Risk Analysis

Guidelines for Use of the Approximate Beta‐Poisson Dose–Response Model

Journal Article

  • Author(s): Gang Xie, Anne Roiko, Helen Stratton, Charles Lemckert, Peter K. Dunn, Kerrie Mengersen
  • Article first published online: 05 Oct 2016
  • DOI: 10.1111/risa.12682
  • Read on Online Library
  • Subscribe to Journal

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 α > 0 and β > 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 β ) α is an approximate formula for the exact beta‐Poisson model. Notwithstanding the required conditions α < < β and β > > 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 | α ̂ , β ̂ ) as a validity measure (r is a random variable that follows a gamma distribution; α ̂ and β ̂ are the maximum likelihood estimates of α and β in the approximate model); and the constraint conditions β ̂ > ( 22 α ̂ ) 0.50 for 0.02 < α ̂ < 2 as a rule of thumb to ensure an accurate approximation (e.g., Pr(0 < r < 1 | α ̂ , β ̂ ) >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 | α ̂ , β ̂ ), 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.

Related Topics

Related Publications

Related Content

Site Footer


This website is provided by John Wiley & Sons Limited, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ (Company No: 00641132, VAT No: 376766987)

Published features on StatisticsViews.com are checked for statistical accuracy by a panel from the European Network for Business and Industrial Statistics (ENBIS)   to whom Wiley and StatisticsViews.com express their gratitude. This panel are: Ron Kenett, David Steinberg, Shirley Coleman, Irena Ograjenšek, Fabrizio Ruggeri, Rainer Göb, Philippe Castagliola, Xavier Tort-Martorell, Bart De Ketelaere, Antonio Pievatolo, Martina Vandebroek, Lance Mitchell, Gilbert Saporta, Helmut Waldl and Stelios Psarakis.