| The main research topic in this dissertation is the development of Bayesian theory of false discoveries and non-discoveries. Considering a more general model that allows the underlying test statistics as well as the associated parameters to be dependent, we first propose procedures controlling the Bayes FDR, which is defined as the average of frequentist FDR with respect to prior distribution.; Our procedure controlling the Bayes FDR (BFDR) is constructed through controlling the posterior FDR (PFDR). This BFDR procedure is developed through a decision theoretic formulation of the original multiple testing problem, allowing us to define the different overall measures combining false discoveries or false non-discoveries for a multiple testing procedure more generally in terms either randomized or non-randomized decisions on the null hypotheses.; We compare our proposed BFDR procedure with the Benjamini and Hochberg (1995) procedure (BH) and Efron (2003) Bayesian FDR procedure (Efron). The proposed BFDR procedure is more powerful, particularly when there is dependence in the tests. Moreover, unlike the BH procedure, which fails to control the frequentist FDR, and hence the Bayes FDR, when the statistics are not all positively dependent, our procedure has more general applicability. Analogously, the Bayes FNR procedure is also proposed when false negatives is the primary concern for multiple testing problem.; When there are directional decisions in simultaneous testing of null hypotheses against two-sided alternatives, a procedure controlling the Bayes directional false discovery rate (BDFDR) is developed through controlling the posterior directional false discovery rate (PDFDR). This is an alternative to Lewis and Thayer (2004) with a better control of the BDFDR. Moreover, it is optimum in the sense of being the non-randomized part of the procedure maximizing the posterior expectation of the directional per comparison power rate (PDPCPR), while controlling the PDFDR. A corresponding empirical Bayes method is proposed in the context of one-way random effects model. Simulation results show that the proposed Bayes and empirical Bayes methods perform much better from a Bayesian perspective than the procedures available in the literature.; *This research is supported by NSF Grant DMS-0306366. |
