On determination of sample size in hierarchical binomial models

Abstract
We consider a two‐ and a three‐stage hierarchical design containing the effects of k clusters with n units per cluster. In the two‐stage model, the conditional distribution of the discrete response Yi is assumed to be independent binomial with mean nθi (I=1,…,k). The success probabilities, θi's, are assumed exchangeable across the k clusters, each arising from a beta distribution. In the three‐stage model, the parameters in the beta distribution are assumed to have independent gamma distributions. The size of each cluster, n, is determined for functions of θi. Lengths of central posterior intervals are computed for various functions of the θi's using Markov chain Monte Carlo and Monte Carlo simulations. Several prior distributions are characterized and tables are provided for n with given k. Methods for sample size calculations under the two‐ and three‐stage models are illustrated and compared for the design of a multi‐institutional study to evaluate the appropriateness of discharge planning rates for a cohort of patients with congestive heart failure. Copyright © 2001 John Wiley & Sons, Ltd.

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