Zhang-Boscardin-Belin (2008) Bayesian Analysis of Multivariate Nominal Measures Using Multivariate Multinomial Probit Models

Summary

Proposes a Bayesian Markov chain Monte Carlo (MCMC) algorithm for the multivariate multinomial probit (MVMNP) model — an extension of the standard multinomial probit to handle $g$ correlated nominal outcomes per subject. The key computational innovation is the parameter-extended Metropolis-Hastings (PX-MH) algorithm, which samples the restricted covariance matrix (with $g$ diagonal elements fixed to 1 for identification) by expanding the constraint into free working parameters, proposing from a Wishart distribution, then collapsing back. Handles missing outcomes naturally via truncated vs. unconstrained normals in the Gibbs steps for latent utilities.

Key Claims

• MNP identification. For a single nominal outcome with $p$ levels, differencing against the reference level gives latent contrasts $z_i = (u_{i1}-u_{ip},\ldots,u_{i(p-1)}-u_{ip}) \sim N(X_i\beta, \Sigma)$ with the first diagonal $\sigma_{11}=1$ fixed for scale identification. Outcome determined by $d_i = \arg\max_j u_{ij}$.
• MVMNP model. For $g$ nominal measures with levels $p_1,\ldots,p_g$, stack the utility contrasts $z_i^T = (z_{i1},\ldots,z_{ig})$ into a joint model $z_i \sim N(X_i\beta, \Sigma)$ where $\Sigma$ has $g$ diagonal elements fixed to 1 (one per measure). Special case: when $X_i = I$ for all $i$, the model reduces to a contingency table correlation model generalizing the log-linear model with all two-way interactions.
• PX-MH algorithm. Direct sampling from $p(\Sigma|\beta,Z,d)$ with the diagonal constraints is intractable. Solution: replace the $g$ fixed-1 diagonal entries with free working parameters $(v_1, v_{p_1}, \ldots)$ to form an unrestricted covariance $W = D\Sigma D$. Propose $W^* \sim \text{Wishart}(m, W^{(t)})$, extract $\Sigma^* = D^{*-1}W^*D^{*-1}$, accept via MH ratio under the PXW prior $p(R,D)$. ~15% acceptance rate in the breast-cancer application.
• One MCMC cycle. (1) $\beta|\Sigma, Z, d$: closed-form normal; (2) each $z_{iqj}|\beta,\Sigma,d_{iq},z_{iq(-j)},z_{i(-q)}$: truncated normal when observed, unconstrained normal when outcome is missing; (3) $\Sigma|\beta,Z,d$: PX-MH step.
• Prior. $\beta \sim N(b, C)$ (default $N(0,100)$); $\Sigma \sim \text{PXW}(m_0, \Lambda)$ (parameter-expanded Wishart family). PXW_ID_Weak ($m_0 = p+1$, $\Lambda = I$) vs. PXW_CS_Strong ($m_0=100$, $\Lambda=\text{CS}(0.4)$, a compound-symmetry (CS) structure) priors compared in simulation.
• Simulated example ($n=2000$, $g=1$ measure, 5 latent dimensions). Both prior scenarios recover regression and correlation parameters; some sensitivity for variance parameters; coverage satisfactory. Posterior under PXW_CS_Strong is tighter (smaller posterior standard deviations (SDs)) but still consistent.
• Cancer-control application ($n=1671$, $g=6$ nominal measures, up to 63.3% missing on patient language). MVMNP estimated on a $10\times10$ latent correlation matrix. Key finding: Spanish language and Hispanic ethnicity highly correlated ($r \approx 0.85$). Negative correlations between non-homemaker working situation and age/referring clinic type. Converged in ~400k iterations after 1k burn-in.
• Comparisons. Univariate results comparable to McCulloch-Rossi (1994) and Imai-van Dyk (2005) IvD algorithm. MVMNP generalises univariate MNP (special case $g=1$) and multiperiod MNP (which allows only AR(1) covariance structures); IvD algorithm is faster for univariate MNP but less flexible in prior specification.

Concepts Introduced or Extended

• Multivariate Probit
• Multinomial Probit

Entities Mentioned

• (none with existing wiki pages)

Quotes

"Our key statistical computing innovation is to use the parameter-extended Metropolis-Hastings (PX-MH) algorithm to sample the covariance matrix with restrictions on the diagonal elements."

"In the special case of contingency table data, the MVMNP model provides additional flexibility over a two-way log-linear model, but is more parsimonious than a saturated model."

My Take

A solid applied methods paper filling a genuine gap: the literature on multinomial probit had been almost entirely univariate, and the restriction to AR(1) covariance structures in multiperiod extensions was arbitrary. The PX-MH algorithm is an elegant solution to the constrained-covariance sampling problem, directly building on the parameter-expansion framework of van Dyk-Meng (2001) and the earlier correlation-matrix PX-MH of Zhang-Boscardin-Belin (2006). The application to breast-cancer early detection demonstrates the missing-data handling strength. The main limitation is convergence speed — the complex MH step induces high autocorrelation in high-dimensional settings, so the algorithm needs many more iterations than a purely Gibbs-based sampler for the unconstrained version. Not a core time-series reference but relevant to the multivariate discrete-choice and multivariate probit cluster.