Testing for zero-modification in count regression models

Count data often exhibit overdispersion and/or require an adjustment for zero outcomes with respect to a Poisson model. Zero-modified Poisson (ZMP) and zero-modified generalized Poisson (ZMGP) regression models are useful model classes for such data. In the literature so far only score tests are used for testing the necessity of this adjustment. We address this problem by using Wald and likelihood ratio tests. We show how poor the performance of the score tests can be in comparison to the performance of Wald and likelihood ratio (LR) tests through a simulation study. In particular, the score test in the ZMP case results in a power loss of 47% compared to the Wald test in the worst case, while in the ZMGP case the worst loss is 87%. Therefore, regardless of the computational advantage of score tests, the loss in power compared to the Wald and LR tests should not be neglected and these much more powerful alternatives should be used instead. We prove consistency and asymptotic normality of the maximum likelihood estimates in ZGMP regression models, which form the basics of Wald and likelihood ratio tests. The usefulness of ZGMP models is illustrated in a data example.