Analyzing Ordered Categorical Data derived from Elliptically Symmetric Distributions

The polychoric correlation is an ML estimator for the correlation parameter between two latent variables. Each latent variable is only observed as an ordered categorical indicator. This estimator is based on an assumption on the joint distribution for the latent variables which in this case is the bivariate standard normal distribution. We perform a simulation study applying the polychoric correlation based on normality if the true distribution is in fact an elliptically symmetric distribution. The results show that the polychoric correlation is robust in the sense that the true correlation between the latent variables is estimated only with small bias if the true distribution is not too leptokurtic and also not too platykurtic. These results imply that in practical applications the polychoric correlation can be applied obtaining meaningful results even if tests suggest that the assumed normal distribution is not appropriate. Basically the same results are obtained if one latent variable is observed directly and the ML-estimator based on normality (polyserial correlation) is applied.