A Generalization of the Polychoric Correlation Coefficient

Abstract

The polychoric correlation coefficient is a measure of association between two ordinal variables. It is based on the assumption that two latent bivariate normally distributed random variables generate couples of ordinal scores. Categories of the two ordinal variables correspond to intervals of the corresponding continuous variables. Thus, measuring the association between ordinal variables means estimating the product moment correlation between the underlying normal variables (Olsonn. 1979). When the hypothesis of latent bivariate normality is empirically or theoretically implausible, other distributional assumptions can be made. In this paper a new and more flexible polychoric correlation coefficient is proposed assuming that the underlying variables are skewnormally distributed (Roscino. 2005). The skew normal (Azzalini and Dalla Valle. 1996) is a family of distributions which includes the normal distribution as a special case, but with an extra parameter to regulate the skewness. As for the original polychoric correlation coefficient, the new coefficient was estimated by the maximization of the log-likelihood function with respect to the thresholds of the continuous variables, the skewness and the correlation parameters. The new coefficient was then tested on samples from simulated populations differing in the number of ordinal categories and the distribution of the underlying variables. The results were compared with those of the original polychoric correlation coefficient.