Abstract：
The polychoric correlation is a popular measure of association for ordinal data. It estimates a latent correlation, i.e., the correlation of a latent vector. This vector is assumed to be bivariate normal, an assumption that cannot always be justified. When bivariate normality does not hold, the polychoric correlation will not necessarily approximate the true latent correlation, even when the observed variables have many categories. We calculate the sets of possible values of the latent correlation when latent bivariate normality is not necessarily true, but at least the latent marginals are known. The resulting sets are called partial identification sets, and are shown to shrink to the true latent correlation as the number of categories increase. Moreover, we investigate partial identification under the additional assumption that the latent copula is symmetric, and calculate the partial identification set when one variable is ordinal and another is continuous. We show that little can be said about latent correlations, unless we have impractically many categories or we know a great deal about the distribution of the latent vector. An open-source R package is available for applying our results.
摘要：
多脉络相关性是序数数据的常用关联度量。它估计了潜在的相关性，即，潜在向量的相关性。假设这个向量是双变量正常的，一个不可能总是合理的假设。当双变量正态不成立时，多脉络相关性不一定接近真正的潜在相关性，即使观察到的变量有很多类别。当潜在双变量正态不一定为真时，我们计算潜在相关性的可能值集合，但至少潜在的边缘是已知的。得到的集合称为部分识别集，并显示随着类别数量的增加而缩小到真正的潜在相关性。此外，我们在潜在系谱是对称的附加假设下研究部分识别，并计算当一个变量为序数而另一个变量为连续时的部分识别集。我们表明，关于潜在相关性几乎没有什么可说的，除非我们有很多类别，或者我们对潜在向量的分布了解很多。一个开源R包可用于应用我们的结果。
