Representative drug sampling: sample size calculations revisited

Calculating the required number of samples to be tested from a consignment of pills suspected of containing drugs can be performed from a Bayesian perspective. Procedures in literature are based on the outstanding work of Aitken. However, in the mathematical treatment of the problem, the limitedness of the consignment is not systematically used. The current Technical Note addresses this problem. A suitable prior distribution for the number of positive pills is derived, being a betabinomial distribution with the consignment size as one of the parameters. A hypergeometric likelihood is used, as sampling generally proceeds without replacement. The betabinomial posterior distribution is mathematically identical to the predictive distribution as reported elsewhere. The currently used large consignment approximation can be derived from the betabinomial posterior, but the quality is not optimal when compared to the exact betabinomial-based results. A new approximation is derived, with better properties, as illustrated in some examples.