I don’t like this cartoon

November 10, 2012
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(This article was originally published at Statistical Modeling, Causal Inference, and Social Science, and syndicated at StatsBlogs.)

Some people pointed me to this:

I am happy to see statistical theory and methods be a topic in popular culture, and of course I’m glad that, contra Feller, the Bayesian is presented as the hero this time, but . . . . I think the lower-left panel of the cartoon unfairly misrepresents frequentist statisticians.

Frequentist statisticians recognize many statistical goals. Point estimates trade off bias and variance. Interval estimates have the goal of achieving nominal coverage and the goal of being informative. Tests have the goals of calibration and power. Frequentists know that no single principle applies in all settings, and this is a setting where this particular method is clearly inappropriate. All statisticians use prior information in their statistical analysis. Non-Bayesians express their prior information not through a probability distribution on parameters but rather through their choice of methods. I think this non-Bayesian attitude is too restrictive, but in this case a small amount of reflection would reveal the inappropriateness of this procedure for this example.

In this comment, Phil defends the cartoon, pointing out that the procedure it describes is equivalent to the classical hypothesis-testing approach that is indeed widely used. Phil (and, by extension, the cartoonist) have a point, but I don’t think a sensible statistician would use this method to estimate such a rare probability. An analogy from a Bayesian perspective would be to use the probability estimate (y+1)/(n+2) with y=0 and n=36 for an extremely unlikely event, for example estimating the rate of BSE infection in a population as 1/38 based on the data that 0 people out of a random sample of 36 are infected. The flat prior is inappropriate in a context where the probability is very low; similarly the test with 1/36 chance of error is inappropriate in a classical setting where the true positive rate is extremely low.

The error represented in the lower-left panel of the cartoon is not quite not a problem with the classical theory of statistics—frequentist statisticians have many principles and hold that no statistical principle is all-encompassing (see here, also the ensuing discussion), but perhaps it is a problem with textbooks on classical statistics, that they typically consider the conditional statistical properties of a test (type 1 and type 2 error rates) without discussing the range of applicability of the method. In the context of probability mathematics, textbooks carefully explain that p(A|B) != p(B|A), and how a test with a low error rate can have a high rate of errors conditional on a positive finding, if the underlying rate of positives is low, but the textbooks typically confine this problem to the probability chapters and don’t explain its relevance to accept/reject decisions in statistical hypothesis testing. Still, I think the cartoon as a whole is unfair in that it compares a sensible Bayesian to a frequentist statistician who blindly follows the advice of shallow textbooks.

As an aside, I also think the lower-right panel is misleading. A betting decision depends not just on probabilities but also on utilities. If the sun as gone nova, money is worthless. Hence anyone, Bayesian or not, should be willing to bet \$50 that the sun has not exploded.

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