Multivariate Correlation Models with Mixed Discrete and Continuous Variables

Abstract

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 $x = (x_0, x_1, \cdots, x_k)$ has a multinomial distribution, and the conditional distribution of $y = (y_1, \cdots, y_p)$ for fixed $x$ is multivariate normal.