Marginalized multilevel models and likelihood inference

Hierarchical or "multilevel" regression models typically parameterize the m ean response conditional on unobserved latent variables or "random" effects and then make simple assumptions regarding their distribution. The interpr etation of a regression parameter in such a model is the change in possibly transformed mean response per unit change in a particular predictor having controlled for all conditioning variables including the random effects. An often overlooked limitation of the conditional formulation for nonlinear m odels is that the interpretation of regression coefficients and their estim ates can be highly sensitive to difficult-to-verify assumptions about the d istribution of random effects, particularly the dependence of the latent va riable distribution on covariates. In this article, we present an alternati ve parameterization for the multilevel model in which the marginal mean, ra ther than the conditional mean given random effects, is regressed on covari ates. The impact of random effects model violations on the marginal and mor e traditional conditional parameters is compared through calculation of asy mptotic relative biases. A simple two-level example from a study of teratog enicity is presented where the binomial overdispersion depends on the binar y treatment assignment and greatly influences likelihood-based estimates of the treatment effect in the conditional model. A second example considers a three-level structure where attitudes toward abortion over time are corre lated with person and district level covariates. We observe that regression parameters in conditionally specified models are more sensitive to random effects assumptions than their counterparts in the marginal formulation.