no free lunch in statistics

Simon and Tibshirani recently posted a short comment on the Reshef et al MIC data mining paper I blogged about a while back:

The proposal of Reshef et. al. (“MIC”) is an interesting new approach for discovering non-linear dependencies among pairs of measurements in exploratory data mining. However, it has a potentially serious drawback. The authors laud the fact that MIC has no preference for some alternatives over others, but as the authors know, there is no free lunch in Statistics: tests which strive to have high power against all alternatives can have low power in many important situations.

They then report some simulation results clearly demonstrating that MIC is (very) underpowered relative to Pearson correlation in most situations, and performs even worse relative to Székely & Rizzo’s distance correlation (which I hadn’t heard about, but will have to look into now). I mentioned low power as a potential concern in my own post, but figured it would be an issue under relatively specific circumstances (i.e., only for certain kinds of associations in relatively small samples). Simon & Tibshirani’s simulations pretty clearly demonstrate that isn’t so. Which, needless to say, rather dampens the enthusiasm for the MIC statistic.

in defense of three of my favorite sayings

Seth Roberts takes issue with three popular maxims that (he argues) people use “to push away data that contradicts this or that approved view of the world”. He terms this preventive stupidity. I’m a frequent user of all three sayings, so I suppose that might make me preventively stupid; but I do feel like I have good reasons for using these sayings, and I confess to not really seeing Roberts’ point.

Here’s what Roberts has to say about the three sayings in question:

1. Absence of evidence is not evidence of absence. Øyhus explains why this is wrong. That such an Orwellian saying is popular in discussions of data suggests there are many ways we push away inconvenient data.

In my own experience, by far the biggest reason this saying is popular in discussions of data (and the primary reason I use it when reviewing papers) is that many people have a very strong tendency to interpret null results as an absence of any meaningful effect. That’s a very big problem, because the majority of studies in psychology tend to have relatively little power to detect small to moderate-sized effects. For instance, as I’ve discussed here, most whole-brain analyses in typical fMRI samples (of say, 15 – 20 subjects) have very little power to detect anything but massive effects. And yet people routinely interpret a failure to detect hypothesized effects as an indication that they must not exist at all. The simplest and most direct counter to this type of mistake is to note that one shouldn’t accept the null hypothesis unless one has very good reasons to think that power is very high and effect size estimates are consequently quite accurate. Which is just another way of saying that absence of evidence is not evidence of absence.

2. Correlation does not equal causation. In practice, this is used to mean that correlation is not evidence for causation. At UC Berkeley, a job candidate for a faculty position in psychology said this to me. I said, “Isn’t zero correlation evidence against causation?” She looked puzzled.

Again, Roberts’ experience clearly differs from mine; I’ve far more often seen this saying used as a way of suggesting that a researcher may be drawing overly strong causal conclusions from the data, not as a way of simply dismissing a correlation outright. A good example of this is found in the developmental literature, where many researchers have observed strong correlations between parents’ behavior and their children’s subsequent behavior. It is, of course, quite plausible to suppose that parenting behavior exerts a direct causal influence on children’s behavior, so that the children of negligent or abusive parents are more likely to exhibit delinquent behavior and grow up to perpetuate the “cycle of violence”. But this line of reasoning is substantially weakened by behavioral genetic studies indicating that very little of the correlation between parents’ and children’s personalities is explained by shared environmental factors, and that the vast majority reflects heritable influences and/or unique environmental influences. Given such findings, it’s a perfectly appropriate rebuttal to much of the developmental literature to note that correlation doesn’t imply causation.

It’s also worth pointing out that the anecdote Roberts provides isn’t exactly a refutation of the maxim; it’s actually an affirmation of the consequent. The fact that an absence of any correlation could potentially be strong evidence against causation (under the right circumstances) doesn’t mean that the presence of a correlation is strong evidence for causation. It may or may not be, but that’s something to be weighed on a case-by-case basis. There certainly are plenty of cases where it’s perfectly appropriate (and even called for) to remind someone that correlation doesn’t imply causation.

3. The plural of anecdote is not data. How dare you try to learn from stories you are told or what you yourself observe!

I suspect this is something of a sore spot for Roberts, who’s been an avid proponent of self-experimentation and case studies. I imagine people often dismiss his work as mere anecdote rather than valuable data. Personally, I happen to think there’s tremendous value to self-experimentation (at least when done in as controlled a manner as possible), so I don’t doubt there are many cases where this saying is unfairly applied. That said, I think Roberts fails to appreciate that people who do his kind of research constitute a tiny fraction of the population. Most of the time, when someone says that “the plural of anecdote is not data,” they’re not talking to someone who does rigorous self-experimentation, but to people who, say, don’t believe they should give up smoking seeing as how their grandmother smoked till she was 88 and died in a bungee-jumping accident, or who are convinced that texting while driving is perfectly acceptable because they don’t personally know anyone who’s gotten in an accident. In such cases, it’s not only legitimate but arguably desirable to point out that personal anecdote is no substitute for hard data.

Orwell was right. People use these sayings — especially #1 and #3 — to push away data that contradicts this or that approved view of the world. Without any data at all, the world would be simpler: We would simply believe what authorities tell us. Data complicates things. These sayings help those who say them ignore data, thus restoring comforting certainty.

Maybe there should be a term (antiscientific method?) to describe the many ways people push away data. Or maybe preventive stupidity will do.

I’d like to be charitable here, since there very clearly are cases where Roberts’ point holds true: sometimes people do toss out these sayings as a way of not really contending with data they don’t like. But frankly, the general claim that these sayings are antiscientific and constitute an act of stupidity just seems silly. All three sayings are clearly applicable in a large number of situations; to deny that, you’d have to believe that (a) it’s always fine to accept the null hypothesis, (b) correlation is always a good indicator of a causal relationship, and (c) personal anecdotes are just as good as large, well-controlled studies. I take it that no one, including Roberts, really believes that. So then it becomes a matter of when to apply these sayings, and not whether or not to use them. After all, it’d be silly to think that the people who use these sayings are always on the side of darkness, and the people who wield null results, correlations, and anecdotes with reckless abandon are always on the side of light.

My own experience, for what it’s worth, is that the use of these sayings is justified far more often than not, and I don’t have any reservation applying them myself when I think they’re warranted (which is relatively often–particularly the first one). But I grant that that’s just my own personal experience talking, and no matter how many experiences I’ve had of people using these sayings appropriately, I’m well aware that the plural of anecdote…