Empirical Bayes False Coverage Rate Controlling Confidence Intervals

Abstract

Benjamini and Yekutieli suggested that it is important to account for multiplicity correction for confidence intervals when only some of the selected intervals are reported. They introduced the concept of the false coverage rate (FCR) for confidence intervals which is parallel to the concept of the false discovery rate in the multiple-hypothesis testing problem and they developed confidence intervals for selected parameters which control the FCR. Their approach requires the FCR to be controlled in the frequentist's sense, i.e. controlled for all the possible unknown parameters. In modern applications, the number of parameters could be large, as large as tens of thousands or even more, as in microarray experiments. We propose a less conservative criterion, the Bayes FCR, and study confidence intervals controlling it for a class of distributions. The Bayes FCR refers to the average FCR with respect to a distribution of parameters. Under such a criterion, we propose some confidence intervals, which, by some analytic and numerical calculations, are demonstrated to have the Bayes FCR controlled at level q for a class of prior distributions, including mixtures of normal distributions and zero, where the mixing probability is unknown. The confidence intervals are shrinkage-type procedures which are more efficient for the θ i s that have a sparsity structure, which is a common feature of microarray data. More importantly, the centre of the proposed shrinkage intervals reduces much of the bias due to selection. Consequently, the proposed empirical Bayes intervals are always shorter in average length than the intervals of Benjamini and Yekutieli and can be only 50% or 60% as long in some cases. We apply these procedures to the data of Choe and colleagues and obtain similar results.

Empirical Bayes Confidence Intervals for Selected Parameters in High-Dimensional Data

“…As demonstrated in Zhao and Hwang (2012) and researchers cited therein, the means of the observations are not unbiased for θ i 's. Similar to Zhao and Hwang (2012), we therefore target capturing the means of the observations, which are θ i 's after suitable biased correction. (Otherwise, all intervals would have extremely low coverage probabilities and extremely high FCR.…”

“…All the parameters θ i 's in these data are set by the experiments and are known. Therefore, it can be used to assess different statistical procedures as carried out in Hwang, Qiu, and Zhao (2009) and Zhao and Hwang (2012) for comparing various confidence intervals and in Zhao (2010) for comparing point estimators. The conclusion reported here is consistent with the simulation results reported in Figures 1-3.…”

“…When (2.29) holds, we say that the intervals are 1 − α empirical Bayes FCR controlling intervals as in Zhao and Hwang (2012). As a by-product of this article, the same proposed intervals described in the last section, each with 1 − α confidence level (not 1 − α/R), can be used to solve the problem.…”

“…Their proposal is a frequentist criterion since the expectation in their FCR definition is with respect to X i while θ i are fixed. Zhao and Hwang (2012) instead proposed controlling for a class of priors π ∈ , the Bayes FCR (FCR π ) to be less than α, that is,…”

“…The goal of this article is to provide post-selection analysis, an area which is receiving more attention in modern statistics research. See Benjamini and Yekutieli (2005), Berk, Brown, and Zhao (2010), Berk et al (2012), Yekutieli (2012), and Zhao and Hwang (2012).…”