Joint Models for Toxicology Studies with Dose‐Dependent Number of Implantations

Many chemicals interfere with the natural reproductive processes in mammals. The chemicals may prevent the fertilization of an egg or keep a zygote from implanting in the uterine wall. For this reason, toxicology studies with pre‐implantation exposure often exhibit a dose‐related trend in the number of observed implantations per litter. Standard methods for analyzing developmental toxicology studies are conditioned on the number of implantations in the litter and therefore cannot estimate this effect of the chemical on the reproductive process. This article presents a joint modeling approach to estimating risk in toxicology studies with pre‐implantation exposure. In the joint modeling approach, both the number of implanted fetuses and the outcome of each implanted fetus is modeled. Using this approach we show how to estimate the overall risk of a chemical that incorporates the risk of lost implantation due to pre‐implantation exposure. Our approach has several distinct advantages over previous methods: (1) it is based on fitting a model for the observed data and, therefore, diagnostics of model fit and selection apply; (2) all assumptions are explicitly stated; and (3) it can be fit using standard software packages. We illustrate our approach by analyzing a dominant lethal assay data set (Luning et al., 1966, Mutation Research, 3, 444–451) and compare our results with those of Rai and Van Ryzin (1985, Biometrics, 41, 1–9) and Dunson (1998, Biometrics, 54, 558–569). In a simulation study, our approach has smaller bias and variance than the multiple imputation procedure of Dunson.