On the problem of identification in multiplicative intensity-rate models with multiple interactions

In this paper we examine a multiplicative intensity model in which a covariate interacts with two other covariates in the same model. We demonstrate, analytically, that in such situations a log-linear parameterization based on two pairs of baseline levels cannot be transformed, uniquely, to the, otherwise equivalent, multiplicative parameterization. We show that the problem lies in an oversight of the conditional independence between the two covariates interacting with a common third covariate. As a solution, therefore, we propose an approach that takes due account of such dependence. Our proposed approach uses a common baseline level for the three covariates involved in interaction while estimating the corresponding relative intensities. The issues addressed are illustrated with a demographic data set involving the estimation of rates of transition to parenthood.[PUBLICATION ABSTRACT]