Multivariate analysis of correlated binary or mixed binary and continuous data, possibly with missing values, can be difficult in practice due to the lack of a natural multivariate distribution. We study the class of Gaussian copula distributions as a viable model in such cases and propose a hybrid approach to likelihood estimation that combines the method of inference functions for margins (IFM) and a parameter-expanded EM (PX-EM) algorithm. A simulation study suggests that the estimates are efficient and reasonably unbiased in finite samples. We illustrate the methodology on mixed data from the St. Louis Risk Research Project that is further complicated by a substantial amount of missing values. This is joint work with Y. Yan (Sun Yat-sen University) and A. R. de Leon (University of Calgary).