Abstract
Consider the problem of selecting independent samples from several populations for the purpose of between-group comparisons. An important aspect of the solution is the determination of clusters where mean levels are equal, often accomplished using multiple comparisons testing. We formulate the hypothesis testing problem of determining equal-mean clusters as a model selection problem. Information from all competing models is combined through Bayesian methods in an effort to provide a more realistic accounting of uncertainty. An example illustrates how the Bayesian approach leads to a logically sound presentation of multiple comparison results.
Highlights
Consider the problem of selecting independent samples from several populations for the purpose of between-group comparisons, either through hypothesis testing or estimation of mean differences
The existence of equal mean levels is considered physically plausible for the multiple comparisons problem, so Bayesian testing of these precise hypotheses will require a measure of prior/posterior belief in H(a,b), and a measure of prior/posterior belief in the effect size δ(a,b) = μb − μa if H(a,b) is not true
The multiple comparisons problem is well known among statistical practitioners
Summary
Consider the problem of selecting independent samples from several populations for the purpose of between-group comparisons, either through hypothesis testing or estimation of mean differences. A companion problem is the estimation of within-group mean levels. Together, these problems form the foundation for the very common analysis of variance framework, and describe essential aspects of stratified sampling, cluster analysis, empirical Bayes, and other settings. The goal of determining which groups have equal means requires testing a collection of related hypotheses. We examine this hypothesis testing problem from a Bayesian viewpoint. The approach is more general, because shrinkage is not necessarily toward an overall mean, but rather toward means deemed likely to be equal
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