This distribution may therefore be suited to accurately reflect the features of the underlying continuous latent variables. Because the means and variances of each subgroup’s normal distribution are allowed to differ, the mixture distribution can be a flexible tool to account for heterogeneous and asymmetric data. That is, a polychoric correlation coefficient can be based on the assumption that the underlying continuous latent variables follow a mixture of two or more normal distributions contingent on the idea that data have been gathered from two or more subpopulations. Although their proposed model was presented as an approach to analyze rating agreement, it could also be applied for modeling mixtures with observed ordinal variables. Uebersax and Grove proposed a latent mixture model in which the distribution of a latent trait measured by an ordinal variable is defined as a combination of two subgroups’ probability density functions. In addition to the skew-normal distribution, the mixture of normal distributions has been considered for the estimation of the polychoric correlation. In line with Roscino and Pollice, their results suggest that assuming an underlying skew-normal distribution generally produces lower bias in the polychoric correlation estimate than assuming an underlying bivariate normal distribution. More recently, Jin and Yang-Wallentin examined the performance of several generalizations in an extensive simulation study, including Azzalini and Dalla Valle’s skew-normal distribution. The polychoric correlation based on this distribution yielded better estimates of the correlation between the underlying continuous latent variables than the original polychoric correlation when the sample size was large, or the number of categories of the ordinal variables was small, or the skewness parameters were discordant. In a study by Roscino and Pollice, a polychoric correlation coefficient was introduced based on the hypothesis that the underlying latent variables follow Azzalini and Dalla Valle’s bivariate skew-normal distribution. ![]() Previous research indeed shows that the polychoric correlation coefficient provides an accurate estimate of the correlation between the two underlying continuous variables when the correct underlying distribution is assumed . Several alternative underlying bivariate distributions have been proposed already, among which are Azzalini and Dalla Valle’s skew-normal distribution and the mixture of normal distributions. When the assumption of underlying bivariate normality does not hold, the polychoric correlation coefficient can be based on other distributional assumptions that represent features of the data more accurately than the bivariate normal distribution. ![]() In accordance with Muthén and Hofacker, Jin and Yang-Wallentin suggested that the estimate of the polychoric correlation should only be used if the underlying bivariate normality assumption holds.
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