Experimental designs for microarray experiments

Abstract

When more than two treatments are compared for each gene in microarray experiments, a good design should allocate the treatments in such a way as to collect more information on treatment comparisons of interest. We compared a balanced incomplete block design, a loop design, and a reference design in terms of power to detect a single differentially expressed gene, where each design uses the same number of slides to compare the same number of treatments. A simple 2-way model is applied. Using a standard F test statistic for each single-gene analysis, the efficiency of each design was related to the size of the non-centrality parameter of the F distribution under a specified alternative. In preparation for the simultaneous analysis of multiple genes, the exact density function of the p-values under the alternative hypothesis was computed. Microarray experiments consider thousands of genes simultaneously and thus the multiple testing issue arises. We can compute individual p-values for each gene, and both the False Discovery Rate (FDR) method and the Westfall-Young resampling method are multiple testing procedures based on ordered p-values. For the FDR method, we derive lower bounds for the probability of rejecting more than k genes, and for the probability that more than j out of m1 truly differentially expressed genes are rejected using the FDR method. We simplify the expression of the lower bound under certain conditions by applying results from the weak convergence of extremal processes: the number of p-values falling into a small neighborhood of 0 has an asymptotic Poisson distribution. These probability estimates can be used to estimate the number of slides needed in experiments. The Westfall-Young method adjusts the observed p-values which are then compared with a pre-determined alpha. Under the independence assumption, the Westfall-Young method is equivalent to the step-down Sidak method. It involves time consuming simulations to use the step-down Sidak method in experimental designs. We apply results from the weak convergence of extremal processes and propose a way to speed up the simulation for the first few k adjusted p-values (where k is a small number relative to the total number of p-values). Their asymptotic distributions are evaluated analytically. An alternative method to adjust p-values suggested by an anonymous reader is also discussed. We numerically illustrate the applications of the procedures discussed in this thesis on a small data set from a spotted microarray experiment. These results are also illustrated in planning a non-spotted Affymetrix array experiment where they help determine the sample size needed to achieve a given level of performance in the FDR or Westfall-Young approaches.