Multivariate Correlation Models with Mixed Discrete and Continuous Variables

A model which frequently arises from experimentation in psychology is one which contains both discrete and continuous variables. The concern in such a model may be with finding measures of association or with problems of inference on some of the parameters. In the simplest such model there is a discrete variable x which takes the values 0 or 1, and a continuous variable y. Such a random variable x is often used in psychology to denote the presence or absence of an attribute. Point-biserial correlation, which is the ordinary product-moment correlation between x and y, has been used as a measure of association. This model, when x has a binomial distribution, and the conditional distribution of y for fixed x is normal, was studied in some detail by Tate 13. In the present paper, we consider a multivariate extension, in which x = (x₀, x₁, , xₖ) has a multinomial distribution, and the conditional distribution of y = (y₁, , yₚ) for fixed x is multivariate normal.