Combining apparently contradictory evidence

I want to write a more formal article about this, but in the meantime here’s a placeholder.

The topic is the combination of apparently contradictory evidence.

Let’s start with a simple example: you have some ratings on a 1-10 scale. These could be, for example, research proposals being rated by a funding committee, or, umm, I dunno, gymnasts being rated by Olympic judges. Suppose there are 3 judges doing the ratings, and consider two gymnasts: one receives ratings of 8, 8, 8; the other is rated 6, 8, 10. Or, forget about ratings, just consider students taking multiple exams in a class. Consider two students: Amy, whose three test scores are 80, 80, 80; and Beth, who had scores 80, 100, 60. (I’ve purposely scrambled the order of those last three so that we don’t have to think about trends. Forget about time trends; that’s not my point here.)

How to compare those two students? A naive reader of test scores will say that Amy is consistent while Beth is flaky; or you might even say that you think Beth is better as she has a higher potential. But if you have some experience with psychometrics, you’ll be wary of overinterpreting results from three exam scores. Inference about an average from N=3 is tough; inference about variance from N=3 is close to impossible. Long story short: from a psychometrics perspective, there’s very little you can say about the relative consistency of Amy and Beth’s test-taking based on just three scores.

Academic researchers will recognize this problem when considering reviews of their own papers that they’ve submitted to journals. When you send in a paper, you’ll typically get a few reviews, and these reviews can differ dramatically in their messages.

Here’s a hilarious example supplied to me by Wolfgang Gaissmaier and Julian Marewski, from reviews of their 2011 article, “Forecasting elections with mere recognition from small, lousy samples: A comparison of collective recognition, wisdom of crowds, and representative polls.”

Here are some positive reviewer comments:

– This is a very interesting piece of work that raises a number of important questions related to public opinion. The major finding — that for elections with large numbers of parties, small non-probability samples looking only at party name recognition do as well as medium-sized probility samples looking at voter intent — is stunning.

– There is a lot to like about this short paper… I’m surprised by the strength of the results… If these results are correct (and I have no real reason to suspect otherwise), then the authors are more than justified in their praise of recognition-based forecasts. This could be an extremely useful forecasting technique not just for the multi-party European elections discussed by the authors, but also in relatively low-salience American local elections.

– This is concise, high-quality paper that demonstrates that the predictive power of (collective) recognition extends to the important domain of political elections.

And now the fun stuff. The negative comments:

– This is probably the strangest manuscript that I have ever been asked to review… Even if the argument is correct, I’m not sure that it tells us anything useful. The fact that recognition can be used to predict the winners of tennis tournaments and soccer matches is unsurprising – people are more likely to recognize the better players/teams, and the better players/teams usually win. It’s like saying that a football team wins 90% (or whatever) of the games in which it leads going into the fourth quarter. So what?

– To be frank, this is an exercise in nonsense. Twofold nonsense. For one thing, to forecast election outcomes based on whether or not voters recognize the parties/candidates makes no sense… Two, why should we pay any attention to unrepresentative samples, which is what the authors use in this analysis? They call them, even in the title, “lousy.” Self-deprecating humor? Or are the authors laughing at a gullible audience?

So, their paper is either “a very interesting piece of work” whose main finding is “stunning”—or it is “an exercise in nonsense” aimed at “a gullible audience.”

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