A Bayesian analysis for the Mann-Whitney statistic

A Bayesian analysis is developed for the Mann-Whitney (U) statistic. The analysis is focused on an Omega parameter, which is defined as the integrated proportion of the population in a generic experimental condition that has values that exceed values in a separate control condition. The likelihood function P(U vertical bar Omega) is approximated for both small and large sample size studies. The likelihood function is free of distributional assumptions about the actual experimental and control variates. A Monte Carlo method is used for small sample sizes, and a conjugate beta approximation model is developed for larger sample sizes. If there are 15 or more observations in each condition, the large sample approximation is highly accurate across a wide range of outcomes. The Bayes-factor efficiency and power efficiency are explored for the new Bayesian method. For properly specified Gaussian data, the power efficiency for the new method is nearly equal to that of the t test. But cases were identified for nonGaussian distributions where the power efficiency was superior for the Bayesian Mann-Whitney procedure. The new Bayesian analysis is thus a robust and powerful alternative to parametric models for examining the data from two independent conditions.