A Puzzle of Proportions: Two Popular Bayesian Tests Can Yield Dramatically Different Conclusions

TL;DR

This paper compares two Bayesian methods for testing the equality of two proportions, revealing that they can lead to significantly different conclusions, especially with extreme observed proportions, and recommends a new default approach.

Contribution

It highlights the differences between two Bayesian tests for proportions and advocates for using the logistic regression perspective as the default method.

Findings

The logistic regression approach yields weaker evidence at distribution extremes.

The contingency table approach can lead to more decisive conclusions.

Different methods can produce markedly different results on the same data.

Abstract

Testing the equality of two proportions is a common procedure in science, especially in medicine and public health. In these domains it is crucial to be able to quantify evidence for the absence of a treatment effect. Bayesian hypothesis testing by means of the Bayes factor provides one avenue to do so, requiring the specification of prior distributions for parameters. The most popular analysis approach views the comparison of proportions from a contingency table perspective, assigning prior distributions directly to the two proportions. Another, less popular approach views the problem from a logistic regression perspective, assigning prior distributions to logit-transformed parameters. Reanalyzing 39 null results from the New England Journal of Medicine with both approaches, we find that they can lead to markedly different conclusions, especially when the observed proportions are at the extremes (i.e., very low or very high). We explain these stark differences and provide recommendations for researchers interested in testing the equality of two proportions and users of Bayes factors more generally. The test that assigns prior distributions to logit-transformed parameters creates prior dependence between the two proportions and yields weaker evidence when the observations are at the extremes. When comparing two proportions, we argue that this test should become the new default.