A One‐Step Gauss‐Newton Estimator for Modelling Categorical Data with Extraneous Variation

We examine data from a study of the effects of prenatal exposure to elevated levels of cadmium and zinc on mortality and physical malformation rates of hamster fetuses. As is common in teratology studies, extraneous variation relative to Poisson and multinomial models arises from examining more than one fetus for each treated female. We establish asymptotic normal properties of a one‐step Gauss‐Newton estimator of the parameters of any sufficiently smooth function that links the expectation of each observed vector of counts to a finite set of covariates, when the data exhibit either overdispersion or underdispersion. The asymptotic properties of this estimator rely on the existence of the first two moments of the observation vectors and on the consistency of initial estimators for parameters in the link function and for any additional parameters used to model extraneous variation. Various patterns of extraneous variation are accommodated by simultaneously adjusting different components of variation. Applications to logistic regression models for multicategory responses with extramultinomial variation are explicitly considered.