Flexible parametrization of variance functions for quantal response data derived from counts

Although the Poisson model has been widely used to fit count data, a well-known drawback is that the Poisson mean equals its variance. Many alternative models for counts that are overdispersed relative to Poisson have been developed to solve this issue, including the negative binomial model. In this article, the negative binomial model with a four-parameter logistic mean is proposed to handle these types of counts, with variance that flexibly depends on the mean. Various parameterizations for the variance are considered, including extra-Poisson variability modeled as an exponentiated B-spline. Thus, the proposed model ably captures the leveling off of the mean, i.e., the "lazy-S" shape often encountered for overdispersed dose-response counts, simultaneously taking into account both overdispersion and natural mortality. Two real datasets illustrate the merits of the proposed approach: media colony counts after tuberculosis decontamination, and the number of monkeys killed by Ache hunters over several hunting trips in the Paraguayan tropical forest.