  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.
