Hedeker (2003) A Mixed-Effects Multinomial Logistic Regression Model

Summary

Hedeker (2003) describes a mixed-effects multinomial logistic regression model for clustered or longitudinal data with nominal (unordered categorical) responses. The model adds subject-level random effects (RE) to the standard multinomial logit, with a flexible parameterization allowing any set of C-1 non-redundant category contrasts rather than requiring reference-cell coding. Estimation uses maximum marginal likelihood (MML) via Gauss-Hermite quadrature and Fisher scoring, implemented in the MIXNO software. A longitudinal study of housing status in homeless mentally ill adults (McKinney Homeless Research Project, MHRP; n=361, 4 time points, 3 categories) illustrates the Helmert contrast approach.

Key Claims

• Model: $p_{ijc} \propto \exp(z_{ijc})$, where $z_{ijc} = \mathbf{w}_{ij}'\boldsymbol{\alpha}_c + \mathbf{x}_{ij}'\mathbf{T}_c\boldsymbol{\theta}_i$, $\boldsymbol{\theta}_i \sim N(\mathbf{0}, \mathbf{I})$ (standardized); each category contrast $c$ has its own fixed-effect vector $\boldsymbol{\alpha}_c$ and Cholesky factor $\mathbf{T}_c$ with $\boldsymbol{\Sigma}_c = \mathbf{T}_c\mathbf{T}_c'$.
• General contrasts: Any $(C-1)\times C$ contrast matrix D is permitted: $z_{ijc} = \mathbf{w}_{ij}'\boldsymbol{\Gamma}\mathbf{d}_c + (\mathbf{x}_{ij}' \otimes \boldsymbol{\theta}_i')\mathbf{J}_{r*}\boldsymbol{\Lambda}\mathbf{d}_c$; reference-cell, Helmert, and deviation contrasts are all special cases.
• Quadrature: $Q^r$ Gauss-Hermite points for $r$ random effects; feasible for $r \lesssim 5$; 10 vs. 20 points produced negligible difference in the example; adaptive quadrature (Liu-Pierce 1994; Pinheiro-Bates 1995) or Gibbs sampling preferred for high-dimensional RE.
• Category-specific intraclass correlation (ICC): Under a random-intercepts model the level-1 logistic variance is $\pi^2/3$, giving $r_c = \hat{\sigma}_c^2/(\hat{\sigma}_c^2 + \pi^2/3)$; a separate ICC is estimated for each category contrast (C-1 total), allowing cluster influence to vary across comparisons.
• IIA assumption: Multinomial logit embeds independence of irrelevant alternatives — the log-odds for any two categories is unaffected by other categories; reasonable when categories are distinct; McFadden (1980) notes robustness even when IIA is implausible.
• MHRP results: ICC ≈ 0.44 (non-street vs. street) and 0.39 (independent vs. community); Section 8 certificates significantly increase independent vs. community housing at all follow-ups (all Section 8 × time interactions p < 0.05); normal vs. uniform RE distribution yields virtually identical fixed-effects conclusions.
• Software: MIXNO (Hedeker 1999, Journal of Statistical Software 4(5): 1–92); free download.

Concepts Introduced or Extended

• Mixed-Effects Multinomial Regression
• Generalized Linear Mixed Model
• Multinomial Probit

Entities Mentioned

• Donald Hedeker

Quotes

"The use of Gibbs sampling and related methods provides an alternative way of handling the integration over the random-effect distribution. While the quadrature solution is relatively fast and computationally tractable for models with few random effects, Gibbs sampling is more advantageous for models with many random effects."

"The model makes an assumption that has been referred to as 'independence of irrelevant alternatives' in the econometric literature. This is because the effect of an explanatory variable comparing two categories is the same regardless of the total number of categories considered."

My Take

The flexible contrast-coding apparatus is the paper's most practically useful contribution — Helmert contrasts encode theoretical structure (off the streets vs. independent vs. community housing) directly into the model, avoiding post-hoc recoding. The MML-quadrature estimator is computationally efficient for small r but scales as $Q^r$; for modern high-dimensional RE problems, lme4 or Stan is preferred. The IIA assumption is the main theoretical limitation relative to Multinomial Probit, which allows correlated utilities via a full covariance matrix; but IIA is often reasonable when categories are substantively distinct. The paper is thin on random-slope models despite the framework allowing them.