In trying to wrap my limited head around apparently unlimited Bayesian statistical practice, Steve Walker pointed me to this article by Andrew Gelman and David Weakliem. The authors critique a study (published in the Journal of Theoretical Biology*), in which it was claimed that highly “attractive” (physically) people have a skewed gender ratio in their children, to the tune of somewhere between 1.05:1 and 1.26:1, girls:boys, depending on how you compute the ratio, based on a sample from about 3000 couples.
Well that’s eye-catching, given that we know that chromosomes in diploids, including the X and Y (gender) chromosomes in humans, typically segregate 1:1 during meiosis. We also know that if you take any large sample of humans, you will get very close to a 1:1 female:male ratio of offspring. The results were interesting enough for Psychology Today to publicized the study, for whatever reason. I mean after all, it’s in J. Theoretical Biology, so it must be valid, presumably with a solid “theoretical” basis, right?
Negative, as Gelman and Weakliem explain.
However, the latter are only using the JTB study to illustrate a larger issue: the existence of certain weaknesses of “frequentist” statistics (i.e. “traditional” methods, developed by Sir Ronald Fisher et al), relative to Bayesian statistics. The one they concentrate on is that frequentist methods are not good at detecting small effects in small sample sizes. OK, small sample sizes most certainly present a limitation to what effect you can detect, no question about it. They then argue that a Bayesian approach can avoid/minimize this problem, which they demonstrate by setting the mean of a “prior” distribution of gender ratios to 1.0, and then evaluating the likelihood of getting a gender ratio of 1.05:1. They find that it’s not very likely (it’s only slightly more likely than not that the ratio is above 1.0): the results of the JTB study are thus probably spurious, which is to say, not representing some truly novel or important biological finding.
I can hear Marv Albert call it now: “And it’s Reverend Bayes for the win, yesssssss!” Hold on please, and calm down Marv, we may not quite be talking about Michael Jordan here.
To wit, what exactly is the basis for setting the mean of the prior distribution to 1.0? Why not 1.047 or 1.26? By Bayesian rules, a prior represents some type of outside information available on the issue at hand. In this case (setting aside the “subjective Bayesianism” that says you can use any value you “believe” in!) that information has to be either (1) observational evidence on chromosome segregation or (2) observed gender ratios in some much larger population of children. Fortunately, the two of these are both super-abundant and one causes the other, so pick either one.
The Reverend wasn’t the one who started us down the road of understanding inheritance patterns, it was instead Father Gregor Mendel, some 100 years later with his pea plants, followed closely by the work of Walther and Bateson who provided explanations of Mendel’s results on a definite physical basis (i.e., the existence of chromosomes and meiosis, and further studies of inheritance patterns respectively). You don’t need Bayesian analysis to tell you that chromosomes segregate 1:1 in humans, which is the key piece of information here. And you most certainly have no underlying reason whatsoever to expect that “attractive” people will deviate from the human norm to the tunes of from 1.05:1 to 1.26:1. The problems with the JTB study are just simply sample size issues, not really “frequentist vs Bayesian” issues at all. The Reverend can (maybe?) help you some by introducing some conditional probability, but only because he can make use of the really critical information provided ultimately by the Father and his intellectual descendents. Do I really need Bayes’ theorem to tell me that a 1.05:1 girl:boy ratio in a small sample size is not likely to represent something scientifically important?
Disclaimer: Notwithstanding any of this, I’m in no way set against Bayesian approaches to statistics, and in fact I strongly favor conditional probability in general, which is what Bayesian approaches are. But I am set against using the wrong reasons to justify why it’s putatively more valid than Fisher’s approaches in certain situations.
* Kanazawa, S. (2007). Beautiful parents have more daughters: A further implication of the generalized Trivers-Willard hypothesis. Journal of Theoretical Biology 244:133–140.