Partial Identification of Latent Correlations with Ordinal Data

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.

Factor analyzing ordinal items requires substantive knowledge of response marginals.

“…Otherwise the underlying correlation matrix is not identified from the ordinal data. For example, if the response marginals are known, but the response copula is unknown, we may calculate the set of possible correlation values an underlying vector may have and still be able to generate the ordinal dataset (Grønneberg et al, 2020; Moss & Grønneberg, 2022). These sets are unfortunately always large, and this is in contrast to the single value required to use cat-LS.…”

“…It is crucial to observe that the underlying correlation matrix is not identified without making strong distributional assumptions concerning X * (Grønneberg et al, 2020; Moss & Grønneberg, 2022). Traditionally, it has been assumed that the response variables are bivariate normally distributed, which yields the polychoric correlation, estimated by ML estimation (Olsson, 1979a).…”

Translation, Cross-Cultural Adaptation to Malay, and psychometric evaluation of the AIM-IAM-FIM questionnaire: Measuring the implementation outcome of a community-based intervention programme

“…Parallel analysis was used to establish the dimensions, and principal component analysis combined with raw varimax rotation was utilised by Factor software to extract the variables' components [27]. The polychoric correlation matrix model was used since the questionnaire used ordinal data [28,29]. Subsequently, the sample underwent both Bartlett's and the Keiser-Meyer-Olkin (KMO) tests to establish whether it was sufficient and appropriate.…”