Wolfinger-O'Connell (1993) Generalized Linear Mixed Models: A Pseudo-Likelihood Approach

Summary

Wolfinger and O'Connell develop the pseudo-likelihood (PL) method for fitting generalized linear mixed models — GLMs extended with random effects and/or correlated errors. The idea is to linearize the model (a first-order Taylor expansion of the link function) into a "modified dependent variable," then fit a weighted Gaussian linear mixed model to it, iterating between updating that working response and estimating the variance components by (restricted) maximum likelihood. This turns GLMM fitting into repeated calls to standard linear-mixed-model machinery, allows flexible covariance structures for both random effects and residual errors, optionally estimates an extra dispersion parameter, and supports both subject-specific and population-averaged inference. It is the estimation engine behind SAS's GLMM procedures.

Key Claims

• The GLMM. With link $g(\cdot)$, $E(y\mid\beta)=g^{-1}(X\alpha+Z\beta)$, fixed effects $\alpha$, random effects $\beta\sim(0,D)$, and (possibly correlated) errors $e$; the conditional variance follows an exponential-family mean-variance relationship, optionally scaled by a dispersion $\phi$.
• Pseudo-likelihood via a working response. Expand $g$ about current estimates to get a modified dependent variable $y^*=\hat\eta+(y-\hat\mu)\,g'(\hat\mu)$ with working weights from the variance function; then $y^*$ satisfies (approximately) a Gaussian linear mixed model $y^*=X\alpha+Z\beta+\epsilon^*$.
• Iterated weighted LMM fitting. Alternate: (i) given variance components, solve the mixed-model (Henderson) equations for $\hat\alpha,\hat\beta$; (ii) given the fit, re-estimate the variance components (in $D$ and the error covariance) by REML on the current pseudo-data; (iii) rebuild $y^*$ and weights. Iterate to convergence — essentially Newton-Raphson / Fisher scoring on the pseudo-likelihood.
• Flexible covariance and dispersion. Because the inner model is a general LMM, arbitrary structured covariances (compound symmetry, AR(1), unstructured, spatial) can be placed on the random effects and the correlated errors; an overdispersion parameter is estimated automatically if desired.
• Two inference targets. The random-effects (conditional) fit gives subject-specific effects; marginalizing gives population-averaged effects — both are available. Illustrated on the salamander mating data (McCullagh-Nelder 1989), the canonical hard binary-GLMM benchmark.

Concepts Introduced or Extended

• Generalized Linear Mixed Model

Entities Mentioned

• Russ Wolfinger
• Michael O'Connell

Quotes

"A pseudo-likelihood estimation procedure is developed to fit this class of mixed models based on an approximate marginal model for the mean response ... implemented via iterated fitting of a weighted Gaussian linear mixed model to a modified dependent variable."

"The method allows for subject-specific and population-averaged inference."

My Take

This is the SAS/GLIMMIX sibling of Breslow-Clayton's penalized quasi-likelihood and the multilevel world's PQL: the same core trick — linearize the link into a working response and hand the problem to a linear-mixed-model solver — but framed so that any structured covariance (R-side correlated errors as well as G-side random effects) drops in for free, which is exactly what longitudinal and spatial applications need. Its practical importance is enormous (it is how a generation of applied statisticians actually fit GLMMs), and its practical limitation is the one the wiki's GLMM page already flags for PQL: because it works off a Gaussian approximation to the conditional likelihood, it is biased for sparse binary/clustered data (the salamander data is the standing warning), which is what later motivates adaptive Gauss-Hermite quadrature, Laplace/INLA, and full MCMC. Read it as the definitional statement of the pseudo-likelihood approach — flexible, fast, and approximate — and the point where "fit a GLMM" became routine software rather than a research project.