Bayesian Gaussian Copula Factor Models for Mixed Data

Abstract

Gaussian factor models have proven widely useful for parsimoniously characterizing dependence in multivariate data. There is rich literature on their extension to mixed categorical and continuous variables, using latent Gaussian variables or through generalized latent trait models accommodating measurements in the exponential family. However, when generalizing to non-Gaussian measured variables, the latent variables typically influence both the dependence structure and the form of the marginal distributions, complicating interpretation and introducing artifacts. To address this problem, we propose a novel class of Bayesian Gaussian copula factor models that decouple the latent factors from the marginal distributions. A semiparametric specification for the marginals based on the extended rank likelihood yields straightforward implementation and substantial computational gains. We provide new theoretical and empirical justifications for using this likelihood in Bayesian inference. We propose new default priors for the factor loadings and develop efficient parameter-expanded Gibbs sampling for posterior computation. The methods are evaluated through simulations and applied to a dataset in political science. The models in this article are implemented in the R package bfa (available from http://stat.duke.edu/jsm38/software/bfa). Supplementary materials for this article are available online.

KEY WORDS:

• Extended rank likelihood
• Factor analysis
• High dimensional
• Latent variables
• Parameter expansion
• Semiparametric

Acknowledgments

We acknowledge the support of the Measurement to Understand Re-Classification of Disease of Cabarrus and Kannapolis (MURDOCK) Study and the NIH (National Institutes of Health) CTSA (Clinical and Translational Science Award) 1UL1RR024128-01, and NIH grant R01 ES017436, without whom this research would not have been possible. The first author would also like to thank Richard Hahn, Jerry Reiter, and Scott Schmidler for helpful discussion.