlier, the relative effect size is directly dependent on the between-subject variance – but that in turn depends on the reliability of individual measurements. If each subject only does one trial, the effect of just one attentional lapse or accidental button press in the task is much more detrimental than when they perform 1000 trials, even if the overall rate of lapses/accidents is the same*.
Why does this matter?
In many experiments, the estimate of between-subject variance will be swamped by the within-subject variability. Returning to the example of gender height differences, this is essentially what would happen if you chose to eyeball each person’s height instead of using a tape measure. I’d suspect that is the case for many experiments in social or personality psychology where each measurement is essentially a single quantity (say, timing the speed with which someone walks out of the lab in a priming experiment) rather than being based on hundreds or thousands of trials as in psychophysics. Notoriously noisy measurements are also doubtless the major limiting factor in most neuroimaging experiments. On the other hand, I assume a lot of questionnaire-type results you might have in psychology (such as IQ or the Big Five personality factors) have actually pretty high test-retest reliability and so you probably do get mostly the between-subject variance.
The problem is that often it is very difficult to determine which scenario we are in. In psychophysics, we are often so extremely dominated by the measurement reliability that a knowledge of the “true” population effect size is actually completely irrelevant. This is a critical issue because you cannot use such an effect size for power analysis: If I take an experiment someone did and base my power analysis on the effect size they reported, I am not really powering my experiment to detect a similar effect but a similar design. (This is particularly useless if I then decide to use a different design…)
So next time you see an unusually large Cohen’s (d>10 or even d>3) ask yourself not simply whether this is a plausible effect but whether this experiment can plausibly estimate the true population effect. If this result is based on a single observation per subject with a highly variable measurement (say, how often Breton drivers stop for female hitchhikers wearing red clothing…), even a d=1 seems incredibly large.
But if it is for a measurement that could have been made more reliable by doubling the amount of data collected in each subject (say, a change in psychometric thresholds), then a very high Cohen’s d is entirely plausible – but it is also pretty meaningless. In this situation, what we should really care about is the absolute effect size (How much does the threshold change? How much does the accuracy drop? etc).
And I must say, I remain unsure whether absolute effect sizes aren’t more useful in general, including for experiments on complex human behaviour, neuroimaging, or drug effects.
* Actually the lapse rate probably increases with a greater number of trials due to subject fatigue, drop in motivation, or out of pure spite. But even that increase is unlikely to be as detrimental as having too few trials.
The other day, my twitter feed got embroiled in another discussion about whether or not p-hacking is deliberate and if it constitutes fraud. Fortunately, I then immediately left for a trip abroad and away from my computer, so there was no danger of me being drawn into this debate too deeply and running the risk of owing Richard Morey another drink. However, now that I am back I wanted to elaborate a bit more on why I think the way our field has often approached p-hacking is both wrong and harmful.
What the hell is p-hacking anyway? When I google it I get this Wikipedia article, which uses it as a synonym for “data dredging”. There we already have a term that seems to me more appropriate. P-hacking refers to when you massage your data and analysis methods until your result reaches a statistically significant p-value. I will put it to you that in practice most p-hacking is not necessarily about hacking p-s but about dredging your data until your results fit a particular pattern. That may be something you predicted but didn’t find or could even just be some chance finding that looked interesting and is amplified this way. However, the p-value is usually probably secondary to the act here. The end result may very well be the same in that you continue abusing the data until a finding becomes significant, but I would bet that in most cases what matters to people is not the p-value but the result. Moreover, while null-hypothesis significance testing with p-values is still by far the most widespread way to make inferences about results, it is not the only way. All this fussing about p-hacking glosses over the fact that the same analytic flexibility or data dredging can be applied to any inference, whether it is based on p-values, confidence intervals, Bayes factors, posterior probabilities, or simple summary statistics. By talking of p-hacking we create a caricature that this is somehow a problem specific to p-values. Whether or not NHST is the best approach for making statistical inferences is a (much bigger) debate for another day – but it has little to do with p-hacking.
What is more, not only is p-hacking not really about p’s but it is also not really about hacking. Here is the dictionary entry for the term ‘hacking‘. I think we can safely assume that when people say p-hacking they don’t mean that peas are physically being chopped or cut or damaged in any way. I’d also hazard a guess that it’s not meant in the sense of “to deal or cope with” p-values. In fact, the only meaning of the term that seems to come even remotely close is this:
Obviously, what is being modified in p-hacking is the significance or impressiveness of a result, rather than a computer program or electronic device, but we can let this slide. I’d also suggest that it isn’t always done in a skillful or clever way either, but perhaps we can also ignore this. However, the verb ‘hacking’ to me implies that this is done in a very deliberate way. It may even, as with computer hacking, carry the connotation of fraud, of criminal intent. I believe neither of these things are true about p-hacking.
That is not to say that p-hacking isn’t deliberate. I believe in many situations it likely is. People no doubt make conscious decisions when they dig through their data. But the overwhelming majority of p-hacking is not deliberately done to createspurious resultsthat the researcher knows to be false. Anyone who does so would be committing actual fraud. Rather, most p-hacking is the result of confirmation bias combined with analytical flexibility. This leads people to sleep walk into creating false positives or – as Richard Feynman would have called it – fooling themselves. Simine Vazire already wrote an excellent post about this a few years ago (and you may see a former incarnation of yours truly in the comment section arguing against the point I’m making here… I’d like to claim that it’s cause I have grown as a person but in truth I only exorcised this personality :P). I’d also guess that a lot of p-hacking happens out of ignorance, although that excuse really shouldn’t fly as easily in 2017 as it may have done in 2007. Nevertheless, I am pretty sure people do not normally p-hack because they want to publish false results.
Some may say that it doesn’t matter whether or not p-hacking is fraud – the outcome is the same: many published results are false. But in my view it’s not so simple. First, the solution to these two problems surely isn’t the same. Preregistration and transparency may very well solve the problem of analytical flexibility and data dredging – but it is not going to stop deliberate fraud, nor is it meant to. Second, actively conflating fraud and data dredging implicitly accuses researchers of being deliberately misleading and thus automatically puts them on the defensive. This is hardly a way to have a productive discussion and convince people to do something about p-hacking. You don’t have to look very far for examples of that playing out. Several protracted discussions on a certain Psychology Methods Facebook group come to mind…
Methodological flexibility is a real problem. We definitely should do something about it and new moves towards preregistration and data transparency are at least theoretically effective solutions to improve things. The really pernicious thing about p-hacking is that people are usually entirely unaware of the fact that they are doing it. Until you have tried to do a preregistered study, you don’t appreciate just how many forks in the road you passed along the way (I may blog about my own experiences with that at some point). So implying, however unintentionally, that people are fraudsters is not helping matters.
Preregistration and data sharing have gathered a lot of momentum over the past few years. Perhaps the opinions of some old tenured folks opposed to such approaches no longer carry so much weight now, regardless how powerful they may be. But I’m not convinced that this is true. Just because there has been momentum now does not mean that these ideas will prevail. It is just as likely that they fizzle out due to lacking enthusiasm or because people begin to feel that the effort isn’t worth it. I seems to me that “open science” very much exists in a bubble and I have bemoaned that before. To change scientific practices we need to open the hearts and minds of sceptics to why p-hacking is so pervasive. I don’t believe we will achieve that by preaching to them. Everybody p-hacks if left to their own devices. Preregistration and open data can help protect yourself against your mind’s natural tendency to perceive patterns in noise. A scientist’s training is all about developing techniques to counteract this tendency, and so open practices are just another tool for achieving that purpose.