Gibbons-Hedeker (1997) Random Effects Probit and Logistic Regression Models for Three-Level Data

Summary

Extends two-level random effects probit and logistic regression to three-level data structures — two nested levels of clustering (e.g., measurements within subjects within classrooms, or patients within clinics within sites). The key computational insight is that conditioning on the cluster-level (level-three) random effect makes subjects within a cluster conditionally independent, reducing the marginal likelihood integral from $(n_i \times r)+1$ dimensions to $r+1$ — tractable for up to 3–4 subject-level random effects. Taylor-series linearization approaches (Longford 1988/1994, Goldstein 1991) are shown by Rodriguez-Goldman (1995) to exhibit considerable bias. Applied to the Television, School and Family Project (TVSFP) smoking cessation study (1,600 students, 135 classrooms, 28 schools): three-level model significantly outperforms two-level alternatives when data have both longitudinal and clustering structure simultaneously.

Key Claims

Linearization bias: Longford (1988, 1994) and Goldstein (1991) first/second-order Taylor-series expansions provide little advantage over ordinary logistic regression ignoring hierarchical structure (Rodriguez-Goldman 1995).
Conditional independence trick: given cluster-level effect $\theta_{(3)}$ , the $n_i$ subjects within cluster $i$ are independent, so the marginal probability factorizes: $h(\mathbf{y}_i) = \int_{\theta_{(3)}} \prod_{j=1}^{n_i} \left[ \int_{\boldsymbol{\theta}_{(2)}} \prod_{k=1}^{n_{ij}} [\Phi(z_{ijk})]^{1-y_{ijk}} [1-\Phi(z_{ijk})]^{y_{ijk}} g(\boldsymbol{\theta}_{(2)}) d\boldsymbol{\theta}_{(2)} \right] g(\theta_{(3)}) d\theta_{(3)}$ Integration dimensionality collapses from $(n_i \times r)+1$ to $r+1$ .
Estimation: Cholesky-orthogonalize $\Sigma_{\beta^*}$ so standardized random effects $\boldsymbol{\theta}^*$ have identity covariance; Fisher scoring for $\boldsymbol{\alpha}$ and unique Cholesky elements $\mathbf{T}$ ; Gauss-Hermite quadrature on standardized $\boldsymbol{\theta}$ space.
Logistic extension: trivially replace $\Phi(z_{ijk})$ with $\Psi(z_{ijk}) = 1/(1+\exp[-z_{ijk}])$ ; residual variance $\pi^2/3$ for standard logistic.
Probit vs. logistic: virtually identical conclusions on significance; logistic fixed-effect marginal maximum likelihood estimates (MMLEs) ~40% larger (scale difference); random-effect standard deviations (SDs) similar; fit virtually identical.
Prior robustness: normal vs. rectangular prior for random effects leaves fixed-effect estimates and fit unchanged; random-effect SDs scale differently (~2/3 reduction for rectangular) but conclusions are unaffected.
TVSFP Analysis 1 (longitudinal + classroom clustering, $N=1{,}600$ ): Class SD = 0.310, Student SD = 0.605; intraclass correlation (ICC) classroom = 6.6%, ICC student = 25.0%; three-level model significantly better than two-level classroom model ( $\chi^2_1=40.40$ , $p<.0001$ ) and two-level student model ( $\chi^2_1=31.95$ , $p<.0001$ ); classroom-curriculum (CC) effect strong and significant; CC×television-media (TV) interaction reaches marginal significance only in the three-level model.
TVSFP Analysis 2 (cross-sectional, classroom within school): three-level not significantly better than two-level classroom model ( $\chi^2_1=1.82$ , $p=\text{n.s.}$ ); ICC school only 2.6%.
Scalability: Gauss-Hermite quadrature intractable beyond ~5–6 total random effects; Gibbs sampling (Geman-Geman 1984) recommended for larger models with no practical limit.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"Recent studies indicate that these approximate solutions exhibit considerable bias and provide little advantage over use of traditional logistic regression analysis ignoring the hierarchical structure."

"Note, however, that conditional on the cluster-effect $\theta_{(3)}$ , the responses from the $n_i$ subjects in cluster $i$ are independent."

My Take

The conditional independence factorization is the pivotal idea — it converts an intractable $(n_i r+1)$ -dimensional integral into a nested $(r+1)$ -dimensional one without approximation. This is the same structural insight that makes factor-based multivariate models tractable (Gibbons-Wilcox-Gok 1998, Chib-Nardari-Shephard 2006). The dual application (Analysis 1 yes, Analysis 2 no) is a useful demonstration that the three-level model is not always necessary: when school-level ICC is small (2.6%), the added complexity adds nothing. The paper's robustness checks (probit/logistic, normal/rectangular prior) are reassuring but undersell the main contribution.