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Substantial statistical research has recently been devoted to the analysis oflarge-scale microarray experiments which provide a measure of the simultaneousexpression of thousands of genes in a particular condition. A typical goal isthe comparison of gene expression between two conditions (e.g., diseased vs.nondiseased) to detect genes which show differential expression. Classicalhypothesis testing procedures have been applied to this problem and more recentwork has employed sophisticated models that allow for the sharing ofinformation across genes. However, many recent gene expression studies have anexperimental design with several conditions that requires an even more involvedhypothesis testing approach. In this paper, we use a hierarchical Bayesianmodel to address the situation where there are many hypotheses that must besimultaneously tested for each gene. In addition to having many hypotheseswithin each gene, our analysis also addresses the more typical multiplecomparison issue of testing many genes simultaneously. We illustrate ourapproach with an application to a study of genes involved in obstructive sleepapnea in humans.

Bayesian testing of many hypotheses $\times$ many genes: A study of sleep apnea