Fitzmaurice-Laird-Rotnitzky (1993) Regression Models for Discrete Longitudinal Responses

Summary

A review of regression methods for longitudinal (repeated) binary responses, where interest centres on the regression parameters for the marginal expectations $E(Y_{it})$ and the within-subject association is treated largely as a nuisance. It contrasts two families: the non-likelihood generalized estimating equations (GEE) approach of Liang and Zeger (1986), and likelihood-based approaches built on a log-linear representation of the joint response probabilities. Its unifying result is a likelihood-based "mixed-parameter" model whose likelihood equations for the regression parameters take exactly the same form as the GEE. The authors compare the two approaches by asymptotic relative efficiency (complete data) and asymptotic bias (incomplete data), and give recommendations for practice.

Key Claims

• Marginal modelling of correlated binary data. For repeated binary outcomes there is no discrete analogue of the multivariate Gaussian that separates mean and covariance into variation-independent parameters, so one models the marginal means (logit link) directly and handles the association separately.
• GEE vs. likelihood. GEE specifies only the first two moments plus a working correlation and yields consistent, robust (sandwich-variance) marginal-coefficient estimates without a full joint distribution; the log-linear / mixed-parameter likelihood approach specifies the full joint law and can be more efficient.
• The mixed-parameter model. A likelihood-based marginal model parameterized so that its score equations for the regression coefficients coincide in form with the GEE — connecting the two literatures and clarifying when they agree.
• Efficiency (complete data). For complete data the GEE and mixed-parameter estimators have comparable asymptotic efficiency across many designs, so the simpler GEE is often adequate.
• Bias under missingness (incomplete data). The approaches differ once responses are missing: under MCAR all estimators are approximately unbiased, but under MAR the "working-independence" GEE estimators can be badly biased for time effects. GEE performs well under MAR when the design has no time-varying covariates; otherwise likelihood-based methods, or a close approximation to $\text{cov}(Y_i)$, are needed.
• Recommendation. When there are missing responses and interest is in time effects, working-independence estimators cannot be recommended; accurately approximating the within-subject correlation reduces bias.

Concepts Introduced or Extended

• Generalized Estimating Equations

Entities Mentioned

• Garrett M. Fitzmaurice
• Nan M. Laird
• Andrea G. Rotnitzky

Quotes

"For many practical situations, it appears that nearly efficient and unbiased estimates of the regression parameters for the marginal expectation can be obtained even when the true association between the responses is only crudely approximated."

My Take

A lucid survey that did real synthesising work: by exhibiting a likelihood-based "mixed-parameter" model whose regression score equations match the GEE, it showed the marginal-model and estimating-equation camps were closer than they looked, and it separated cleanly the two regimes that actually matter — complete data (where GEE's simplicity costs little efficiency) and missing data (where ignoring the correlation biases time effects under MAR). The MCAR/MAR bias analysis is the part that has aged best; it prefigures the missing-data-robustness agenda that Rotnitzky in particular went on to develop. On the wiki it deepens the GEE page's treatment of the marginal-vs-likelihood trade-off and connects it to model misspecification and missing-data mechanisms.