Empirical Bayes models of Poisson clinical trials and sample size determination

Bayesian methods are often used to reduce the sample sizes and/or increase the power of clinical trials. The right choice of the prior distribution is a critical step in Bayesian modeling. If the prior not completely specified, historical data may be used to estimate it. In the empirical Bayesian analysis, the resulting prior can be used to produce the posterior distribution. In this paper, we describe a Bayesian Poisson model with a conjugate Gamma prior. The parameters of Gamma distribution are estimated in the empirical Bayesian framework under two estimation schemes. The straightforward numerical search for the maximum likelihood (ML) solution using the marginal negative binomial distribution is unfeasible occasionally. We propose a simplification to the maximization procedure. The Markov Chain Monte Carlo method is used to create a set of Poisson parameters from the historical count data. These Poisson parameters are used to uniquely define the Gamma likelihood function. Easily computable approximation formulae may be used to find the ML estimations for the parameters of gamma distribution. For the sample size calculations, the ML solution is replaced by its upper confidence limit to reflect an incomplete exchangeability of historical trials as opposed to current studies. The exchangeability is measured by the confidence interval for the historical rate of the events. With this prior, the formula for the sample size calculation is completely defined.