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

Area of Science:

• Statistics
• Biostatistics
• Econometrics

Background:

• Overdispersion is prevalent in count data, often necessitating alternatives to the standard Poisson model.
• Negative binomial (NB) and generalized Poisson (GP) models are common alternatives for handling overdispersion.
• Existing tests (Wald, LRT, score) for overdispersion have limitations, particularly in small sample sizes.

Purpose of the Study:

• To propose and evaluate a score test for overdispersion within the generalized Poisson (GP) regression framework.
• To compare the power of the proposed score test against Likelihood Ratio Test (LRT) and Wald tests.
• To introduce a bootstrap-adjusted score test to improve significance level accuracy in small samples.

Main Methods:

• Development of a score test statistic for overdispersion comparing Poisson and GP models.
• Simulation studies to assess the empirical power and significance levels of asymptotic score tests.
• Implementation of a bootstrap procedure to refine the score test's performance with small sample sizes.

Main Results:

• The asymptotic score test shows higher empirical power but underestimates the nominal significance level, especially for small samples.
• The bootstrap-adjusted score test demonstrates significance levels closer to the nominal size.
• The bootstrap test consistently exhibits greater power compared to the asymptotic score test.

Conclusions:

• The bootstrap score test is recommended for comparing GP and Poisson models due to its improved accuracy and power.
• A weak indication of model inadequacy from the asymptotic score test (e.g., p < 0.10) warrants using the bootstrap procedure.
• The Vuong test is demonstrated as a method for model selection between GP and NB2 models.