Gay brains or bad statistics

Peter Coles (who is himself gay) looked at the story Gay men and heterosexual women have similarly shaped brains, research shows, going back to the actual research paper.

In Cerebral Asymmetry: is it all in the Mind? he argues that the statistics behind the conclusion are suspect:

The problem in this case is that the tests done by Savic & Lindstrom all depend on the quantity being analysed (AI) having a
normal (Gaussian) distribution. This is very often
a reasonable hypothesis for biometric data, but unfortunately in this case the construction of the asymmetry index is such that it is
expected to have a very non-Gaussian shape as is commonly the case for distributions of
variables formed as ratios. In fact, the ratio of two normal
variates has a peculiar distribution with very long tails. Many statistical analyses appeal to the
Central Limit Theorem to justify the assumption of normality,
but distributions with very long tails (such as the Cauchy distribution)
violate the conditions of this Theorem, namely that the distribution must have finite variance. The asymmetry index is probably therefore
an inappropriate choice of variable for the tests that Savic & Lindstrom perform. In particular the significance levels
(or p-values) quoted in their paper are very low (of order 0.0008,
for example, in the ANOVA test) which is surprising for such small samples. These probabilities are obtained by assuming the observations
have Gaussian statistics, and they would be much lower for a distribution with longer tails.

This problem with the Cauchy distribution is well known among the mathematically aware. I learned about it as an undergraduate at Carleton in 1972. Unfortunately, many users of statistics are not aware of these subtleties, and so questionable results are published and then sensationalized in the mass media.

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