Hierarchical or ‘‘multilevel’’
regression models typically parameterize the mean response conditional on
unobserved latent variables or ‘‘random’’ effects
and then make simple assumptions regarding their distribution. The
interpretation 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
models is that the interpretation of regression coefficients and their
estimates can be highly sensitive to difficult-to-verify assumptions about the
distribution of random effects, particularly the dependence of the latent
variable distribution on covariates. In this article, we present an alternative
parameterization for the multilevel model in which the marginal mean, rather
than the conditional mean given random effects, is regressed on covariates. The
impact of random effects model violations on the marginal and more traditional
conditional parameters is compared through calculation of asymptotic relative
biases. A simple two-level example from a study of teratogenicity is presented
where the binomial overdispersion depends on the binary 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 correlated 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.