Zeger-Liang-Albert (1988) Models for Longitudinal Data: A Generalized Estimating Equation Approach

Summary

This paper draws the now-standard distinction between two ways to extend generalized linear models to longitudinal / correlated data and shows both can be fit by a generalized estimating equation (GEE) approach. Subject-specific (SS) models put the heterogeneity across subjects explicitly into the regression parameters (as in random-effects / mixed GLMs), so coefficients describe how a given individual's response depends on covariates. Population-averaged (PA, or marginal) models target the aggregate response of the population, so coefficients describe how the population-average response depends on covariates. The authors fit both classes for discrete and continuous outcomes via GEE, and show that when the subject-specific effects are Gaussian there are simple closed-form relationships between the PA and SS parameters. The methods are illustrated on longitudinal data for mothers' smoking and children's respiratory disease. (Biometrics 44(4): 1049–1060.)

Key Claims

• Two targets, two questions. For correlated longitudinal outcomes there are two distinct modeling goals: subject-specific (model the heterogeneity — "how does this subject's expected response change with $x$?") and population-averaged / marginal (model the mean over the population — "how does the average response change with $x$?"). These answer genuinely different scientific questions and generally give different coefficients.
• GEE fits both. Building on Liang–Zeger (1986), a generalized estimating equation — a quasi-score equation with a working correlation structure and a robust (sandwich) variance — estimates the regression coefficients of either model class consistently, for discrete or continuous outcomes, without specifying a full joint distribution.
• PA and SS coefficients are systematically related under Gaussian mixing. If the subject-specific effects follow a Gaussian distribution, the PA (marginal) parameters are explicit functions of the SS parameters and the random-effect variance. For nonlinear links (e.g. logistic) the marginal coefficients are attenuated (shrunk toward zero) relative to the subject-specific coefficients by a factor increasing in the random-effect variance; for the identity link they coincide. So the SS-vs-PA gap is a feature of the nonlinearity, not a modeling error.
• Which to use is a substantive choice. PA models are natural when the target of inference is a population contrast (e.g. public-health effect of an exposure); SS models are natural when the target is an individual-level mechanism or when subject-level prediction is wanted. The GEE machinery makes the estimation comparable across the two.
• Robustness. As in the marginal GEE, consistency of the coefficient estimates and validity of the sandwich standard errors require correct specification of the marginal mean model only, not of the working correlation.

Concepts Introduced or Extended

• Generalized Estimating Equations
• Generalized Linear Mixed Model
• Multilevel Model

Entities Mentioned

• Scott L. Zeger
• Kung-Yee Liang
• Paul S. Albert

Quotes

"Two approaches are considered: subject-specific (SS) models in which heterogeneity in regression parameters is explicitly modelled; and population-averaged (PA) models in which the aggregate response for the population is the focus."

"When the subject-specific parameters are assumed to follow a Gaussian distribution, simple relationships between the PA and SS parameters are available."

My Take

This is the paper that made "marginal vs. conditional" (population-averaged vs. subject-specific) a required distinction in longitudinal-data analysis, and its lasting practical message is that the attenuation between the two is real and computable, not a discrepancy to be explained away: a logistic random-effects coefficient and a GEE coefficient estimate different things and differ in magnitude by a factor governed by the random-effect variance. It complements Liang–Zeger's estimating-equation machinery by clarifying what the estimand is under each modeling choice, and it sits directly upstream of the GLMM vs. marginal-model debates and the Bayesian hierarchical treatments (e.g. Zeger–Karim's Gibbs sampler) that followed. The main caveat is that the clean PA–SS relationships rely on the Gaussian-mixing assumption; with non-Gaussian or misspecified random effects the exact correspondence breaks down.