Empirical Bayes and fiducial effect-size estimation for small numbers of tests

Abstract

Estimation of an effect size or other parameter of interest (POI), such as an average of differential abundance levels of metabolites or of the differential expression levels of genes, may be improved by shrinking toward a null-hypothesis value to the extent of a probability that the null hypothesis is true. For example, the local false discovery rate (LFDR) is a null-hypothesis posterior probability that is estimable via empirical Bayes methods without specifying the hyperprior distributions needed for a hierarchical Bayesian approach.

We compared the following (estimated) null-hypothesis probabilities as degrees of shrinkage in order to improve POI interval estimates (IEs) and POI point estimates (PEs): a histogram-based estimator (HBE) of the LFDR, a binomial-based estimator of the LFDR, a maximum-likelihood estimator of the LFDR, an expected LFDR (ELFDR), and a fiducial probability (FP).

In multiple-hypothesis testing, the ELFDR yields reliable IEs while the HBE gives the best PEs. For single-hypothesis testing, the FP generates an IE that outperforms the confidence interval and generates a PE performing at least as well as the unbiased estimator.

We apply these POI estimators to the abundance levels of 20 plasma proteins in women with breast cancer.