Edwards and Allenby (2003) Multivariate Analysis of Multiple Response Data

Summary

Edwards and Allenby propose a Bayesian Markov chain Monte Carlo (MCMC) estimator for the multivariate binomial probit model applied to "pick any/J" survey data — questions where respondents check all that apply from J items. The key contribution is a general strategy for handling identification restrictions: rather than directly sampling the constrained correlation matrix $R$ , they fit an unrestricted covariance matrix $\Omega$ and postprocess each MCMC draw by standardizing ( $C = \text{diag}(\Omega)^{-1/2}$ , then $\alpha = C\gamma$ and $R = C\Omega C'$ ). This guarantees a positive-definite $R$ with lower mean squared error (MSE) than tetrachoric correlations in small samples. Three marketing applications demonstrate the approach: market-structure principal component analysis (PCA) on scotch whiskey brand usage ( $J=21$ , $n=2218$ ), cross-selling scoring via conditional normals on financial products ( $J=39$ , $n=417$ ), and membership attrition canonical correlation on an American Marketing Association (AMA) open-ended telephone survey ( $n=120$ ).

Key Claims

The identification restriction in the multivariate binomial probit (unit-variance latent responses ⟹ $R$ is a correlation matrix) can be handled by fitting an unconstrained covariance $\Omega$ under an inverted-Wishart prior and postprocessing each draw. This is valid because a proper prior guarantees a proper posterior even when the likelihood is non-identified.
The Bayesian estimator always produces positive-definite (PD) $R$ (by construction via the prior); tetrachoric correlations can yield non-PD matrices in small samples and are completely undefined when any cell count is zero.
Bayes has lower MSE than tetrachoric across nearly all sample sizes and off-diagonal correlation levels (Tables 1–3); bias is comparable and shrinks toward zero at modest n.
Financial product ownership data requires 11 of 39 eigenvalues > 1, ruling out any low-dimensional factor model — a finding inaccessible to existing methods limited to dim ≤ 4.
The postprocessing strategy is general: for any non-identified likelihood with a proper prior, information concentrates on the identified subspace of the posterior and can be extracted without deriving the transformation analytically in advance. The draws of $\beta_1$ and $\beta_2$ in $y = \beta_1\beta_2 x + \varepsilon$ wander indefinitely, but $\beta_1\beta_2$ converges normally.

Concepts Introduced or Extended

Multivariate Probit — new high-dimensional MCMC algorithm via postprocessed unconstrained covariance draws; enables J > 50
Data Augmentation — univariate sequential truncated-normal augmentation for multivariate binomial latent responses
Gibbs Sampler — three-block Gibbs on $(y_k, \Omega, \gamma)$ followed by $C$ -transform postprocessing

Entities Mentioned

Quotes

"A Bayesian analysis equivalently treats the distinction between theoretical nonidentification...and empirical nonidentification (e.g., caused by data collinearity) because the analysis is conditional on both the model and the data."

"The proposed methodology is especially useful in conducting exploratory analyses of data for which well-formulated hypotheses are not available and for which the problem dimension is large."

My Take

The postprocessing strategy is genuinely elegant and general — the paper correctly notes it extends far beyond the binomial probit context to any model with identification restrictions exploitable through postprocessing. The key limitation is slow convergence when the restriction is unknown (chains wander over the non-identified directions), and the two-step nature means standard errors on derived quantities like eigenvectors are not available without additional delta-method work. The paper wisely positions this as an exploratory tool; when a low-dimensional factor structure is known in advance, the one-step Hahn-Carvalho-Scott sparse factor probit is preferable.