Regression and Dimension Reduction for Multivariate Mixed-Type Data via Semiparametric Gaussian Copula

Abstract

Clinical and epidemiological studies encode participant information in multivariate vectors with mixed type variables on continuous, truncated, ordinal, and binary scales. Semiparametric Gaussian Copula (SGC) assumes that observed data is generated by latent multivariate normal random variables which marginals are monotonically transformed and then truncated/ordinalized/binarized. In SGC, the latent correlation matrix fully determines the dependence structure and it is estimated through an inversion of ``bridges'' between Kendall's Tau rank correlations of observed variables and latent correlations. By employing SGC, we develop regression (SGC-Reg), principal component analysis (SGC-PCA), and principal component regression (SGC-PCR) for latent representations of observed data. To build our framework, we make several key contributions: i) establishing novel bridging results for general ordinal type variables, ii) developing regression estimation on the latent space and deriving asymptotic normality of estimators, iii) developing a computationally efficient algorithm that reduces calculation complexity of all steps including calculation of asymptotic covariance matrix from $O(n^4)$ to $O(n\log n)$, iv) developing methods to predict latent representations of observed data and perform imputation of missing data, and v) developing principal component analysis and principal component regression on the latent space. We apply our framework to study the association between a 5-year mortality and 61 frailty-related measures composed of 29 continuous, 17 ordinal, and 15 binary variables in 9478 participants of 1999-2010 waves of National Health and Nutrition Examination Survey (NHANES).