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

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Accepted Manuscript (AM)

##### Author(s)

Xie, Gang

Roiko, Anne

Stratton, Helen

Lemckert, Charles

Dunn, Peter K

Mengersen, Kerrie

##### Year published

2017

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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 math formula and math formula, which involves the Kummer confluent hypergeometric function. Evaluation of a hypergeometric function is a computational challenge. Denoting math formula as the probability of infection at a given mean dose d, the widely used dose–response model math formula is an approximate formula for the exact beta-Poisson model. Notwithstanding the required conditions math formula and math formula, issues related to the validity and ...

View more >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 math formula and math formula, which involves the Kummer confluent hypergeometric function. Evaluation of a hypergeometric function is a computational challenge. Denoting math formula as the probability of infection at a given mean dose d, the widely used dose–response model math formula is an approximate formula for the exact beta-Poisson model. Notwithstanding the required conditions math formula and math formula, 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 | math formula, math formula) as a validity measure (r is a random variable that follows a gamma distribution; math formula and math formula are the maximum likelihood estimates of α and β in the approximate model); and the constraint conditions math formula for math formula as a rule of thumb to ensure an accurate approximation (e.g., Pr(0 < r < 1 | math formula, math formula) >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 | math formula, math formula), 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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View more >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 math formula and math formula, which involves the Kummer confluent hypergeometric function. Evaluation of a hypergeometric function is a computational challenge. Denoting math formula as the probability of infection at a given mean dose d, the widely used dose–response model math formula is an approximate formula for the exact beta-Poisson model. Notwithstanding the required conditions math formula and math formula, 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 | math formula, math formula) as a validity measure (r is a random variable that follows a gamma distribution; math formula and math formula are the maximum likelihood estimates of α and β in the approximate model); and the constraint conditions math formula for math formula as a rule of thumb to ensure an accurate approximation (e.g., Pr(0 < r < 1 | math formula, math formula) >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 | math formula, math formula), 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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##### Journal Title

Risk Analysis

##### Volume

37

##### Issue

7

##### Copyright Statement

© 2017 Society for Risk Analysis. This is the peer reviewed version of the following article: Guidelines for Use of the Approximate Beta-Poisson Dose-Response Model, Risk Analysis, Vol 37(7) pp. 1388-1402, 2016, which has been published in final form at DOI. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving (http://olabout.wiley.com/WileyCDA/Section/id-828039.html)

##### Note

This publication has been entered into Griffith Research Online as an Advanced Online Version.

##### Subject

Public Health and Health Services not elsewhere classified