Exact calculation of probabilities of false positives and false negatives for low background counting

The purpose of this paper is to demonstrate the derivation and use of exact formulas, and their algorithms, for calculating the probabilities alpha of Type I errors, and beta of Type II errors, for low blank total counts (Poisson-distributed). The calculations are carried out to examine the alpha and beta probabilities at low blank levels of the decision level (DL) and minimum detectable amount (MDA) formulations as adopted in the Health Physics Society Standard, "Performance Criteria for Radiobioassay." These formulations are consistent with those published by L.A. Currie, which have received wide acceptance in defining lower limits of detection (LLD). Although Currie's formulation was derived assuming a normal distribution in net counts, the behavior of the distribution of net counts at low count levels, which is a distribution of the difference between two Poisson variates, is such that the MDA formulation in the standard could be considered acceptable for the purpose of providing one simple formulation of MDA. The derivations in this note can also be useful in other problems involving differences in Poisson variates, such as those in certain population studies.