Large-scale multiple testing of composite null hypotheses under heteroskedasticity

SummaryHeteroskedasticity poses several methodological challenges in designing valid and powerful procedures for simultaneous testing of composite null hypotheses. In particular, the conventional practice of standardizing or rescaling heteroskedastic test statistics in this setting may severely affect the power of the underlying multiple testing procedure. Additionally, when the inferential parameter of interest is correlated with the variance of the test statistic, methods that ignore this dependence may fail to control the Type I error at the desired level. We propose a new heteroskedasticity-adjusted multiple testing procedure that avoids data reduction by standardization and directly incorporates the side information from the variances into the testing procedure. Our approach relies on an improved nonparametric empirical Bayes deconvolution estimator that offers a practical way of capturing the dependence between the inferential parameter of interest and the variance of the test statistic. We develop theory to establish that the proposed procedure is asymptotically valid and optimal for false discovery rate control. Simulation results demonstrate that our method outperforms existing procedures, with substantial power gains across many settings at the same false discovery rate level. The method is illustrated with an application involving the detection of engaged users on a mobile game app.