Conduct and Interpret a Canonical Correlation

What is Canonical Correlation analysis?

The Canonical Correlation is a multivariate analysis of correlation.  Canonical is the statistical term for analyzing latent variables (which are not directly observed) that represent multiple variables (which are directly observed).  The term can also be found in canonical regression analysis and in multivariate discriminant analysis.

Canonical Correlation analysis is the analysis of multiple-X multiple-Y correlation.  The Canonical Correlation Coefficient measures the strength of association between two Canonical Variates.

A Canonical Variate is the weighted sum of the variables in the analysis.  The canonical variate is denoted CV.  Similarly to the discussions on why to use factor analysis instead of creating unweighted indices as independent variables in regression analysis, canonical correlation analysis is preferable in analyzing the strength of association between two constructs.  This is such because it creates an internal structure, for example, a different importance of single item scores that make up the overall score (as found in satisfaction measurements and aptitude testing).

For multiple x and y the canonical correlation analysis constructs two variates CVX1 = a1x1 + a2x2 + a3x3 + … + anxnand CVY1 = b1y1 + b2y2 + b3y3 + … + bmym.  The canonical weights a1…an and b1…bn are chosen so that they maximize the correlation between the canonical variates CVX1 and CVY1.  A pair of canonical variates is called a canonical root. This step is repeated for the residuals to generate additional duplets of canonical variates until the cut-off value = min(n,m) is reached; for example, if we calculate the canonical correlation between three variables for test scores and five variables for aptitude testing, we would extract three pairs of canonical variates or three canonical roots.  Note that this is a major difference from factor analysis.  In factor analysis the factors are calculated to maximize between-group variance while minimizing in-group variance.  They are factors because they group the underlying variables.

Canonical Variants are not factors because only the first pair of canonical variants groups the variables in such way that the correlation between them is maximized.  The second pair is constructed out of the residuals of the first pair in order to maximize correlation between them.  Therefore the canonical variants cannot be interpreted in the same way as factors in factor analysis.  Also the calculated canonical variates are automatically orthogonal, i.e., they are independent from each other.

Similar to factor analysis, the central results of canonical correlation analysis are the canonical correlations, the canonical factor loadings, and the canonical weights.  They can also be used to calculate d, the measure of redundancy.  The redundancy measurement is important in questionnaire design and scale development.  It can answer questions such as, “When I measure a five item satisfaction with the last purchase and a three item satisfaction with the after sales support, can I exclude one of the two scales for the sake of shortening my questionnaire?” Statistically it represents the proportion of variance of one set of variables explained by the variant of the other set of variables.

The canonical correlation coefficients test for the existence of overall relationships between two sets of variables, and redundancy measures the magnitude of relationships.  Lastly Wilk’s lambda (also called U value) and Bartlett’s V are used as a Test of Significance of the canonical correlation coefficient.  Typically Wilk’s lambda is used to test the significance of the first canonical correlation coefficient and Bartlett’s V is used to test the significance of all canonical correlation coefficients.

Hello Samuel,

In addition to Ajit's comprehensive answer, I think you'll find this resource to be a very useful one:

Sherry, A., & Henson, R. K. (2005). Conducting and interpreting canonical correlation analysis in personality research: A user-friendly primer. Journal of Personality Assessment, 84, 27-48. doi:10.1207/s15327752jpa8401_09

Good luck with your work!

Vid-supported doc, maybe.

Editorial volume provided by MultReg text (1990s, 2003).

Dear Dr. David Morse and Dr. Matthew Kerry

Thanks a lot for your support all. I am moving in to it, thanks.

Thanks Professor Ajit kumar Roy for the Brief Explanation. I re-read it, and it helps me a lot.