Pendergast et al. (1996) A Survey of Methods for Analyzing Clustered Binary Response Data

Summary

A comprehensive survey of statistical methods for analyzing correlated binary responses arising from clustered designs — repeated measures on the same subject, bilateral data (e.g., two eyes), family studies, and cluster-randomized trials. The paper organizes the methods landscape into four families: naive/response-feature approaches (ignore correlation), conditionally specified models (specify joint distribution via pairwise terms), marginal models (generalized estimating equations, GEE), and cluster-specific random-effects models. The central tension is between marginal population-average coefficients (GEE) and cluster-specific subject-specific coefficients (random effects), which diverge in magnitude for nonlinear link functions. GEE, introduced by Liang and Zeger (1986), is highlighted as the most robust and widely applicable marginal approach.

Key Claims

• Four families of methods for clustered binary data: (1) naive/response-feature (ignore clustering — inflates Type I error and yields biased standard errors (SEs)); (2) conditionally specified (Besag 1974 auto-logistic, Rosner 1984 beta-binomial, Zhao-Prentice 1990 quadratic exponential); (3) marginal/GEE (Liang-Zeger 1986); (4) cluster-specific random effects (Stiratelli et al. 1984, Clayton 1978).
• GEE (Liang-Zeger 1986) working correlation matrix: $V_i = \phi\, A_i^{1/2} R_i(\alpha) A_i^{1/2}$, where $R_i(\alpha)$ can be exchangeable, AR-1, unstructured, or independence. The estimating equations $\sum_i D_i' V_i^{-1}(y_i - \mu_i) = 0$ yield $\hat\beta$ consistent for any choice of $R_i(\alpha)$ provided the marginal mean model is correctly specified.
• Robust sandwich variance estimator $\hat{V}(\hat\beta) = M^{-1}\hat{B}M^{-1}$ provides valid large-sample inference regardless of whether the working correlation is correct; only the marginal mean model need be correct.
• GEE2 (Prentice 1988): extension that jointly models first and second moments via augmented estimating equations for $(\beta, \alpha)$ simultaneously; more efficient than standard GEE when association is of interest, but less robust to misspecification of the correlation model.
• Marginal vs. cluster-specific interpretation: GEE $\hat\beta$ has a population-average (marginal) interpretation; random-effects $\hat\beta^{re}$ has a cluster-specific (subject-specific) interpretation. For logistic link, $|\hat\beta^{re}| > |\hat\beta^{GEE}|$ by a factor that grows with the variance of the random effect $\sigma_b^2$; for linear links the two coincide.
• Conditional inference: conditioning on the cluster total $y_{i+} = \sum_j y_{ij}$ eliminates random cluster effects without modeling their distribution; valid for exchangeable designs; sacrifices efficiency but robust to distributional misspecification of random effects.
• Goodness of fit: GEE has no likelihood, so standard deviance and likelihood-ratio (LR)–based GOF tests do not apply. Score tests for correlation parameters $\alpha$ and permutation-based methods are available but less powerful than likelihood-ratio tests for parametric alternatives.
• Overdispersion in clustered binary data can be handled via: (a) extra-binomial variance in marginal models, (b) beta-binomial marginal distribution (Rosner 1984), or (c) random intercept in logistic regression (Stiratelli et al. 1984).
• The auto-logistic model (Besag 1974) specifies the joint distribution via conditional probabilities $P(y_{ij} | y_{i,-j}) \propto \exp(\alpha_j y_{ij} + \sum_{k \neq j} \psi_{jk} y_{ij} y_{ik})$; valid for general pairwise association structure but the conditional likelihood may differ from the marginal likelihood implied by the parameters.
• The quadratic exponential family (Zhao-Prentice 1990) specifies a flexible joint distribution by including pairwise product terms $\sum_{j<k} \psi_{jk} y_{ij} y_{ik}$ in the log-likelihood; GEE2 reduces to this model when the association parameters are estimated from second-moment equations.

Concepts Introduced or Extended

• Generalized Estimating Equations

Entities Mentioned

None of the six authors has a dedicated wiki entity page.

Quotes

"A central distinction in clustered data analysis is between population-averaged and cluster-specific models. The choice depends on the scientific question: population-averaged models answer questions about average effects in the population; cluster-specific models answer questions about effects for a particular individual or cluster."

My Take

A well-organized survey that established the four-family taxonomy now standard in biostatistics textbooks (Diggle-Liang-Zeger 1994; Fitzmaurice-Laird-Ware 2004). The GEE exposition is clear and the marginal vs. cluster-specific distinction is explained more carefully than in most applied references. The goodness-of-fit gap for GEE remains a genuine limitation; the paper is honest about it. The paper predates Bayesian random-effects approaches (Zeger-Karim 1991 used Gibbs sampling but GEE2 for correlation) and does not discuss Markov chain Monte Carlo (MCMC)–based full-likelihood inference for clustered binary data, which has since become practical.