Carlin-Wolfe-Brown-Gelman (2001) Multilevel Models for Longitudinal Binary Outcomes

Summary

A case-study comparison of three model families for longitudinal binary outcomes, applied to self-reported regular smoking in the Victorian Adolescent Health Cohort Study (waves 1992–1995). The three models are: a semiparametric generalized estimating equations (GEE)/marginal model, a standard multilevel logistic-normal model (subject-specific), and a discrete latent-class mixture model. Model checking via Gelman's posterior predictive distributions shows the mixture model captures bimodal smoking trajectories that the other two miss.

Key Claims

• Marginal vs. subject-specific effects. For binary outcomes under a logistic link, population-averaged (marginal) effects from GEE and individual-level (subject-specific) effects from the logistic-normal model differ by a shrinkage factor proportional to the random-effect variance. They coincide only under a linear (identity) link. Practitioners must choose which estimand is of interest before selecting a model.
• Three model families. (1) Marginal/GEE: estimates population-averaged odds-ratio effects; does not assume a random-effects distribution; robust to misspecification of within-subject correlation structure. (2) Logistic-normal: estimates subject-specific log-odds trajectories; integrates over a normal random intercept; assumes normal heterogeneity. (3) Discrete mixture: finite mixture of latent trajectory classes with class-specific fixed effects; nonparametric approximation to the random-effects distribution.
• Posterior predictive model checking. Replicate the data many times from the fitted posterior predictive distribution; compare observed summary statistics (e.g. marginal smoking rate by wave, proportion always-smoking) against the distribution of replicated statistics. Extreme observed values relative to replicated distribution indicate model failure.
• GEE and logistic-normal fail the predictive check. Neither can reproduce the observed proportion of persistent non-smokers or the heavy-smoking tail; both yield replicated smoking rates that are too diffuse at the individual level. The discrete mixture with three latent classes passes all checks.
• Identifiability of discrete mixture. With $K=3$ latent classes the model is identified from the longitudinal structure; class membership probabilities and class-specific trajectories are estimated jointly via expectation-maximization (EM) or Bayesian simulation.

Concepts Introduced or Extended

• Bayesian Hierarchical Model
• Posterior Predictive Check
• Generalized Estimating Equations
• Mixture Model

Entities Mentioned

• Andrew Gelman

Quotes

"The appropriate application and interpretation of these models remains somewhat unclear, especially when compared with the computationally more straightforward semiparametric or 'marginal' approach."

My Take

This is a model-selection paper in applied biostatistics, not econometrics. Its relevance to the wiki is methodological: the marginal-vs-subject-specific distinction is directly analogous to random-effects vs. population-averaged estimation in panel probit models (echoing Rossi-Allenby 2003 on individual vs. aggregate effects), and the posterior predictive checking methodology is Gelman-Meng-Stern (1996) applied systematically. The discrete mixture "wins" here because smoking trajectories are genuinely bimodal, not because mixtures are universally superior; the lesson is that predictive checks should drive model selection rather than theoretical elegance alone.