Ellipsoidally symmetric extensions of the general location model for mixed categorical and continuous data

The general location model (Olkin Krzanowski, 1980, 1982; Little & Schluchter, 1985) has categorical variables marginally distributed as a multinomial and continuous variables conditionally normally distributed with different means across cells defined by the categorical variables but a common covariance matrix across cells. Two extensions of the general location model are obtained. The first replaces the common covariance matrix with different but proportional covariance matrices, where the proportionality constants are to be estimated. The second replaces the multivariate normal distributions of the first extension with multivariate t distributions, where the degrees of freedom can also vary across cells and are to be estimated. The t distribution is just one example of more general ellipsoidally symmetric distributions that can be used in place of the normal. These extensions can provide more accurate fits to real data and can be viewed as tools for robust inference. Moreover, the models can be very useful for multiple imputation of ignorable missing values. Maximum likelihood estimation using the AECM algorithm (Meng & van Dyk, 1997) is presented, as is a monotone-data Gibbs sampling scheme for drawing parameters and missing values from their posterior distributions. To illustrate the techniques, a numerical example is presented.