13.5 - Obtain the Canonical Coefficients
13.5 - Obtain the Canonical CoefficientsPage 2 of the SAS output provides the estimated canonical coefficients \(\left(a_{ij}\right)\) for the sales variables:
Canonical Correlation Analysis
Raw Canonical Coefficients for the Sales Variables
| \(\bf{U}_1\) sales1 | sales2 | sales3 | |
|---|---|---|---|
| growth | 0.0623778783 | -0.174070306 | -0.377152934 |
| profit | 0.020925642 | 0.2421640883 | 0.1035150082 |
| net | 0.0782581746 | -0.23829403 | 0.3834150736 |
Using the coefficient values in the first column, the first canonical variable for sales is determined using the following formula:
\(U_1 = 0.0624X_{growth}+0.0209X_{profit}+0.0783X_{new}\)
Likewise, the estimated canonical coefficients \(\left(b_{ij}\right)\) for the test scores are located in the next table in the SAS output:
Raw Canonical Coefficients for the Test Scores
| \(\bf{V}_1\) scores1 | scores2 | scores3 | |
|---|---|---|---|
| create | 0.0697481411 | -0.192391323 | 0.2465565859 |
| mech | 0.0307382997 | 0.201574382 | -0.141895279 |
| abs | 0.0895641768 | -0.495763258 | -0.280224053 |
| math | 0.0628299739 | 0.0683160677 | 0.0113325936 |
Using the coefficient values in the first column, the first canonical variable for test scores is determined using a similar formula:
\(V_1 = 0.0697Y_{create}+0.0307Y_{mech}+0.0896Y_{abstract}+0.0628Y_{math}\)
In both cases, the magnitudes of the coefficients give the contributions of the individual variables to the corresponding canonical variable. However, just like in principal components analysis, these magnitudes also depend on the variances of the corresponding variables. Unlike principal components analysis, however, standardizing the data has no impact on the canonical correlations.