Multiple Testing Of Composite Null Hypotheses With High Dimensional Count Data: A Local False Discovery Rate Approach
High throughput sequencing technology is routinely used to generate high dimensional (HD) count data. One objective is to determine which variables are associated with a covariate via the testing of multiple null hypotheses, say $H_m:β_m = 0$ for $m = 1, 2, ..., M$ where each $β_m$ is a regression parameter. The standard approach computes a $p$-value for each null hypothesis and applies some false discovery rate (FDR) method to determine which null hypotheses are rejected, i.e. which variables can be discovered as associated with the covariate. However, even if data are ``well modeled'', this approach tends to provide a misleading list of discoveries in that some ``uninteresting'' negligible associations ($β_m \approx 0$) are discovered, while other interesting associations are not. This paper formally addresses this issue using Poisson Mixture models and a Local FDR (PMLFDR) procedure for testing of multiple composite null hypotheses $H_m:β_m\in[-ε,ε]$. It is shown that PMLFDR procedures allow for consistently estimable expressions for overdispersion, provide asymptotically optimal FDR control, and are robust with respect to model misspecification. Simulation studies and real data analysis illustrate that PMLFDR procedures perform favorably compared to established procedures in that they a) are flexible enough to model most HD count data regardless of the technology used in generation b) allow for continuous and categorical covariates and c) yield more meaningful discoveries without sacrificing FDR control.