Testing Approaches for Overdispersion in Poisson Regression versus the Generalized Poisson Model

• Published in: Biometrical journal Vol. 49; no. 4; pp. 565 - 584
• Main Authors: Yang, Zhao, Hardin, James W., Addy, Cheryl L., Vuong, Quang H.
• Format: Journal Article
• Language: English
• Published: Berlin WILEY-VCH Verlag 01.08.2007; WILEY‐VCH Verlag; Wiley-VCH
• Subjects: Applications; Biology, psychology, social sciences; Biometry - methods; Bootstrap; Computer Simulation; Count data; Data Interpretation, Statistical; Epidemiologic Methods; Exact sciences and technology; General topics; Generalized Poisson model; Linear inference, regression; Mathematics; Models, Biological; Models, Statistical; Multivariate analysis; Negative binomial model; Overdispersion; Poisson Distribution; Poisson model; Probability and statistics; Sciences and techniques of general use; Score test; Statistics; Vuong test; Biometrics; Wald test; Score test; Statistical distribution; Sample size; Overdispersion; Gaussian distribution; Significance level; Negative binomial model; Multivariate analysis; Statistical simulation; Negative binomial distribution; Hypothesis test; Significance test; Statistical test; Poisson model; Vuong test; Distribution function; Bootstrap; Jackknife method; Approximation theory; Test power; Sample survey; Likelihood function; Score statistics; Generalized Poisson model; Count data; Life science; Asymptotic behavior; Likelihood ratio; Statistical method; Statistical regression; Sampling theory; Regression model; Asymptotic normality; Likelihood ratio test; Small sample; Resampling method
• ISSN: 0323-3847; 1521-4036
• DOI: 10.1002/bimj.200610340

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. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)