Interpretation of local false discovery rates under the zero assumption

Abstract

In large-scale studies with parallel signal-plus-noise observations, the local false discovery rate is a summary statistic that is often presumed to be equal to the posterior probability that the signal is null. We prefer to call the latter quantity the local null-signal rate to emphasize our view that a null signal and a false discovery are not identical events. The local null-signal rate is commonly estimated through empirical Bayes procedures that build on the `zero density assumption,' which attributes the density of observations near zero entirely to null signals. In this paper, we argue that this strategy does not furnish estimates of the local null-signal rate, but instead of a quantity we call the complementary local activity rate (clar). Although it is likely to be small, an inactive signal is not necessarily zero. The clar dominates both the local null-signal rate and the local false sign rate and is a weakly continuous functional of the signal distribution. As a consequence, it takes on sensible values when the signal is sparse but not exactly zero. Our findings clarify the interpretation of local false discovery rates estimated under the zero density assumption.