Metameters in nonlinear random effects and frailty models

Random effects (RE) models are commonly used in clustered data to account for intra-cluster correlation, overdispersion and to structure cluster-specific inference. For this modeling approach, the joint distribution is built up from the conditional distributions with parameters specified for each cluster. The cluster-specific parameters are assumed to follow a distribution, which can be regarded as "borrow strength across clusters". Regression parameters in these models have a cluster-specific interpretation in that they quantify covariate effects for each cluster, but not necessarily for the overall population. These "population-averaged" predictions and inferences are often desired and in most nonlinear RE models the cluster-specific functional shape is different from that for the population average. Thus, cluster-specific regression parameters usually do not have a direct population interpretation.; In this thesis, we develop a modeling strategy for cluster-specific (conditional) and population-averaged (marginal) inferences that produces identical functional forms. We first consider RE models with random intercept and introduce the bridge density function for the random effect. This density depends on the functional form of the cluster-specific model (e.g., logistic) and the conditional functional shape is retained in the marginal scale. Thus, regression parameters in these models retain an explicit population-averaged interpretation. The bridge function is found closely related to the self-decomposable distribution. The RE logistic model is discussed in some detail. Secondly, we extend the bridge function approach to nonlinear RE models with random vectors and thereby propose a general framework for connecting cluster specific and population-averaged inferences. By appropriately controlling for random effects or covariates or both, the conditional functional shapes can be retained in the marginal scale. By using a reparameterization, marginal specific RE models can be obtained for direct population inferences, while still allow for subject-specific predictions and inferences. The approach applies to nonlinear RE models such as quantal responses and proportional hazards with random frailties.; Parameter estimation is carried out using maximum likelihood with Gaussian-Hermite quadrature and generalized estimating equations (GEE). We apply and compare the models on several real data sets. Limited simulations are carried out to compare the bridge function and the commonly used Gaussian distribution.