Hedeker (2007) Multilevel Models for Ordinal and Nominal Variables

Summary

Chapter 6 of Handbook of Multilevel Analysis (de Leeuw & Meijer eds., Springer, 2007), pp. 239–276. Presents a unified maximum marginal likelihood (MML) framework — Gauss-Hermite quadrature (GHQ) + Fisher scoring, Cholesky-parameterized random-effect covariance — for multilevel models with ordinal responses (proportional odds, partial proportional odds, complementary log-log) and nominal responses (general contrast matrix). Bridges the gap between biostatistics multilevel models and item response theory (IRT), showing that two-parameter logistic (2PL) IRT is the multilevel logistic model with item-specific discrimination parameters. Applications throughout use the McKinney Homeless Research Project (MHRP), a longitudinal housing-assistance study of 361 subjects.

Key Claims

Proportional odds (McCullagh 1980): $\log[P_{ijc}/(1-P_{ijc})] = \gamma_c - [x'\beta + z'T\theta_j]$ , $c=1,\ldots,C-1$ ; a single regression coefficient vector $\beta$ and a single intraclass correlation (ICC) $= \hat\sigma^2/(\hat\sigma^2 + \pi^2/3)$ characterise the full ordinal response — the "threshold concept" of one latent continuum.
Partial proportional odds: subset $x^*_{ij}$ of $h$ covariates with category-varying coefficients $\beta_c$ ; likelihood-ratio (LR) $\chi^2_7=52.14$ rejects proportionality in the MHRP ordinal model; crossing of regression lines implies possible inadmissible cell probabilities for extreme $x^*$ values.
Complementary log-log / discrete-time survival: $\log[-\log(1-P_{ijc})] = \gamma_c + x'\beta + z'T\theta_j$ (eq. 6.6); coefficients are invariant to interval length under proportional hazards (PH) assumption. Ordinal formulation (eq. 6.6) and pooled-dichotomous formulation (eq. 6.7) are equivalent when covariates are time-invariant; the pooled form handles time-varying covariates more naturally.
Nominal model with general contrasts: $\eta_{ijc} = x'\Gamma d_c + (z'\otimes\theta')J'_{r*}\Lambda d_c$ (eq. 6.16); $\mathbf{D}$ is a $(C-1)\times C$ contrast matrix; $\Lambda = [v(T_1)\cdots v(T_{C-1})]$ stacks Cholesky factors; reference-cell and Helmert contrasts are special cases. The "extremal concept" (argmax over $C$ latent tendencies) differs fundamentally from the threshold concept.
Category-specific ICC: $r_c = \hat\sigma^2_c/(\hat\sigma^2_c + \pi^2/3)$ for the nominal model — one per contrast; MHRP gives $r_1=0.19$ , $r_2=0.62$ , with separate-variance model favoured by $\chi^2_1=49.2$ .
IRT connections: Rasch model = multilevel logistic with homogeneous discrimination; 2PL = heterogeneous $T=[a_1\cdots a_m]'$ (item-specific standard deviations, SDs); ordinal items → Samejima difference models; nominal items → Bock (1972) nominal IRT model.
Heterogeneous variance / twin designs: $z_{ij}$ as a $g\times 1$ group-indicator vector yields group-specific SDs; applied to binary data, this framework recovers polychoric/tetrachoric correlations.
Computation: marginal and penalized quasi-likelihood (MQL/PQL) can yield biased estimates; Laplace (as in the Hierarchical Linear Models, HLM, software) is fast and accurate; standard GHQ with $Q^r$ points becomes burdensome for $r>5$ random effects; adaptive quadrature with 3 pts/dim outperforms ordinary GHQ; simulated maximum likelihood estimation (MLE) and Bayesian/Gibbs Markov chain Monte Carlo (MCMC) are alternatives for large $r$ .
MHRP extended results (Helmert contrasts): Section 8 housing certificates increase independent-vs-community housing at all follow-up waves ( $p<0.05$ ); no S8×time effect on non-street vs. street contrast; ordinal ICC=0.39 vs. nominal $r_1=0.19$ , $r_2=0.62$ .

Concepts Introduced or Extended

Entities Mentioned

Quotes

"The multilevel logistic regression model with heterogeneous discrimination parameters across items is the two-parameter logistic (2PL) IRT model." (p. 262, paraphrased)

"The proportional odds model uses a 'threshold concept': a single underlying latent variable is cut by thresholds; the nominal model uses an 'extremal concept': the observed category is the argmax of $C$ latent tendencies." (p. 254, paraphrased)

My Take

An exceptionally clear unified treatment of the MML quadrature approach across ordinal and nominal outcomes, with the IRT bridge making the chapter valuable beyond the multilevel-analysis audience. The partial proportional odds section is honest about the crossing-lines problem that arises when the proportionality assumption is relaxed. The discrete-time survival connection via cloglog is underemphasised in most multilevel textbooks. The computation section is now dated (2007) relative to modern adaptive quadrature implementations, but the conceptual framework remains current.