A note on the use of marginal likelihood and conditional likelihood in analyzing clustered data

In the analysis of clustered data, when a generalized linear model with a random intercept term is fitted using maximum marginal likelihood and maximum conditional likelihood, respectively, a discrepancy between estimated regression coefficients from the two methods has been observed. This discrepancy happens when some cluster-level confounders are omitted from the model. Here we offer a straightforward explanation for the discrepancy in terms of different modeling assumptions underlying the use of the two likelihood functions. Specifically, the marginal likelihood approach requires a full distributional assumption on random effects, and this assumption is violated when some cluster-level confounders are omitted from the model. We also propose to use residual plots to uncover the problem.