Gibbons-Hedeker (2000) Applications of Mixed-Effects Models in Biostatistics

Summary

A broad-scope tutorial on mixed-effects regression models for biostatistics, covering continuous, binary, ordinal, and multivariate outcomes in two- and three-level hierarchical designs. The paper walks through four substantive datasets — National Institute of Mental Health (NIMH) Treatment of Depression Collaborative Research Program (TDCRP), NIMH Schizophrenia collaborative study, British coal miners respiratory disease, and National Opinion Research Center (NORC) social attitudes survey — to demonstrate the MIXREG and MIXOR software implementations. Key methodological contributions include pattern-mixture models for nonignorable dropout, partial proportional odds models for ordinal outcomes with heterogeneous covariate effects, and Gaussian factorization of the covariance matrix for numerical stability with near-zero variance components.

Key Claims

• NIMH TDCRP (two-level continuous): Mixed-effects model reveals significant experience × treatment interaction (p < .004) invisible to fixed-effects analysis of variance (ANOVA); intraclass correlation (ICC): ICC_therapist = 0.02 (not significant), ICC_patient = 0.27. Empirical Bayes expected a posteriori (EAP) estimates shrink extreme therapist effects toward the mean, providing more stable predictions than ordinary least squares (OLS).
• NIMH TDCRP (three-level): Treating therapists as a third level adds therapist-level variance ($\hat{\sigma}^2_{\text{therapist}}$) and subjects-within-therapist variance; conditional independence trick reduces integration dimensions.
• NIMH Schizophrenia (binary): Three-drug randomized controlled trial (RCT); random effects probit with $\hat{\sigma}_{\beta_0} = 0.860$, $\hat{\sigma}_{\beta_1} = 0.630$; significant treatment × time interaction for all three drugs; fixed-effects model misses all but Fluphenazine main effect.
• Coal miners (multivariate probit): Five respiratory symptoms (wheeze, phlegm, breathlessness, cough, dyspnea) jointly modeled via two-factor Bock-Gibbons structure; factor 1 = chronic obstructive (wheeze/phlegm/dyspnea), factor 2 = acute (breathlessness/cough); dust exposure loads on both.
• NORC (ordinal partial proportional odds): Partial proportional odds model for social attitudes toward premarital sex (7 categories); likelihood-ratio (LR) χ² = 36.14, df = 4, p < .001 for proportionality violation; age and religion have heterogeneous threshold effects.
• Pattern-mixture for dropout: Missing data pattern (dropout time) entered as between-subjects covariate in the mixed-effects model; no separate dropout mechanism required; identifies which patterns drive results.
• Gaussian factorization: $\boldsymbol{\Sigma}_\beta = \boldsymbol{\Lambda}\mathbf{D}\boldsymbol{\Lambda}'$ with diagonal $\mathbf{D}$ having exponential-transformed entries; preferred over Cholesky decomposition when variance components approach zero — avoids convergence failures.
• EAP vs. OLS: Empirical Bayes shrinkage estimators $\tilde{\boldsymbol{\beta}}_i = [\mathbf{X}_i'(\sigma^2_\varepsilon \boldsymbol{\Omega}_i)^{-1}\mathbf{X}_i + \boldsymbol{\Sigma}_\beta^{-1}]^{-1}\mathbf{X}_i'(\sigma^2_\varepsilon \boldsymbol{\Omega}_i)^{-1}(\mathbf{y}_i - \mathbf{W}_i\boldsymbol{\alpha})$ have lower MSE than OLS when true random effects distribution is normal.

Concepts Introduced or Extended

• Mixed-Effects Regression
• Random Effects Probit
• Pattern-Mixture Model
• Multivariate Probit
• Generalized Estimating Equations

Entities Mentioned

• Robert D. Gibbons
• Donald Hedeker

Quotes

"The mixed-effects regression model (MRM) provides a useful general framework for the analysis of clustered and longitudinal data in biomedical research." (p. 70)

"The EAP estimate... provides a compromise between the overall population mean (James-Stein shrinkage) and the individual's own OLS estimate... the degree of shrinkage depends on how well the population distribution predicts the individual's data." (p. 79)

"Ignoring the clustering of patients within therapists leads to incorrect standard errors and potentially erroneous conclusions." (p. 84)

My Take

An excellent applied reference that ties together four distinct modeling branches — continuous MRM, binary random effects probit model (REPM), multivariate probit, and ordinal partial proportional odds — under a unified mixed-effects umbrella. The pattern-mixture treatment of dropout is practically important but presented tersely; the Diggle-Kenward (1994) selection model approach is the main competitor not discussed in depth. The MIXREG/MIXOR software focus makes results reproducible, though the programs are vintage Fortran and superseded by R packages (lme4, ordinal). The coal miners multivariate probit example is the clearest exposition of the Bock-Gibbons factor structure in an applied setting.