Methods for combining rates from several studies

When several independent groups have conducted studies to estimate a procedure's success rate, it is often of interest to combine the results of these studies in the hopes of obtaining a better estimate for the true unknown success rate of the procedure. In this paper we present two hierarchical methods for estimating the overall rate of success. Both methods take into account the within-study and between-study variation and assume in the first stage that the number of successes within each study follows a binomial distribution given each study's own success rate. They differ, however, in their second stage assumptions. The first method assumes in the second stage that the rates of success from individual studies form a random sample having a constant expected value and variance. Generalized estimating equations (GEE) are then used to estimate the overall rate of success and its variance. The second method assumes in the second stage that the success rates from different studies follow a beta distribution. Both methods use the maximum likelihood approach to derive an estimate for the overall success rate and to construct the corresponding confidence intervals. We also present a two-stage bootstrap approach to estimating a confidence interval for the success rate when the number of studies is small. We then perform a simulation study to compare the two methods. Finally, we illustrate these two methods and obtain bootstrap confidence intervals in a medical example analysing the effectiveness of hyperdynamic therapy for cerebral vasospasm.