Revisiting Interpretation of Canonical Correlation Analysis: A Tutorial and Demonstration of Canonical Commonality Analysis

Abstract

In the face of multicollinearity, researchers face challenges interpreting canonical correlation analysis (CCA) results. Although standardized function and structure coefficients provide insight into the canonical variates produced, they fall short when researchers want to fully report canonical effects. This article revisits the interpretation of CCA results, providing a tutorial and demonstrating canonical commonalty analysis. Commonality analysis fully explains the canonical effects produced by using the variables in a given canonical set to partition the variance of canonical variates produced from the other canonical set. Conducting canonical commonality analysis without the aid of software is laborious and may be untenable, depending on the number of noteworthy canonical functions and variables in either canonical set. Commonality analysis software is identified for the canonical correlation case and we demonstrate its use in facilitating model interpretation. Data from Holzinger and Swineford (1939) are employed to test a hypothetical theory that problem-solving skills are predicted by fundamental math ability.

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Notes

¹A FORTRAN IV computer program to accomplish commonality analysis was developed by Morris (1976). However, the program was written for a mainframe computer and is now obsolete.

²Statistical software to replicate the demonstration is provided in Appendices A and B. Appendix A provides R code to invoke canonical correlation (cca; Butts, 2009) and canonical commonality analysis (canonCommonality; Nimon & Roberts, 2009) functions. R is an open-source system for statistical computation and graphics that can be run on Unix, Windows, and MacOS platforms (Hornik, 2009). Instructions for downloading, installing, and running the R platform can be found on the R project home page (http://www.r-project.org/). Appendix B provides SPSS syntax to invoke canonical correlation (CANCORR; SPSS Inc., 2008) and commonality analysis (cc) macros. The SPSS canonical correlation macro (canonical correlation.sps) is part of the SPSS Base system and can be found in the installation directory (SPSS Inc., 2008). The SPSS commonality analysis macro is available from Kim Nimon and at http://profnimon.com/CommonalityCoefficients.sps. Both sets of software automate the steps of conducting a canonical commonality analysis previously described.

³SAS software (SAS Institute Inc., 2008) and SYSTAT (2004) support APS regression, which is one of the steps in conducting canonical commonalty analysis. To yield canonical commonality data, SAS and SYSTAT users need to derive the appropriate canonical commonalty coefficient formulas for each canonical variate they wish to interpret and populate the resulting formulas with the appropriate R ² values from the APS regression.