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.