# 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
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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 ${P}_{I}\phantom{\rule{0.33em}{0ex}}\left(d\right)$ as the probability of infection at a given mean dose d, the widely used dose–response model ${P}_{I}\phantom{\rule{0.33em}{0ex}}\phantom{\rule{0.33em}{0ex}}\left(d\right)=\phantom{\rule{0.33em}{0ex}}1-\phantom{\rule{0.33em}{0ex}}{\left(1+\frac{d}{\beta }\right)}^{-\alpha }$ 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 | $\stackrel{̂}{\phantom{\rule{0.16em}{0ex}}\alpha }$, $\stackrel{̂}{\beta }$) as a validity measure (r is a random variable that follows a gamma distribution; $\stackrel{̂}{\alpha }$ and $\stackrel{̂}{\beta }$ are the maximum likelihood estimates of α and β in the approximate model); and the constraint conditions $\stackrel{̂}{\beta }>{\left(22\phantom{\rule{0.33em}{0ex}}\stackrel{̂}{\alpha }\right)}^{0.50}$ for $0.02<\stackrel{̂}{\alpha }<2$ as a rule of thumb to ensure an accurate approximation (e.g., Pr(0 < r < 1 | $\stackrel{̂}{\phantom{\rule{0.16em}{0ex}}\alpha }$, $\stackrel{̂}{\beta }$) >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 | $\stackrel{̂}{\alpha }$, $\stackrel{̂}{\beta }$), 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.

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