Gibbons-Wilcox-Gok (1998) Health Service Utilization and Insurance Coverage: A Multivariate Probit Analysis

Summary

Applies the full-information multivariate probit model of Bock and Gibbons (1996) to joint health service utilization (5 binary outcomes) among 4,658 U.S. Medicare enrollees aged 65+, using data from the 1987–1988 National Medical Expenditure Survey (NMES). The key extension over prior work is generalization to continuous (ungrouped) covariates, empirical nonparametric prior estimation, and marginal/pattern probability inference with bootstrapped standard errors. A two-factor solution fits the five services best: Factor 1 loads on ER/hospital/home health (acute care); Factor 2 loads on provider/outpatient visits (routine care). Medicare + Medicaid enrollees show substantially higher acute care utilization; excellent self-reported health dramatically reduces all services.

Key Claims

• Joint model of $p=5$ binary outcomes via latent response vector $\mathbf{y}_i = \mathbf{B}\mathbf{x}_i + \mathbf{\Lambda}\boldsymbol{\theta} + \boldsymbol{\zeta}$, where $\boldsymbol{\theta}$ is an $m$-dimensional latent factor ($m<p$) and $\boldsymbol{\zeta}$ is uncorrelated residual.
• Response probability conditional on latent factor: $P(u_{ij}=1|\boldsymbol{\theta}) = \Phi(z_{ij})$ where $z_{ij} = (\beta_j \mathbf{x}_i + \lambda_j \boldsymbol{\theta})/d_j$ and $d_j^2 = 1 - \sum_k \lambda_{jk}^2$.
• Marginal probability integrates out $\boldsymbol{\theta}$ via $m$-fold Gauss-Hermite quadrature; all $2^p = 32$ response pattern probabilities estimable.
• Estimation via hybrid EM (initializes $\Lambda$) + Fisher scoring (provides standard errors); identification achieved by fixing $m(m+1)/2$ elements of $\Lambda$ in lower-triangular form.
• Empirical nonparametric prior (Bock–Aitkin 1981) replaces normal prior; here yields nearly identical results (log-likelihood −9,491 vs. −9,496) with 100 quadrature weights.
• Sampling weights incorporated multiplicatively into the log-likelihood: $\log L = \sum_i r_i \log P(\mathbf{u}_i)$.
• Model selection via change of log-likelihood (deviance): 2-factor model significantly improves on 1-factor ($\chi^2_4 = 24.98$, $p < .0001$); 3-factor adds nothing.
• Estimated inter-service correlations moderate to high: largest $r_{ER,HOSP} = 0.665$; smallest $r_{MED,HOME} = 0.130$.
• Medicare + Medicaid group: ~20% higher home health care, ~10% higher ER and hospitalization relative to private-only coverage.
• Excellent self-reported health reduces all utilization substantially; each chronic condition raises all five service probabilities ($p < .001$).
• Black/Hispanic enrollees have lower provider-visit rates (MLE $-0.287$, $p < .001$) but not higher ER rates — contradicting common assumptions.

Concepts Introduced or Extended

• Multivariate Probit
• Binary Probit
• Generalized Estimating Equations

Entities Mentioned

• Robert D. Gibbons
• Virginia Wilcox-Gok

Quotes

"Data from all five service utilization variables are used to perform a single test of insurance coverage effects taking into account their intercorrelation and effects of covariates — the major contribution of this article."

"The estimated correlation among the five health service utilization variables is then taken into account when testing the hypothesis of no insurance coverage effects, providing an advance over traditional approaches such as univariate logistic regression."

My Take

The paper's methodological contribution is solid and underappreciated: full-information marginal maximum likelihood (MML) for jointly modeled binary outcomes is strictly more powerful than running $p$ separate logistic regressions, and the pattern probability estimates are unavailable from generalized estimating equations (GEE) or partial-information procedures (Muthén 1979). The hybrid EM + Fisher scoring solution is computationally practical and the empirical prior robustness check is reassuring. The application is well-chosen — Medicare utilization patterns are inherently multivariate and the correlation structure (acute vs. routine care dimensions) is substantively interpretable. The main limitation is frequency-data vintage: 1987 NMES predates managed care penetration, so the insurance-coverage effect sizes may not generalize to post-2000 Medicare.