Article,
Marginalized Multilevel Models and Likelihood Inference
Statistical Science, (2000)
Abstract
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.
