Weighted multiple hypothesis testing procedures.

Multiple hypothesis testing is commonly used in genome research such as genome-wide studies and gene expression data analysis (Lin, 2005). The widely used Bonferroni procedure controls the family-wise error rate (FWER) for multiple hypothesis testing, but has limited statistical power as the number of hypotheses tested increases. The power of multiple testing procedures can be increased by using weighted p-values (Genovese et al., 2006). The weights for the p-values can be estimated by using certain prior information. Wasserman and Roeder (2006) described a weighted Bonferroni procedure, which incorporates weighted p-values into the Bonferroni procedure, and Rubin et al. (2006) and Wasserman and Roeder (2006) estimated the optimal weights that maximize the power of the weighted Bonferroni procedure under the assumption that the means of the test statistics in the multiple testing are known (these weights are called optimal Bonferroni weights). This weighted Bonferroni procedure controls FWER and can have higher power than the Bonferroni procedure, especially when the optimal Bonferroni weights are used. To further improve the power of the weighted Bonferroni procedure, first we propose a weighted Sidák procedure that incorporates weighted p-values into the Sidák procedure, and then we estimate the optimal weights that maximize the average power of the weighted Sidák procedure under the assumption that the means of the test statistics in the multiple testing are known (these weights are called optimal Sidák weights). This weighted Sidák procedure can have higher power than the weighted Bonferroni procedure. Second, we develop a generalized sequential (GS) Sidák procedure that incorporates weighted p-values into the sequential Sidák procedure (Scherrer, 1984). This GS idák procedure is an extension of and has higher power than the GS Bonferroni procedure of Holm (1979). Finally, under the assumption that the means of the test statistics in the multiple testing are known, we incorporate the optimal Sidák weights and the optimal Bonferroni weights into the GS Sidák procedure and the GS Bonferroni procedure, respectively. Theoretical proof and/or simulation studies show that the GS Sidák procedure can have higher power than the GS Bonferroni procedure when their corresponding optimal weights are used, and that both of these GS procedures can have much higher power than the weighted Sidák and the weighted Bonferroni procedures. All proposed procedures control the FWER well and are useful when prior information is available to estimate the weights.