Empirical null and false discovery rate inference for exponential families

TL;DR

This paper introduces a mode matching method for estimating an empirical null distribution within exponential families, improving false discovery rate inference in large-scale multiple testing scenarios.

Contribution

It proposes a novel mode matching approach for fitting empirical nulls in exponential families, enhancing FDR estimation accuracy in large-scale tests.

Findings

Standard FDR estimates are biased in the tails.

Correlation among test statistics affects covariance estimates.

Application to genome-wide association and brain imaging studies demonstrates effectiveness.

Abstract

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 $\chi^2$ statistics are shown in a family-based genome-wide association study from the Framingham Heart Study and an anatomical brain imaging study of dyslexia in children.