Statistics is an Excellent Servant and a Bad Master

I.

Imagine a universe where every cognitive scientist receives extensive training in how to deal with demand characteristics. 

(Demand characteristics describe any situation in a study where a participant either figures out what a study is about, or thinks they have, and changes what they do in response. If the participant is friendly and helpful, they may try to give answers that will make the researchers happy; if they have the opposite disposition, they might intentionally give nonsense answers to ruin the experiment. This is a big part of why most studies don’t tell participants what condition they’re in, and why some studies are run double-blind.)

In the real world, most students get one or two lessons about demand characteristics when they take their undergrad methods class. When researchers are talking about a study design, sometimes we mention demand, but only if it seems relevant.

Let’s return to our imaginary universe. Here, things are very different. Demand characteristics are no longer covered in undergraduate methods courses — instead, entire classes are exclusively dedicated to demand characteristics and how to deal with them. If you major in a cognitive science, you’re required to take two whole courses on demand — Introduction to Demand for the Psychological Sciences and Advanced Demand Characteristics

Often there are advanced courses on specific forms of demand. You might take a course that spends a whole semester looking at the negative-participant role (also known as the “screw-you effect”), or a course on how to use deception to avoid various types of demand. 

If you apply to graduate school, how you did in these undergraduate courses will be a major factor determining whether they let you in. If you do get in, you still have to take graduate-level demand courses. These are pretty much the same as the undergrad courses, except they make you read some of the original papers and work through the reasoning for yourself. 

When presenting your research in a talk or conference, you can usually expect to get a couple of questions about how you accounted for demand in your design. Students are evaluated based on how well they can talk about demand and how advanced the techniques they use are.

Every journal requires you to include a section on demand characteristics in every paper you submit, and reviewers will often criticize your manuscript because you didn’t account for demand in the way they expected. When you go up for a job, people want to know that you’re qualified to deal with all kinds of demand characteristics. If you have training in dealing with an obscure subtype of demand, it will help you get hired.

It would be pretty crazy to devote such a laser focus to this one tiny aspect of the research process. Yet this is exactly what we do with statistics.

II. 

Science is all about alternative explanations. We design studies to rule out as many stories as we can. Whatever stories remain are possible explanations for our observations. Over time, we whittle this down to a small number of well-supported theories. 

There’s one alternative explanation that is always a concern. For any relationship we observe, there’s a chance that what we’re seeing is just noise. Statistics is a set of tools designed to deal with this problem. This holds a special place in science because “it was noise” is a concern for every study in every field, so we always want to make sure to rule it out.    

But of course, there are many alternative explanations that we need to be concerned with. Whenever you’re dealing with human participants, demand characteristics will also be a possible alternative. Despite this, we don’t jump down people’s throats about demand. We only bring up these issues when we have a reason to suspect that it is a problem for the design we’re looking at.

There will always be more than one way to look at any set of results. We can never rule out every alternative explanation — the best we can do is account for the most important and most likely alternatives. We decide which ones to account for by using our judgement, by taking some time to think about what alternatives we (and our readers) will be most concerned about. 

The right answer will look different for different experiments. But the wrong answer is to blindly throw statistics at every single study. 

Statistics is useful when a finding looks like it could be the result of noise, but you’re not sure. For example, let’s say we’re testing a new treatment for a disease. We have a group of 100 patients who get the treatment and a control group of 100 people who don’t get the treatment. If 52/100 people recover when they get the treatment, compared to 42/100 recovering in the control group, does that mean the treatment helped? Or is the difference just noise? I can’t tell with just a glance, but a simple chi-squared test can tell me that p = .013, meaning there’s only a 1.3% chance that we would see something like this from noise alone.

That’s helpful, but it would be pointless to run a statistical test if we saw 43/100 people recover with the treatment, compared to 42/100 in the control group. I can tell that this is very consistent with noise (p > .50) just by looking at it. And it would be pointless to run a statistical test if we saw 98/100 people recover with the treatment, compared to 42/100 in the control group. I can tell that this is very inconsistent with noise (p < .00000000000001) just by looking at it. If something passes the interocular trauma test (the conclusion hits you between the eyes), you don’t need to pull out another test.

This might sound outlandish today, but you can do perfectly good science without any statistics at all. After all, statistics is barely more than a hundred years old. Sir Francis Galton came up with the concept of the standard deviation in the 1860s, and the story with the ox didn’t happen until 1907. It took until the 1880s to dream up correlation. Karl Pearson was born in 1857 but didn’t do most of his statistics work until around the turn of the century. Fisher wasn’t even born until 1890. He introduced the term variance for the first time in 1918, but both that term and the ANOVA didn’t gain popularity until the publication of his book in 1925.

This means that Galileo, Newton, Kepler, Hooke, Pasteur, Mendel, Lavoisier, Maxwell, von Helmholtz, Mendeleev, etc. did their work without anything that resembled modern statistics, and that Einstein, Curie, Fermi, Bohr, Heisenberg, etc. etc. did their work in an age when statistics was still extremely rudimentary. We don’t need statistics to do good research.

This isn’t an original idea, or even a particularly new one. When statistics was young, people understood this point better. For an example, we can turn to Sir Austin Bradford Hill. He was trained by Karl Pearson (who, among other things, invented the chi-squared test we used earlier), was briefly president of the Royal Statistical Society, and was sometimes referred to as the world’s leading medical statistician. As early as the 1920s, he was pioneering the introduction of the randomized clinical trial in medicine. As far as opinions on statistics go, the man was pretty qualified. 

While you may not know his name, you’re probably familiar with his work. He was one of the researchers who demonstrated the connection between cigarette smoking and lung cancer, and in 1965 he gave a speech about his work on the topic. Most of the speech was a discussion of how one can infer a causal relationship from largely correlational data, as he had done with the smoking-lung cancer connection, a set of considerations that came to be known as the Bradford Hill criteria

But near the end of the speech, he turns to a discussion of tests of significance, as he calls them, and their limitations:

No formal tests of significance can answer [questions of cause and effect]. Such tests can, and should, remind us of the effects that the play of chance can create, and they will instruct us in the likely magnitude of those effects. Beyond that they contribute nothing to the ‘proof’ of our hypothesis. 

Nearly forty years ago, amongst the studies of occupational health that I made for the Industrial Health Research Board of the Medical Research Council was one that concerned the workers in the cotton-spinning mills of Lancashire (Hill 1930). … All this has rightly passed into the limbo of forgotten things. What interests me today is this: My results were set out for men and women separately and for half a dozen age groups in 36 tables. So there were plenty of sums. Yet I cannot find that anywhere I thought it necessary to use a test of significance. The evidence was so clear cut, the differences between the groups were mainly so large, the contrast between respiratory and non-respiratory causes of illness so specific, that no formal tests could really contribute anything of value to the argument. So why use them?

Would we think or act that way today? I rather doubt it. Between the two world wars there was a strong case for emphasizing to the clinician and other research workers the importance of not overlooking the effects of the play of chance upon their data. Perhaps too often generalities were based upon two men and a laboratory dog while the treatment of choice was deducted from a difference between two bedfuls of patients and might easily have no true meaning. It was therefore a useful corrective for statisticians to stress, and to teach the needs for, tests of significance merely to serve as guides to caution before drawing a conclusion, before inflating the particular to the general. 

I wonder whether the pendulum has not swung too far – not only with the attentive pupils but even with the statisticians themselves. To decline to draw conclusions without standard errors can surely be just as silly? Fortunately I believe we have not yet gone so far as our friends in the USA where, I am told, some editors of journals will return an article because tests of significance have not been applied. Yet there are innumerable situations in which they are totally unnecessary – because the difference is grotesquely obvious, because it is negligible, or because, whether it be formally significant or not, it is too small to be of any practical importance. What is worse, the glitter of the t-table diverts attention from the inadequacies of the fare. Only a tithe, and an unknown tithe, of the factory personnel volunteer for some procedure or interview, 20% of patients treated in some particular way are lost to sight, 30% of a randomly-drawn sample are never contacted. The sample may, indeed, be akin to that of the man who, according to Swift, ‘had a mind to sell his house and carried a piece of brick in his pocket, which he showed as a pattern to encourage purchasers.’ The writer, the editor and the reader are unmoved. The magic formulae are there. 

Of course I exaggerate. Yet too often I suspect we waste a deal of time, we grasp the shadow and lose the substance, we weaken our capacity to interpret the data and to take reasonable decisions whatever the value of P. And far too often we deduce ‘no difference’ from ‘no significant difference.’ Like fire, the chi-squared test is an excellent servant and a bad master.

III.

We grasp the shadow and lose the substance. 

As Dr. Hill notes, the blind use of statistical tests is a huge waste of time. Many designs don’t need them; many arguments don’t benefit from them. Despite this, we have long disagreements about which of two tests is most appropriate (even when both of them will be highly significant), we spend time crunching numbers when we already know what we will find, and we demand that manuscripts have their statistics arranged just so — even when it doesn’t matter.

This is an institutional waste of time as well as a personal one. It’s weird that students get so much training in statistics. Methods are almost certainly more important, but most students are forced to take multiple stats classes, while only one or two methods classes are even offered. This is also true at the graduate level. Methods and theory courses are rare in graduate course catalogs, but there is always plenty of statistics.

Some will say that this is because statistics is so much harder to learn than methods. Because it is a more difficult subject, it takes more time to master. Now, it’s true that students tend to take several courses in statistics and come out of them remembering nothing at all about statistics. But this isn’t because statistics is so much more difficult. 

We agree that statistical thinking is very important. What we take issue with is the neurotic focus on statistical tests, which are of minor use at best. The problem is that our statistics training spends multiple semesters on tests, while spending little to no time at all on statistical thinking. 

This also explains why students don’t learn anything in their statistics classes. Students can tell, even if only unconsciously, that the tests are unimportant, so they have a hard time taking them seriously. They would also do poorly if we asked them to memorize a phone book — so much more so if we asked them to memorize the same phone book for three semesters in a row.

The understanding of these tests is based on statistical thinking, but we don’t teach them that. We’ve become anxious around the tests, and so we devote more and more of the semester to them. But this is like becoming anxious about planes crashing and devoting more of your pilot training time to the procedure for making an emergency landing. If the pilots get less training in the basics, there will be more emergency landings, leading to more anxiety and more training, etc. — it’s a vicious cycle. If you just teach students statistical thinking to begin with, they can see why it’s useful and will be able to easily pick up the less-important tests later on, which is exactly what I found when I taught statistics this way.

The bigger problem is turning our thinking over to machines, especially ones as simple as statistical tests.

Your new overlord.

Sometimes a test is useful, sometimes it is not. We can have discussions about when a test is the right choice and when it is the wrong one. Researchers aren’t perfect, but we have our judgement and damn it, we should be expected to use it. We may be wrong sometimes, but that is better than letting the p-values call all the shots. 

We need to stop taking tests so seriously as a criterion for evaluating papers. There’s a reason, of course, that we are on such high alert about these tests — the concept of p-hacking is only a decade old, and questionable statistical practices are still being discovered all the time. 

But this focus on statistical issues tends to obscure deeper problems. We know that p-hacking is bad, but a paper with perfect statistics isn’t necessarily good — the methods and theory, even the basic logic, can be total garbage. In fact, this is part of how we got in the p-hacking situation in the first place: by using statistics as the main way of telling if a paper is any good or not! 

Putting statistics first is how we end up with studies with beautifully preregistered protocols and immaculate statistics, but deeply confounded methods, on topics that are unimportant and frankly uninteresting. This is what Hill meant when he said that “the glitter of the t-table diverts attention from the inadequacies of the fare”. Confounded methods can produce highly significant p-values without any p-hacking, but that doesn’t mean the results of such a study are of any value at all. 

This is why I find proposals to save science by revising statistics so laughable. Surrendering our judgement to Bayes factors instead of p-values won’t do anything to solve our problems. Changing the threshold of significance from .05 to .01, or .005, or even .001 won’t make for better research. We shouldn’t try to revise statistics, we should use it less often. 


Thanks to Adam Mastroianni, Grace Rosen, and Alexa Hubbard for reading drafts of this piece.

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