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Open Access Methodology article

Empirical Bayes estimation of posterior probabilities of enrichment: A comparative study of five estimators of the local false discovery rate

Zhenyu Yang1, Zuojing Li2 and David R Bickel1*

Author Affiliations

1 Ottawa Institute of Systems Biology, Department of Biochemistry, Microbiology, and Immunology, Department of Mathematics and Statistics, University of Ottawa, 451 Smyth Road, Ottawa, Ontario, K1H 8M5 Canada

2 School of Foundation, Shenyang Pharmaceutical University, No. 103 Wenhua Road, Shenyang, Liaoning, 110016, China

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BMC Bioinformatics 2013, 14:87  doi:10.1186/1471-2105-14-87

Published: 6 March 2013



In investigating differentially expressed genes or other selected features, researchers conduct hypothesis tests to determine which biological categories, such as those of the Gene Ontology (GO), are enriched for the selected features. Multiple comparison procedures (MCPs) are commonly used to prevent excessive false positive rates. Traditional MCPs, e.g., the Bonferroni method, go to the opposite extreme: strictly controlling a family-wise error rate, resulting in excessive false negative rates. Researchers generally prefer the more balanced approach of instead controlling the false discovery rate (FDR). However, the q-values that methods of FDR control assign to biological categories tend to be too low to reliably estimate the probability that a biological category is not enriched for the preselected features. Thus, we study an application of the other estimators of that probability, which is called the local FDR (LFDR).


We considered five LFDR estimators for detecting enriched GO terms: a binomial-based estimator (BBE), a maximum likelihood estimator (MLE), a normalized MLE (NMLE), a histogram-based estimator assuming a theoretical null hypothesis (HBE), and a histogram-based estimator assuming an empirical null hypothesis (HBE-EN). Since NMLE depends not only on the data but also on the specified value of Π0, the proportion of non-enriched GO terms, it is only advantageous when either Π0 is already known with sufficient accuracy or there are data for only 1 GO term. By contrast, the other estimators work without specifying Π0 but require data for at least 2 GO terms. Our simulation studies yielded the following summaries of the relative performance of each of those four estimators. HBE and HBE-EN produced larger biases for 2, 4, 8, 32, and 100 GO terms than BBE and MLE. BBE has the lowest bias if Π0 is 1 and if the number of GO terms is between 2 and 32. The bias of MLE is no worse than that of BBE for 100 GO terms even when the ideal number of components in its underlying mixture model is unknown, but has high bias when the number of GO terms is small compared to the number of estimated parameters. For unknown values of Π0, BBE has the lowest bias for a small number of GO terms (2-32 GO terms), and MLE has the lowest bias for a medium number of GO terms (100 GO terms).


For enrichment detection, we recommend estimating the LFDR by MLE given at least a medium number of GO terms, by BBE given a small number of GO terms, and by NMLE given either only 1 GO term or precise knowledge of Π0.