On the Relationship Between Factor Loadings and Component Loadings When Latent Traits and Specificities are Treated as Latent Factors

Most existing studies on the relationship between factor analysis (FA) and principal component analysis (PCA) focus on approximating the common factors by the first few components via the closeness between their loadings. Based on a setup in Bentler and de Leeuw (Psychometrika 76:461–470, 2011), this study examines the relationship between FA loadings and PCA loadings when specificities are treated as latent factors. In particular, we will examine the closeness between the two types of loadings when the number of observed variables (p) increases. Parallel to the development in Schneeweiss (Multivar Behav Res 32:375–401, 1997), an average squared canonical correlation (ASCC) is used as the criterion for measuring the closeness. We show that the ASCC can be partitioned into two parts, the first of which is a function of FA loadings and the inverse correlation matrix, and the second of which is a function of unique variances and the inverse correlation matrix of the observed variables. We examine the behavior of these two parts as p approaches infinity. The study gives a different perspective on the relationship between PCA and FA, and the results add additional insights on the selection of the two types of methods in the analysis of high dimensional data.