Guidelines for Use of the Approximate Beta-Poisson Dose-Response Model

For dose-response analysis in quantitative microbial risk assessment (QMRA), the exact beta-Poisson model is a two-parameter mechanistic dose-response model with parameters alpha > 0 and beta > 0, which involves the Kummer confluent hypergeometric function. Evaluation of a hypergeometric function is a computational challenge. Denoting P-I (d) as the probability of infection at a given mean dose d, the widely used dose-response model P-I (d) = 1 - (1 + d/beta)(-alpha) is an approximate formula for the exact beta-Poisson model. Notwithstanding the required conditions alpha << beta and beta >> 1, issues related to the validity and approximation accuracy of this approximate formula have remained largely ignored in practice, partly because these conditions are too general to provide clear guidance. Consequently, this study proposes a probability measure Pr(0 < r < 1 vertical bar (alpha) over cap, (beta) over cap) as a validity measure (r is a random variable that follows a gamma distribution; (alpha) over cap and (beta) over cap are the maximum likelihood estimates of alpha and beta in the approximate model); and the constraint conditions (beta) over cap > (22 (alpha) over cap)(0.50) for 0.02 < <(alpha)over cap> < 2 as a rule of thumb to ensure an accurate approximation (e.g., Pr(0 < r < 1 vertical bar <(alpha)over cap>, (beta) over cap) > 0.99). This validity measure and rule of thumb were validated by application to all the completed beta-Poisson models (related to 85 data sets) from the QMRA community portal (QMRA Wiki). The results showed that the higher the probability Pr(0 < r < 1 vertical bar (alpha) over cap, (beta) over cap), the better the approximation. The results further showed that, among the total 85 models examined, 68 models were identified as valid approximate model applications, which all had a near perfect match to the corresponding exact beta-Poisson model dose-response curve.