EMPIRICAL NULL AND FALSE DISCOVERY RATE INFERENCE FOR EXPONENTIAL FAMILIES BY ARMIN SCHWARTZMAN1 Harvard School of Public Health and Dana-Farber Cancer Institute

In large scale multiple testing, the use of an empirical null distribution rather than the theoretical null distribution can be critical for correct inference. This paper proposes a “mode matching” method for fitting an empirical null when the theoretical null belongs to any exponential family. Based on the central matching method for z-scores, mode matching estimates the null density by fitting an appropriate exponential family to the histogram of the test statistics by Poisson regression in a region surrounding the mode. The empirical null estimate is then used to estimate local and tail false discovery rate (FDR) for inference. Delta-method covariance formulas and approximate asymptotic bias formulas are provided, as well as simulation studies of the effect of the tuning parameters of the procedure on the bias-variance trade-off. The standard FDR estimates are found to be biased down at the far tails. Correlation between test statistics is taken into account in the covariance estimates, providing a generalization of Efron’s “wing function” for exponential families. Applications with χ2 statistics are shown in a family-based genomewide association study from the Framingham Heart Study and an anatomical brain imaging study of dyslexia in children.