Score Tests for Extra-Zero Models in Zero-Inflated Negative Binomial Models

Abstract

When overdispersion is present in count data, a negative binomial (NB) model is commonly used in place of the standard Poisson model. However, the model is sometimes not adequate because of the occurrence of excess zeros and a zero-inflated negative binomial (ZNB) model may be more appropriate. This article proposes a general score test statistic for comparing a ZNB regression model to the NB model and the test is extended to a composite score test. Simulation results indicate that the test performs reasonably well and has a sampling distribution under the null hypothesis (NB model) approximated by the usual χ² distribution. Use of the test is illustrated on a set of apple shoot propagation data. The composite score test is found to indicate suitable models.

Keywords:

• Count data
• Overdispersion
• Score test
• Zero-inflation

Mathematics Subject Classification:

• 62-07
• 62J99

Acknowledgments

The authors would like to thank the referees for their comments, which helped them to improve the article.

Notes

For the constant λ models we have λ = exp(1.25) = 3.5. The explanatory variables are: x ₁, a two-level factor with third-fifth observations in the first group; x ₂, a variable with values uniformly distributed on (1, 3). With these values λ varies from 1.5–6.

x ₁ denotes a two-level factor with third-fifth observations in first group. With these values λ varies from 2.0–5.75.

x ₁ denotes a two-level factor with third-fifth observations in first group. With these values λ varies from 2.00–5.75, ω ₁ varies from 0.15–0.20, and ω ₂ varies from 0.25–0.55.

x ₂ is a variate taking on n values uniformly distributed on (1, 3). With these values λ varies from 1.5–6.

x ₂ is a variate taking on n values uniformly distributed on (1,3). With these values λ varies from 1.5–6, ω ₁ varies from 0.05–0.24, and ω ₂ varies from 0.26–0.65.

P is a two-level factor for photoperiod; H is a four-level factor for the BAP levels; Lin(H) is a linear trend over the levels of H (on the log-concentration scale for BAP); H*P represents a full interaction model.

P is a two-level factor for photoperiod; H is a four-level factor for the BAP levels; Lin(H) is a linear trend over the levels of H (on the log-concentration scale for BAP); H*P represents a full interaction model.

^†fails to converge.