In canonical correlation analysis, a variate is a weighted sum of linear variables in the analysis. There can be many linear combinations of variables in the data set to form several variates, but canonical correlation analysis maximizes the first two covariates since they are the most used in the analysis. For instance, for variables Y and X, canonical correlation analysis will develop two variates CVX1 = a1x1 + … + anxn and CVY1 = b1y1 + b2y2 + … + bmym. In this case, the canonical weights (a1…an and b1…bn) were selected so that they can be used to maximize the correlation between the two variates. The maximization is continued for residuals of the preceding variates to obtain the next until the limit is reached (n, m). Important to note is that canonical variants cannot be used as factors since only the 1st pair is maximized and the rest of the pairs extracted from their residuals. Finally, calculated variates are automatically orthogonal (independent) (Garson, 2015).

            Other than considering the variates correlation in a canonical analysis, other parts can also be interpreted for more information. First, the test of significance for all correlations can be interpreted. Usually, Wilk’s Lambda, Pillai’s trace or Roy’s largest root can be used to check the statistical significance of the coefficients. At α=0.05 if their p<0.05, then the coefficients are statistically significant. However, Wilk’s Lambda is the most commonly used test for statistical significance of the correlation coefficients. Second, depending on the analysis software used, the correlation between test scores and canonical variables can be established. This art can also be used to in place of the usual canonical correlation as well (Wilks, 2013).

References

Garson, D. (2015). GLM Multivariate, MANOVA, and Canonical Correlation. Asheboro, NC: Statistical Associates Publishers.

Wilks, D. (2013). Probabilistic canonical correlation analysis forecasts, with application to tropical Pacific sea-surface temperatures. International Journal of Climatology, 34(5), 1405-1413. 

          In this respect, we also want to examine communality, adequacy, and redundancy. Each provide us with information about the goodness of predictors in variates, the relationship between variates and predictors, as well as the ability of variates to explain outcome variables. Given what we know of each concept, what would you look for more specifically when examining for redundancy?

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