Geweke-Gowrisankaran-Town (2003) Bayesian Inference for Hospital Quality in a Selection Model

Summary

Geweke, Gowrisankaran, and Town (2003) develop a Bayesian simultaneous-equation model to estimate hospital quality in the presence of non-random patient assignment. Patients select hospitals partly on unobserved severity of illness, so standard probit rankings confound quality with case mix. The model couples a multinomial probit for hospital choice (instrumented by patient-hospital distance) with a binary probit for 10-day in-hospital mortality; errors in the two equations are linked via hospital-specific "severity correlations" $\rho_j$, allowing the model to learn which hospitals attract sicker-than-average patients. Applied to 74,848 Medicare pneumonia admissions in Los Angeles County (1989–1992), the paper finds selection is massive — the selection component of mortality variance is eight times the independent component — and that correcting for it reverses the ranking of several hospital types relative to standard probit.

Key Claims

• Simultaneous-equation selection model: Two latent-variable equations share correlated errors. Hospital choice equation: $y_{ih}^* = x_i'\gamma_h + \delta_h\varepsilon_i + \eta_{ih}$ with $y_{ih}^* > \max_{h'\neq h} y_{ih'}^*$ indicating patient $i$ chooses hospital $h$. Mortality equation: $m_i^* = x_i'\beta + q_{s(i)} + \varepsilon_i$ with $m_i = \mathbf{1}[m_i^*>0]$. The scalar $\varepsilon_i$ (mortality shock) enters both equations: $\delta_j$ is the population regression coefficient of $\varepsilon_i$ on $(\eta_{i1},\dots,\eta_{iJ})$, so $\delta_j>0$ means hospital $j$ attracts patients with above-average severity.
• Identification instrument: Patient-specific driving distances to all 114 hospitals. Conditional on patient covariates, distance predicts hospital choice but has no direct effect on mortality, satisfying the exclusion restriction.
• Markov chain Monte Carlo (MCMC) algorithm: Gibbs sampling with data augmentation extending Albert-Chib (1993b) binary probit and Geweke-Keane-Runkle (1994, 1997) multinomial probit. Mortality latent variable $m_i^*$ and hospital choice latent utilities $y_{ih}^*$ are drawn conditional on all other parameters; the mortality shock $\varepsilon_i$ is then drawn from its full conditional, collapsing the joint latent structure. Hospital quality indicators $q_j$ and severity correlations $\delta_j$ are drawn separately. Full run: 12,000 iterations, 2,000 burn-in; inefficiency factors for $q_j$ range 3–30.
• Hierarchical prior on quality: Hospital quality indicators $q_j$ drawn from a normal distribution with mean and variance depending on ownership (public, private non-teaching, private teaching) and bed-size category (≤150, 151–200, 201–300, >300 beds). This shrinks small-hospital estimates toward their group mean while allowing the data to determine inter-group differences.
• Selection is massive: Posterior mean of $\delta'\Sigma\delta = 8.7$ vs. independent mortality variance component of 1. About 90% of between-hospital mortality variation is explained by which patients each hospital attracts (selection), not by quality.
• Quality is U-shaped in bed size: Smallest hospitals (≤150 beds) and largest (>300 beds) have highest quality; the 151–200 bed category has the lowest. Public hospitals rank worst; private teaching hospitals rank best.
• Standard probit rankings are misleading: Spearman correlation between selection-corrected and standard-probit quality rankings is only 0.517. High-quality hospitals attract sicker patients ($q_j$ and $\rho_j = \delta_j/\sqrt{\delta_j^2+1}$ have posterior correlation 0.517), so without correction they appear to have worse outcomes.
• Robustness: Tighter/looser hyperpriors on quality variance change rankings only modestly (correlation with base model ≈ 0.80). Removing instruments (variant A with only patient covariates) collapses identification: correlation with base model drops to 0.34.
• Computation advantage: Full maximum likelihood (ML) estimation of this model is computationally infeasible (authors estimate months per parameter vector evaluation for 114 hospitals); MCMC with data augmentation makes inference tractable.

Concepts Introduced or Extended

• Selection Model
• Binary Probit
• Multinomial Probit
• Bayesian Hierarchical Model

Entities Mentioned

• John Geweke
• Gautam Gowrisankaran
• Robert J. Town

Quotes

"The analysis indicates that selection is an important phenomenon. The posterior mean of the variance of the 'selection component' in the mortality equation is 8.7, compared with 1.0 for the variance of the independent component."

"These results indicate the importance of controlling for patient selection when assessing hospital quality. If one fails to account for patient selection, the resulting quality estimates will be substantially different from those based on the correct model."

My Take

The paper makes two distinct contributions: a methodological one (a tractable Bayesian MCMC estimator for a nonlinear simultaneous-equation selection model) and an empirical one (showing that the selection problem in hospital quality measurement is quantitatively enormous, not just theoretically possible). The identification strategy via distance instruments is standard in health economics but the Bayesian treatment allows honest propagation of uncertainty about $\delta_j$ through to quality estimates — something classical two-step estimators cannot do cleanly. The U-shaped quality–size relationship and the reversal of public-vs-private rankings are substantively interesting. A limitation is that the distance instrument may be weak for patients with multiple hospitals nearby; the paper does not report first-stage F-statistics, relying instead on the Bayesian posterior to discipline identification. Originally circulated as a 2001 NBER working paper; published as Geweke, Gowrisankaran, and Town (2003) in Econometrica 71(4): 1215–1238 (the citation of record).