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 classes of models for such
data. In the literature so far only score tests are used for
testing the necessity of this adjustment. For this testing problem
we show how poor the performance of the corresponding score test
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 also prove consistency and
asymptotic normality of the maximum likelihood estimators in the
above mentioned regression models to give a theoretical
justification for Wald and likelihood ratio tests.