  Many clinical and epidemiological studies encode collected participant-level
information via a collection of continuous, truncated, ordinal, and binary
variables. To gain novel insights in understanding complex interactions between
collected variables, there is a critical need for the development of flexible
frameworks for joint modeling of mixed data types variables. We propose
Semiparametric Gaussian Copula Regression modeling (SGCRM) that allows to model
a joint dependence structure between observed continuous, truncated, ordinal,
and binary variables and to construct conditional models with these four data
types as outcomes with a guarantee that derived conditional models are mutually
consistent. Semiparametric Gaussian Copula (SGC) mechanism assumes that
observed SGC variables are generated by - i) monotonically transforming
marginals of latent multivariate normal random variable and ii)
dichotimizing/truncating these transformed marginals. SGCRM estimates the
correlation matrix of the latent normal variables through an inversion of
"bridges" between Kendall's Tau rank correlations of observed mixed data type
variables and latent Gaussian correlations. We derive a novel bridging result
to deal with a general ordinal variable. In addition to the previously
established asymptotic consistency, we establish asymptotic normality of the
latent correlation estimators. We also establish the asymptotic normality of
SGCRM regression estimators and provide a computationally efficient way to
calculate asymptotic covariances. We propose computationally efficient methods
to predict SGC latent variables and to do imputation of missing data. Using
National Health and Nutrition Examination Survey (NHANES), we illustrate SGCRM
and compare it with the traditional conditional regression models including
truncated Gaussian regression, ordinal probit, and probit models.
