Simultaneous testing of multiple null hypotheses has now become an integral part of statistical analysis of data arising from modern scientific investigations. Often the test statistics in such multiple testing problem are correlated. The research in this dissertation is motivated by the scope of improving or extending existing methods to incorporate correlation in the data. Sarkar (2008) proposes controlling the pairwise false discovery rate (Pairwise-FDR), which inherently takes into account the dependence among the p-values, thereby making it a more robust, less conservative and more powerful under dependence than the usual notion of FDR. In this dissertation, we further investigate the performance of Pairwise-FDR under a dependent mixture model. In particular, we consider a step-up method to control the Pairwise-FDR under this model assuming that the correlation between any two p-values is the same (exchangeable). We also suggest improving this method by incorporating an estimate of the number of pairs of true null hypotheses developed under this model. Efron (2007, Journal of the American Statistical Association 102, 93-103) proposed a novel approach to incorporate dependence among the null p-values into a multiple testing method controlling false discoveries. In this dissertation, we try to investigate the scope of utilizing this approach by proposing alternative versions of adaptive Bonferroni and BH methods which estimates the number of true null hypotheses from the empirical null distribution introduced by Efron. These newer adaptive procedures have been numerically shown to perform better than existing adaptive Bonferroni or BH methods within a wider range of dependence. A gene expression microarray data set has been used to highlight the difference in results obtained upon applying the proposed and other adaptive BH methods. Another approach to address the presence of correlation is motivated by the scope of utilizing the dependence structure of the data towards further improving some multiple testing methods while maintaining control of some error rate. The dependence structure of the data is incorporated using pairwise weights. In this dissertation we propose a weighted version of the pairwise FDR (Sarkar, 2008) using pairwise weights and a method controlling the weighted pairwise- FDR. We give a discussion on the application of such weighted procedure and suggest some weighting schemes that generates pairwise weights. / Statistics

The false discovery rate (FDR) introduced by Benjamini and Hochberg (1995) is perhaps the most standard error controlling measure being used in a wide variety of applications involving multiple hypothesis testing. There are two approaches to control the FDR - the fixed error rate approach of Benjamini and Hochberg (BH, 1995) where a rejection region is determined with the FDR below a fixed level and the estimation based approach of Storey (2002) where the FDR is estimated for a fixed rejection region before it is controlled. In this proposal, we concentrate on both these approaches and propose new, improved versions of some FDR controlling procedures available in the literature. A number of adaptive procedures have been put forward in the literature, each attempting to improve the method of Benjamini and Hochberg (1995), the BH method, by incorporating into this method an estimate of number true null hypotheses. Among these, the method of Benjamini, Krieger and Yekutieli (2006), the BKY method, has been receiving lots of attention recently. In this proposal, a variant of the BKY method is proposed by considering a different estimate of number true null hypotheses, which often outperforms the BKY method in terms of the FDR control and power. Storey's (2002) estimation based approach to controlling the FDR has been developed from a class of conservatively biased point estimates of the FDR under a mixture model for the underlying p-values and a fixed rejection threshold for each null hypothesis. An alternative class of point estimates of the FDR with uniformly smaller conservative bias is proposed under the same setup. Numerical evidence is provided to show that the mean squared error (MSE) is also often smaller for this new class of estimates. Compared to Storey's (2002), the present class provides a more powerful estimation based approach to controlling the FDR. / Statistics