Gilks et al. (1993) Random-Effects Models for Longitudinal Data Using Gibbs Sampling

Summary

Gilks, Wang, Yvonnet and Coursaget generalize the Laird-Ware (1982) linear random-effects model for longitudinal data to accommodate multiple, not-necessarily-nested random effects, and show how to estimate it by Gibbs sampling. Where Laird-Ware handled a single subject-level random effect (irregular number and spacing of observations) via the EM algorithm, this model lets several overlapping groupings of individuals each contribute a source of heterogeneity. The method is illustrated on long-term response to hepatitis B vaccination and extended to handle censoring in the dependent variable.

Key Claims

• Multiple random effects. A linear multiple-random-effects model that simultaneously accommodates group-specific heterogeneity for several groupings of individuals — the nested models of Laird-Ware (1982), Goldstein (1986), and Longford (1987) are special cases.
• Gibbs-sampling estimation. Rather than EM, the model is estimated by Gibbs sampling (Geman-Geman 1984), cycling through the full conditionals of fixed effects, random effects, and variance components — a fully Bayesian route well suited to the unbalanced, irregularly-spaced data typical of longitudinal epidemiology.
• Censoring handled naturally. The methodology extends "easily and effectively" to censoring in the dependent variable, by treating censored responses as latent quantities imputed within the Gibbs loop.
• Why random effects matter. They model the effect of unmeasured covariates that induce non-independence among an individual's repeated responses; ignoring them biases inference when balance is unattainable (loss to follow-up, irregular contact).

Concepts Introduced or Extended

• Mixed-Effects Regression

Entities Mentioned

• Wally R. Gilks

Quotes

"We propose a generalisation of [the Laird-Ware] model to accommodate multiple random effects, and show how Gibbs sampling can be used to estimate it … the methodology can be easily and effectively extended to deal with censoring in the dependent variable."

My Take

This is the applied twin of the same group's "Modelling Complexity" survey from the same year: where that paper made the general case for Gibbs sampling in medicine, this one works a specific, valuable extension — pushing the workhorse Laird-Ware mixed model beyond a single nested random effect and beyond fully-observed responses. The multiple-non-nested-random-effects generalization is the genuinely useful modelling move (crossed rather than strictly nested groupings are common and awkward for EM), and casting estimation as Gibbs sampling is what makes the censoring extension almost free. It is a clean early example of the pattern that came to define BUGS-era biostatistics: once you can sample the full conditionals, model features that were painful under likelihood maximization (crossed effects, censoring, missing data) become incremental additions rather than new theory.