On determination of sample size in hierarchical binomial models

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 Y i is assumed to be independent binomial with mean n θ i ( I =1,…, k ). The success probabilities, θ i '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 θ i . Lengths of central posterior intervals are computed for various functions of the θ i '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.