Simultaneous canonical correlation analysis with invariant canonical loadings

Abstract

In canonical correlation analysis (CCA), the substantive interpretations of the canonical variates are of primary interest to the applied researchers. However, there are two different interpretive approaches used by different researchers—the weight-based approach and the loading-based approach, of which the latter is favored by the majority of researchers in practice. For those who choose the loading-based approach and apply CCA simultaneously to multiple samples, they may wish to test the invariance of the canonical loadings. In this paper, three covariance structure models are defined for CCA. In particular, the first model (i.e., the CCA-W model) corresponds directly with regular CCA, including the canonical correlations and canonical weights as the parameters, while the third model (i.e., the CCA-L model) is in alignment with the loading-based interpretive approach, including the canonical correlations and canonical loadings as the parameters. The CCA-L model is further extended to the unrestricted and restricted SCCA-L models, of which the latter allows one to test the invariance of the canonical loadings. A real example drawn from the sociological literature is provided to illustrate the restrictive SCCA-L model, and some strategies to calculate good starting values for the restrictive SCCA-L model are discussed.