Empirical Bayes confidence intervals for a series of related experiments

When data from different studies are analyzed, often one objective of the analysis is to obtain improved parameter estimates and confidence intervals for each individual study. James-Stein or empiricial Bayes estimation is well studied to producing estimates. This paper applies recent results on the construction of confidence intervals for the parameters of the individual studies. A naive empirical Bayes approach to confidence intervals assumes that the estimated posterior for each study can be used directly for inferences. The naive approach generally leads to confidence intervals with less than the desired coverage. Morris (1983, Journal of the American Statistical Association 78, 47-59) suggested one approach to the construction of empirical Bayes confidence intervals based on the use of a Bayes hyperprior distribution. Laird and Louis (1987, Journal of the American Statistical Association 82, 739-757) define the empirical Bayes bootstrap for drawing bootstrap samples from the compound (empirical Bayes) model and present a procedure for obtaining empirical Bayes confidence intervals which uses the bootstrap samples to adjust for uncertainty in the estimate of the prior. This paper compares the empirical performance of classical, Morris's, and bootstrap intervals on a random sample of bioassays from the National Cancer Institute data base on potential chemical carcinogens.