### Abstract

Overdispersion is a common phenomenon in Poisson modeling, and the negative binomial (NB) model is frequently used to account for overdispersion. Testing approaches (Wald test, likelihood ratio test (LRT), and score test) for overdispersion in the Poisson regression versus the NB model are available. Because the generalized Poisson (GP) model is similar to the NB model, we consider the former as an alternate model for overdispersed count data. The score test has an advantage over the LRT and the Wald test in that the score test only requires that the parameter of interest be estimated under the null hypothesis. This paper proposes a score test for overdispersion based on the GP model and compares the power of the test with the LRT and Wald tests. A simulation study indicates the score test based on asymptotic standard Normal distribution is more appropriate in practical application for higher empirical power, however, it underestimates the nominal significance level, especially in small sample situations, and examples illustrate the results of comparing the candidate tests between the Poisson and GP models. A bootstrap test is also proposed to adjust the underestimation of nominal level in the score statistic when the sample size is small. The simulation study indicates the bootstrap test has significance level closer to nominal size and has uniformly greater power than the score test based on asymptotic standard Normal distribution. From a practical perspective, we suggest that, if the score test gives even a weak indication that the Poisson model is inappropriate, say at the 0.10 significance level, we advise the more accurate bootstrap procedure as a better test for comparing whether the GP model is more appropriate than Poisson model. Finally, the Vuong test is illustrated to choose between GP and NB2 models for the same dataset.

Original language | English (US) |
---|---|

Pages (from-to) | 565-584 |

Number of pages | 20 |

Journal | Biometrical Journal |

Volume | 49 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2007 |

### Fingerprint

### Keywords

- Bootstrap
- Count data
- Generalized poisson model
- Negative binomial model
- Overdispersion
- Poisson model
- Score test
- Vuong test

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Biometrical Journal*,

*49*(4), 565-584. https://doi.org/10.1002/bimj.200610340

**Testing approaches for overdispersion in poisson regression versus the generalized poisson model.** / Yang, Zhao; Hardin, James W.; Addy, Cheryl L.; Vuong, Quang.

Research output: Contribution to journal › Article

*Biometrical Journal*, vol. 49, no. 4, pp. 565-584. https://doi.org/10.1002/bimj.200610340

}

TY - JOUR

T1 - Testing approaches for overdispersion in poisson regression versus the generalized poisson model

AU - Yang, Zhao

AU - Hardin, James W.

AU - Addy, Cheryl L.

AU - Vuong, Quang

PY - 2007/8

Y1 - 2007/8

N2 - Overdispersion is a common phenomenon in Poisson modeling, and the negative binomial (NB) model is frequently used to account for overdispersion. Testing approaches (Wald test, likelihood ratio test (LRT), and score test) for overdispersion in the Poisson regression versus the NB model are available. Because the generalized Poisson (GP) model is similar to the NB model, we consider the former as an alternate model for overdispersed count data. The score test has an advantage over the LRT and the Wald test in that the score test only requires that the parameter of interest be estimated under the null hypothesis. This paper proposes a score test for overdispersion based on the GP model and compares the power of the test with the LRT and Wald tests. A simulation study indicates the score test based on asymptotic standard Normal distribution is more appropriate in practical application for higher empirical power, however, it underestimates the nominal significance level, especially in small sample situations, and examples illustrate the results of comparing the candidate tests between the Poisson and GP models. A bootstrap test is also proposed to adjust the underestimation of nominal level in the score statistic when the sample size is small. The simulation study indicates the bootstrap test has significance level closer to nominal size and has uniformly greater power than the score test based on asymptotic standard Normal distribution. From a practical perspective, we suggest that, if the score test gives even a weak indication that the Poisson model is inappropriate, say at the 0.10 significance level, we advise the more accurate bootstrap procedure as a better test for comparing whether the GP model is more appropriate than Poisson model. Finally, the Vuong test is illustrated to choose between GP and NB2 models for the same dataset.

AB - Overdispersion is a common phenomenon in Poisson modeling, and the negative binomial (NB) model is frequently used to account for overdispersion. Testing approaches (Wald test, likelihood ratio test (LRT), and score test) for overdispersion in the Poisson regression versus the NB model are available. Because the generalized Poisson (GP) model is similar to the NB model, we consider the former as an alternate model for overdispersed count data. The score test has an advantage over the LRT and the Wald test in that the score test only requires that the parameter of interest be estimated under the null hypothesis. This paper proposes a score test for overdispersion based on the GP model and compares the power of the test with the LRT and Wald tests. A simulation study indicates the score test based on asymptotic standard Normal distribution is more appropriate in practical application for higher empirical power, however, it underestimates the nominal significance level, especially in small sample situations, and examples illustrate the results of comparing the candidate tests between the Poisson and GP models. A bootstrap test is also proposed to adjust the underestimation of nominal level in the score statistic when the sample size is small. The simulation study indicates the bootstrap test has significance level closer to nominal size and has uniformly greater power than the score test based on asymptotic standard Normal distribution. From a practical perspective, we suggest that, if the score test gives even a weak indication that the Poisson model is inappropriate, say at the 0.10 significance level, we advise the more accurate bootstrap procedure as a better test for comparing whether the GP model is more appropriate than Poisson model. Finally, the Vuong test is illustrated to choose between GP and NB2 models for the same dataset.

KW - Bootstrap

KW - Count data

KW - Generalized poisson model

KW - Negative binomial model

KW - Overdispersion

KW - Poisson model

KW - Score test

KW - Vuong test

UR - http://www.scopus.com/inward/record.url?scp=34548095721&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34548095721&partnerID=8YFLogxK

U2 - 10.1002/bimj.200610340

DO - 10.1002/bimj.200610340

M3 - Article

VL - 49

SP - 565

EP - 584

JO - Biometrical Journal

JF - Biometrical Journal

SN - 0323-3847

IS - 4

ER -