A novel application of a bivariate regression model for binary and continuous outcomes to studies of fetal toxicity

Summary.  Public health concerns over the occurrence of birth defects and developmental abnormalities that may occur as a result of prenatal exposure to drugs, chemicals and other environmental factors has led to an increasing number of developmental toxicity studies. Because fetal pups are commonly evaluated for multiple outcomes, data analysis frequently involves a joint modelling approach. We focus on modelling clustered binary and continuous outcomes in the setting where both outcomes are potentially observable in all offspring but, owing to practical limitations, the continuous outcome is only observed in a subset of offspring. The subset is not a simple random sample but is selected by the experimenter under a prespecified probability model. Although joint models for binary and continuous outcomes have been developed when both outcomes are available for every fetus, many existing approaches are not directly applicable when the continuous outcome is not observed in a simple random sample. We adapt a likelihood‐based approach for jointly modelling clustered binary and continuous outcomes when the continuous response is missing by design and missingness depends on the binary trait. The approach takes into account the probability that a fetus is selected in the subset. Through the use of a partial likelihood, valid estimates can be obtained by a simple modification to the partial likelihood score. Data involving the herbicide 2,4,5‐trichlorophenoxyacetic‐acid are analysed. Simulation results confirm the approach.