### Abstract

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 PI(d) as the probability of infection at a given mean dose d, the widely used dose-response model PI(d)=1-(1+dβ)-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 | αˆ, βˆ) as a validity measure (r is a random variable that follows a gamma distribution; αˆ and βˆ are the maximum likelihood estimates of α and β in the approximate model); and the constraint conditions βˆ>(22αˆ)0.50 for 0.02<αˆ<2 as a rule of thumb to ensure an accurate approximation (e.g., Pr(0 < r < 1 | αˆ, βˆ) >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 | αˆ, βˆ), 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 language | English |
---|---|

Pages (from-to) | 1388-1402 |

Number of pages | 15 |

Journal | Risk Analysis |

Volume | 37 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1 Jul 2017 |

Externally published | Yes |

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*Risk Analysis*,

*37*(7), 1388-1402. https://doi.org/10.1111/risa.12682

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*Risk Analysis*, vol. 37, no. 7, pp. 1388-1402. https://doi.org/10.1111/risa.12682

**Guidelines for Use of the Approximate Beta-Poisson Dose-Response Model.** / Xie, Gang; Roiko, Anne; Stratton, Helen; Lemckert, Charles; Dunn, Peter K.; Mengersen, Kerrie.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Guidelines for Use of the Approximate Beta-Poisson Dose-Response Model

AU - Xie, Gang

AU - Roiko, Anne

AU - Stratton, Helen

AU - Lemckert, Charles

AU - Dunn, Peter K.

AU - Mengersen, Kerrie

PY - 2017/7/1

Y1 - 2017/7/1

N2 - 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 PI(d) as the probability of infection at a given mean dose d, the widely used dose-response model PI(d)=1-(1+dβ)-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 | αˆ, βˆ) as a validity measure (r is a random variable that follows a gamma distribution; αˆ and βˆ are the maximum likelihood estimates of α and β in the approximate model); and the constraint conditions βˆ>(22αˆ)0.50 for 0.02<αˆ<2 as a rule of thumb to ensure an accurate approximation (e.g., Pr(0 < r < 1 | αˆ, βˆ) >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 | αˆ, βˆ), 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.

AB - 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 PI(d) as the probability of infection at a given mean dose d, the widely used dose-response model PI(d)=1-(1+dβ)-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 | αˆ, βˆ) as a validity measure (r is a random variable that follows a gamma distribution; αˆ and βˆ are the maximum likelihood estimates of α and β in the approximate model); and the constraint conditions βˆ>(22αˆ)0.50 for 0.02<αˆ<2 as a rule of thumb to ensure an accurate approximation (e.g., Pr(0 < r < 1 | αˆ, βˆ) >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 | αˆ, βˆ), 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.

KW - A rule of thumb

KW - Beta-Poisson dose-response model

KW - Experimental dose-response data

KW - QMRA

KW - experimental dose–response data

KW - beta-Poisson dose–response model

UR - http://www.scopus.com/inward/record.url?scp=84995422415&partnerID=8YFLogxK

UR - http://onlinelibrary.wiley.com/doi/10.1111/risa.12682/full

UR - http://www.mendeley.com/research/guidelines-approximate-betapoisson-doseresponse-model

U2 - 10.1111/risa.12682

DO - 10.1111/risa.12682

M3 - Article

VL - 37

SP - 1388

EP - 1402

JO - Risk Analysis

JF - Risk Analysis

SN - 0272-4332

IS - 7

ER -