doi:http://dx.doi.org/10.5705/ss.2010.255
Abstract: Multiple testing procedures, such as the False Discovery Rate control, often rely on estimating the proportion of true null hypotheses. This proportion is directly related to the minimum of the density of the p-value distribution. We propose a new estimator for the minimum of a density that is based on constrained multinomial likelihood functions. The proposed method involves partitioning the support of the density into several intervals, and estimating multinomial probabilities that are a function of the density. The motivation for this new approach comes from multiple testing settings where the test statistics are dependent, since this framework can be extended to the case of the dependent observations by using weighted univariate likelihoods. The optimal weights are obtained using the theory of estimating equations, and depend on the discretized pairwise joint distributions of the observations. We discuss how optimal weights can be estimated when the test statistics have multivariate normal distribution, and their correlation matrix is available or estimated. We evaluate the performance of the proposed estimator in simulations that mimic the testing for differential expression using microarray data.
Key words and phrases: Constrained multinomial likelihood, correlated tests, FDR, minimum of a density, multiple testing under dependence, proportion of null hypotheses.