Goldstein-Rasbash (1996) Improved Approximations for Multilevel Models with Binary Responses

Summary

This paper diagnoses and fixes a serious bias in the quasi-likelihood estimation of multilevel (generalized linear mixed) models with binary responses. Rodriguez and Goldman (1995) had shown by simulation that the standard linearization procedures — first-order marginal quasi-likelihood (MQL) — can badly underestimate the fixed effects and variance components when the true random-effect variances are large, the response is binary, and there are few level-1 units per level-2 unit. Goldstein and Rasbash introduce an improved linearization — expanding the random part about the current estimated residuals rather than about zero, and adding second-order terms (predictive / penalized quasi-likelihood, PQL) — which largely removes the bias in the difficult regime, and which was already implemented in available multilevel software.

Key Claims

• The model. A two-level logit model $\pi_{ij}=\{1+\exp(-[X_{ij}\beta+u_j])\}^{-1}$ with $y_{ij}\sim\text{Bin}(1,\pi_{ij})$ and community random effect $u_j\sim\mathcal N(0,\sigma_u^2)$; the exposition extends to random coefficients, more levels, and proportion responses.
• The estimation strategy. Linearize the logit by Taylor expansion so the model becomes an approximate two-level Normal model, then apply quasi-likelihood via iterative generalized least squares (IGLS) (Goldstein 1986, 1991), using the binomial variance for the level-1 variation.
• Where MQL fails. The standard procedure expands the random part about zero (first-order MQL); Rodriguez-Goldman (1995) show this produces large downward biases in both fixed effects and variance components precisely when random-effect variances are large, clusters are small, and responses are binary.
• The fix: expand about the residuals (PQL). Expanding the random part about the current estimated residuals $\hat u_j$ rather than zero, and retaining second-order terms in the Taylor expansion, yields predictive quasi-likelihood (PQL); it largely eliminates the Rodriguez-Goldman biases in simulation.
• Practical and implemented. The improvement requires no new software framework — it is available in the existing multilevel programs (ML3/MLn), so the better estimates come at essentially no extra cost to the analyst.

Concepts Introduced or Extended

• Generalized Linear Mixed Model
• Multilevel Model

Entities Mentioned

• Harvey Goldstein
• Jon Rasbash

Quotes

"An improved approximation is introduced which largely eliminates the biases in the situation described by Rodriguez and Goldman."

"For the second order expansion for the random part we expand about zero, and we show below how this is modified to obtain improved estimates."

My Take

This is a small, targeted methods note whose importance is calibration: it takes a specific, documented failure — MQL's collapse for sparse binary multilevel data — and shows that a better expansion point (the residuals, not zero) plus a second-order term fixes most of it, without abandoning the fast IGLS machinery of the multilevel-modeling (MLwiN) tradition. It sits alongside the wiki's other GLMM-estimation entries as the frequentist-quasilikelihood counterpart to the Bayesian routes: where Fong-Rue-Wakefield answer the same clustered-binary difficulty with INLA and careful priors, Goldstein-Rasbash answer it by repairing the linearization. The honest limitation, which later work (and the move toward adaptive quadrature, Laplace, and MCMC) reflects, is that PQL is still an approximation — better than MQL but not exact — and for the very hardest sparse-binary cases only full-likelihood or Bayesian methods remove the bias entirely.