Recruitment prediction for multi-centre clinical trials based on a hierarchical Poisson-gamma model: asymptotic analysis and improved intervals

We analyse predictions of future recruitment to a multi-centre clinical trial based on a maximum-likelihood fitting of a commonly used hierarchical Poisson-Gamma model for recruitments at individual centres. We consider the asymptotic accuracy of quantile predictions in the limit as the number of recruitment centres grows large and find that, in an important sense, the accuracy of the quantiles does not improve as the number of centres increases. When predicting the number of further recruits in an additional time period, the accuracy degrades as the ratio of the additional time to the census time increases, whereas when predicting the amount of additional time to recruit a further $n^+_\bullet$ patients, the accuracy degrades as the ratio of $n^+_\bullet$ to the number recruited up to the census period increases. Our analysis suggests an improved quantile predictor. Simulation studies verify that the predicted pattern holds for typical recruitment scenarios in clinical trials and verify the much improved coverage properties of prediction intervals obtained from our quantile predictor. In the process of extending the applicability of our methodology, we show that in terms of the accuracy of all integer moments it is always better to approximate the sum of independent gamma random variables by a single gamma random variable matched on the first two moments than by the moment-matched Gaussian available from the central limit theorem.