Testing for treatment related trend with partially exchangeable clustered data

When testing for treatment related effect, it is common to focus on data that describe some primary response, even though an experiment might yield data on secondary, less conspicuous effects. An example of such an experiment can be found in developmental toxicity studies, where a primary response might be the presence of skeletal malformation in a fetus, and a secondary effect might be reduced weight. In such studies, weight reduction may occur not only among malformed fetuses, but also among normal fetuses. In this paper, we use a likelihood ratio procedure to construct treatment related trend tests for developmental experiments with such outcomes. We assume that within each cluster, data are partially exchangeable and we use the presence or absence of malformations as categorical covariates. Two models are considered for the covariance structures: In the general model, partially exchangeable covariance matrices are used for each cluster within dose groups. In the homogeneous model, a common exchangeable covariance matrix is used for all clusters. Maximum likelihood estimates (MLEs) of unknown parameters in the general model are obtained by directly maximizing the log-likelihood function through careful numerical consideration, while MLEs of parameters of teh homogeneous model are obtained through an iterative procedure. Two applications to developmental toxicity data are described.