Robert D. Gibbons

Overview

Robert D. Gibbons is Professor in the Departments of Biostatistics and Psychiatry at the University of Illinois at Chicago (UIC). His primary methodological contributions are in full-information multivariate probit modeling for jointly observed binary outcomes, building on his collaboration with R. Darrell Bock (Bock-Gibbons 1996, Biometrics). He has also contributed to multilevel and mixed-effects models for clustered and longitudinal binary data in biostatistics and psychiatric epidemiology applications.

Key Contributions / Features

• Bock-Gibbons (1996): With R. Darrell Bock, developed the full-information multivariate probit model for grouped data: $p$ binary outcomes driven by $m < p$ latent factors via $\mathbf{y}_i = \mathbf{B}\mathbf{x}_i + \mathbf{\Lambda}\boldsymbol{\theta} + \boldsymbol{\zeta}$; marginal MML via hybrid EM + Fisher scoring; Gauss-Hermite quadrature for orthant probability integration; identification via lower-triangular $\Lambda$.
• Gibbons-Wilcox-Gok (1998): Extended Bock-Gibbons to continuous (ungrouped) covariates, empirical nonparametric prior (Bock-Aitkin 1981), sampling weights, and bootstrapped pattern probability standard errors; applied to 5-outcome Medicare utilization data (NMES 1987); two-factor solution: acute care (ER/hospital/home health) and routine care (provider/outpatient).
• Gibbons-Hedeker (1997): Extended random-effects probit and logistic regression to three-level hierarchical data (measurement occasions within subjects within clusters). Key insight: conditioning on the cluster-level random effect $\theta_{(3)}$ makes subjects within a cluster conditionally independent, collapsing the marginal likelihood integral from $(n_i \times r)+1$ to $r+1$ dimensions — tractable for Gauss-Hermite quadrature with up to 3–4 subject-level random effects. Cholesky orthogonalization of $\Sigma_{\beta^*}$ standardizes the quadrature space. Applied to TVSFP smoking cessation (1,600 students / 135 classrooms / 28 schools): ICC classroom = 6.6%, ICC student = 25.0%; CC×TV interaction significant only in the three-level model. Biometrics 53(4): 1527–1537.
• Gibbons-Hedeker (2000): Applications of mixed-effects models in biostatistics (Sankhyā 62: 70–103) — unified tutorial on MIXREG/MIXOR covering continuous MRM (NIMH TDCRP: ICC_therapist = 0.02, ICC_patient = 0.27; experience × treatment interaction p < .004 invisible to fixed-effects ANOVA), binary random-effects probit (NIMH Schizophrenia: $\hat{\sigma}_{\beta_0}=0.860$, $\hat{\sigma}_{\beta_1}=0.630$; all three drugs show significant treatment × time), multivariate probit (coal miners: 5 symptoms / 2 factors), ordinal partial proportional odds (NORC: LR χ² = 36.14 df = 4 p < .001); Gaussian factorization $\boldsymbol{\Sigma}_\beta = \boldsymbol{\Lambda}\mathbf{D}\boldsymbol{\Lambda}'$ for numerical stability; pattern-mixture model for MNAR dropout.

Related

• Multivariate Probit
• Binary Probit
• Mixed-Effects Regression
• Pattern-Mixture Model
• Virginia Wilcox-Gok
• Gibbons-Wilcox-Gok (1998) Health Service Utilization and Insurance Coverage
• Generalized Estimating Equations
• Donald Hedeker
• Random Effects Probit
• Gibbons-Hedeker (1997) Random Effects Probit and Logistic Regression Models for Three-Level Data
• Gibbons-Hedeker (2000) Applications of Mixed-Effects Models in Biostatistics