Guidelines for use of the approximate beta-poisson dose-response model

Gang Xie, Anne Roiko, Helen Stratton, Charles Lemckert, Peter Dunn, Kerrie Mengersen

    Research output: Contribution to journalArticle

    6 Citations (Scopus)


    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 urn:x-wiley:02724332:media:risa12682:risa12682-math-0001 and urn:x-wiley:02724332:media:risa12682:risa12682-math-0002, which involves the Kummer confluent hypergeometric function. Evaluation of a hypergeometric function is a computational challenge. Denoting urn:x-wiley:02724332:media:risa12682:risa12682-math-0003 as the probability of infection at a given mean dose d, the widely used dose–response model urn:x-wiley:02724332:media:risa12682:risa12682-math-0004 is an approximate formula for the exact beta‐Poisson model. Notwithstanding the required conditions urn:x-wiley:02724332:media:risa12682:risa12682-math-0005 and urn:x-wiley:02724332:media:risa12682:risa12682-math-0006, 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 | urn:x-wiley:02724332:media:risa12682:risa12682-math-0007, urn:x-wiley:02724332:media:risa12682:risa12682-math-0008) as a validity measure (r is a random variable that follows a gamma distribution; urn:x-wiley:02724332:media:risa12682:risa12682-math-0009 and urn:x-wiley:02724332:media:risa12682:risa12682-math-0010 are the maximum likelihood estimates of α and β in the approximate model); and the constraint conditions urn:x-wiley:02724332:media:risa12682:risa12682-math-0011 for urn:x-wiley:02724332:media:risa12682:risa12682-math-0012 as a rule of thumb to ensure an accurate approximation (e.g., Pr(0 < r < 1 | urn:x-wiley:02724332:media:risa12682:risa12682-math-0013, urn:x-wiley:02724332:media:risa12682:risa12682-math-0014) >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 | urn:x-wiley:02724332:media:risa12682:risa12682-math-0015, urn:x-wiley:02724332:media:risa12682:risa12682-math-0016), 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.
    Original languageEnglish
    Pages (from-to)1388-1402
    Number of pages15
    JournalRisk Analysis
    Issue number7
    Early online date05 Oct 2016
    Publication statusPublished - Jul 2017

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