A generalization of the polychoric correlation coefficient

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 la- tent bivariate normality is empirically or theoretically implausible, other dis- tributional assumptions can be made. In this paper a new and more °exible polychoric correlation coe±cient is proposed assuming that the underlying variables are skew-normally 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 reg- ulate the skewness. As for the original polychoric correlation coe±cient, the new coe±cient was estimated by the maximization of the log-likelihood func- tion with respect to the thresholds of the continuous variables, the skewness and the correlation parameters. The new coe±cient was then tested on sam- ples from simulated populations di®ering in the number of ordinal categories and the distribution of the underlying variables. The results were compared with those of the original polychoric correlation coe±cient.