Parallel Analysis With Categorical Variables: Impact of Category Probability Proportions on Dimensionality Assessment Accuracy

Parallel analysis (PA) is regarded as one of the most accurate methods to determine the number of factors underlying a set of variables. Commonly, PA is performed on the basis of the variables' product-moment correlation matrix. To improve dimensionality assessments for dichotomous or ordered categorical variables, it has been proposed to replace product-moment correlations with more appropriate coefficients, such as tetrachoric or polychoric correlations. While similar modifications have proven useful for various factor analytic approaches, PA results were not consistently improved. The present article outlines a main reason for this result. Specifically, it explains the dependency of PA results on differing proportions of category probabilities when using tetrachoric or polychoric correlations and shows how to adjust for it by generating appropriate reference eigenvalues. The accuracy of dimensionality assessments of PA accounting for category probability proportions versus not accounting for them is investigated using simulation studies. The results show that the category probability adjusted approach distinctly improves dimensionality assessments. Translational Abstract Assessing the number of latent variables that sufficiently account for the joint variability among a set of observed variables has been an important research topic since Spearman (1904) proposed his factor analytic approach for measuring general intelligence. As a consequence, a variety of dimensionality assessment approaches have been proposed. Parallel analysis (PA) is considered as one of the most accurate methods. It compares the eigenvalues of a sample correlation matrix to the expected eigenvalues of independent variables. In the classical approach introduced by Horn (1965) all variables are assumed as metric. However, because categorical variables are prominent in psychology, it has been proposed to replace PA on the basis of product-moment correlations with PA based on tetrachoric or polychoric correlations, which are designed to account for the variables' ordinal level of measurement. Although this modification is theoretically reasonable, PA results were not consistently improved. The present article outlines a main reason for this outcome. Specifically, it explains the dependency of PA results on the distribution of category probabilities on the basis of the theoretical properties of the different correlation coefficients. Furthermore, it demonstrates which PA approach is most robust to unbalanced category probabilities using simulation studies. As the results show, PA based on tetrachoric/polychoric correlations is superior if, and only if, reference eigenvalues are generated based on variables matching the univariate category probabilities of the empirical sample. Program code for generating appropriate eigenvalues is provided.