ELLIPSOIDALLY SYMMETRICAL EXTENSIONS OF THE GENERAL LOCATION MODEL FOR MIXED CATEGORICAL AND CONTINUOUS DATA

The general location model (Olkin & Tate, 1961; Krzanowski, 1980, 1982 ; Little & Schluchter, 1985) has categorical variables marginally dist ributed as a multinomial and continuous variables conditionally normal ly distributed with different means across cells defined by the catego rical variables but a common covariance matrix across cells. Two exten sions of the general location model are obtained. The first replaces t he common covariance matrix with different but proportional covariance matrices, where the proportionality constants are to be estimated. Th e second replaces the multivariate normal distributions of the first e xtension with multivariate t distributions, where the degrees of freed om can also vary across cells and are to be estimated. The t distribut ion is just one example of more general ellipsoidally 'symmetric distr ibutions 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 fo r robust inference. Moreover, the models;can be very useful for multip le imputation of ignorable missing values. Maximum likelihood estimati on using the AECM algorithm (Meng & van Dyk, 1997) is presented, as is a monotone-data Gibbs sampling scheme for drawing parameters and miss ing values from their posterior distributions. To illustrate the techn iques, a numerical example is presented.