The usability of statistics; or, what happens when you think that (p=.05) != (p=.06)
The difference between significant and not significant is not itself significant.
That is the title of a 2006 paper by statisticians Andrew Gelman and Hal Stern. It is also the theme of a new review article in Nature Neuroscience by Sander Nieuwenhuis, Birte U Forstmann, and Eric-Jan Wagenmakers (via Gelman’s blog). The review examined several hundred papers in behavioral, systems, and cognitive neuroscience. Of all the papers that tried to compare two effects, about half of them made this error instead of properly testing for an interaction.
I don’t know how often the error makes it through to published papers in social and personality psychology, but I see it pretty regularly as a reviewer. I call it when I see it; sometimes other reviewers call it out too, sometimes they don’t.
I can also remember making this error as a grad student – and my advisor correcting me on it. But the funny thing is, it’s not something I was taught. I’m quite sure that nowhere along the way did any of my teachers say you can compare two effects by seeing if one is significant and the other isn’t. I just started doing it on my own. (And now I sometimes channel my old advisor and correct my own students on the same error, and I’m sure nobody’s teaching it to them either.)
If I wasn’t taught to make this error, where was I getting it from? When we talk about whether researchers have biases, usually we think of hot-button issues like political bias. But I think this reflects a more straightforward kind of bias — old habits of thinking that we carry with us into our professional work. To someone without scientific training, it seems like you should be able to ask “Does X cause Y, yes or no?” and expect a straightforward answer. Scientific training teaches us a couple of things. First, the question is too simple: it’s not a yes or no question; the answer is always going to come with some uncertainty; etc. Second, the logic behind the tool that most of us use – null hypothesis significance testing (NHST) – does not even approximate the form of the question. (Roughly: “In a world where X has zero effect on Y, would we see a result this far from the truth less than 5% of the time?”)
So I think what happens is that when we are taught the abstract logic of what we are doing, it doesn’t really pervade our thinking until it’s been ground into us through repetition. For a period of time – maybe in some cases forever – we carry out the mechanics of what we have been taught to do (run an ANOVA) but we map it onto our old habits of thinking (“Does X cause Y, yes or no?”). And then we elaborate and extrapolate from them in ways that are entirely sensible by their own internal logic (“One ANOVA was significant and the other wasn’t, so X causes Y more than it causes Z, right?”).
One of the arguments you sometimes hear against NHST is that it doesn’t reflect the way researchers think. It’s a sort of usability argument: NHST is the butterfly ballot of statistical methods. In principle, I don’t think that argument carries the day on its own (if we need to use methods and models that don’t track our intuitions, we should). But it should be part of the discussion. And importantly, the Nieuwenhuis et al. review shows us how using unintuitive methods can have real consequences.