Hypothesis testing is foundational to the discipline of statistics. Procedures exist which control for individual Type I error rates and more global or family-wise error rates for a series of hypothesis tests. However, the ability of scientists to produce very large data sets with increasing ease has led to a rapid rise in the number of statistical tests performed, often with small sample sizes. This is seen particularly in the area of biotechnology and the analysis of microarray data. This thesis considers this high-dimensional context with particular focus on the effects of dependence on existing multiple hypothesis testing procedures. / While dependence is often ignored, there are many existing techniques employed currently to deal with this context but these are typically highly conservative or require difficult estimation of large correlation matrices. This thesis demonstrates that, in this high-dimensional context when the distribution of the test statistics is light-tailed, dependence is not as much of a concern as in the classical contexts. This is achieved with the use of a moving average model. One important implication of this is that, when this is satisfied, procedures designed for independent test statistics can be used confidently on dependent test statistics. / This is not the case however for heavy-tailed distributions, where we expect an asymptotic Poisson cluster process of false discoveries. In these cases, we estimate the parameters of this process along with the tail-weight from the observed exceedences and attempt to adjust procedures. We consider both conservative error rates such as the family-wise error rate and more popular methods such as the false discovery rate. We are able to demonstrate that, in the context of DNA microarrays, it is rare to find heavy-tailed distributions because most test statistics are averages.

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Copy-number variation (CNV) has been implicated in many complex diseases. It is of great interest to detect and locate such regions through genetic association testings. However, the association testings are complicated by the fact that CNVs usually span multiple markers and thus such markers are correlated to each other. To overcome the difficulty, it is desirable to pool information across the markers. In this thesis, we propose a kernel-based method for aggregation of marker-level tests, in which first we obtain a bunch of p-values through association tests for every marker and then the association test involving CNV is based on the statistic of p-values combinations. In addition, we explore several aspects of its implementation. Since p-values among markers are correlated, it is complicated to obtain the null distribution of test statistics for kernel-base aggregation of marker-level tests. To solve the problem, we develop two proper methods that are both demonstrated to preserve the family-wise error rate of the test procedure. They are permutation based and correlation base approaches. Many implementation aspects of kernel-based method are compared through the empirical power studies in a number of simulations constructed from real data involving a pharmacogenomic study of gemcitabine. In addition, more performance comparisons are shown between permutation-based and correlation-based approach. We also apply those two approaches to the real data. The main contribution of the dissertation is the development of marker-level association testing, a comparable and powerful approach to detect phenotype-associated CNVs. Furthermore, the approach is extended to high dimension setting with high efficiency.

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