**Canonical Correlations example**

The data for this example are in Table 9.12 in Johnson and Wichern. They consist of 3 measurements on the sales performance of 50 salespeople for a large firm and 4 test scores.

The data begin:

I used SAS to compute the canonical correlation analysis between the first 3 variables and the remaining 4. Here is the SAS code.

data sales; infile "T9-12.DAT"; input growth profit new create mech abst math; proc cancorr; var growth profit new; with create mech abst math; run;And here is the output.

The SAS System 1 17:49 Monday, March 9, 1998 Canonical Correlation Analysis Adjusted Approx Squared Canonical Canonical Standard Canonical Correlation Correlation Error Correlation 1 0.994483 0.994021 0.001572 0.988996 2 0.878107 0.872097 0.032704 0.771071 3 0.383606 0.366795 0.121835 0.147153 Eigenvalues of INV(E)*H = CanRsq/(1-CanRsq) Eigenvalue Difference Proportion Cumulative 1 89.8745 86.5063 0.9621 0.9621 2 3.3682 3.1956 0.0361 0.9982 3 0.1725 . 0.0018 1.0000Notice the first canonical correlates account for most of the correlation between the two sets of variables.

The SAS System 2 17:49 Monday, March 9, 1998 Canonical Correlation Analysis Test of H0: The canonical correlations in the current row and all that follow are zero Likelihood Ratio Approx F Num DF Den DF Pr > F 1 0.00214847 87.3915 12 114.0588 0.0001 2 0.19524127 18.5263 6 88 0.0001 3 0.85284669 3.8822 2 45 0.0278 Multivariate Statistics and F Approximations S=3 M=0 N=20.5 Statistic Value F Num DF Den DF Pr > F Wilks' Lambda 0.00214847 87.392 12 114.0588 0.0001 Pillai's Trace 1.90722020 19.635 12 135 0.0001 Hotelling-Lawley Trace 93.41517506 324.358 12 125 0.0001 Roy's Greatest Root 89.87446318 1011.088 4 45 0.0001 NOTE: F Statistic for Roy's Greatest Root is an upper bound.Clearly all 3 of the canonical correlations are non-zero. The multivariate tests are of .

The SAS System 3 17:49 Monday, March 9, 1998 Canonical Correlation Analysis Raw Canonical Coefficients for the 'VAR' Variables V1 V2 V3 GROWTH 0.0623778783 -0.174070306 -0.377152934 PROFIT 0.020925642 0.2421640883 0.1035150082 NEW 0.0782581746 -0.23829403 0.3834150736 Raw Canonical Coefficients for the 'WITH' Variables W1 W2 W3 CREATE 0.0697481411 -0.192391323 0.2465565859 MECH 0.0307382997 0.201574382 -0.141895279 ABST 0.0895641768 -0.495763258 -0.280224053 MATH 0.0628299739 0.0683160677 0.0113325936 The SAS System 4 17:49 Monday, March 9, 1998 Canonical Correlation Analysis Standardized Canonical Coefficients for the 'VAR' Variables V1 V2 V3 GROWTH 0.4577 -1.2772 -2.7673 PROFIT 0.2119 2.4517 1.0480 NEW 0.3688 -1.1229 1.8067 Standardized Canonical Coefficients for the 'WITH' Variables W1 W2 W3 CREATE 0.2755 -0.7600 0.9739 MECH 0.1040 0.6823 -0.4803 ABST 0.1916 -1.0607 -0.5996 MATH 0.6621 0.7199 0.1194The standardized coefficients are easiest to interpret. For instance a quantity which is not too different from the average of the 3 sales indices is strongly correlated with a weighted average of the psychological test scores which puts most of the weight on Math. The second set of correlates focus on the relation between profitability minus the average of the other two sales indices correlated with (Math + Mech) - (Abstract + Creativity). The last one is not particularly meaningful to me but it is a very small part of the correlation structure between the two sets of variables.

The SAS System 5 17:49 Monday, March 9, 1998 Canonical Structure Correlations Between the 'VAR' Variables and Their Canonical Variables V1 V2 V3 GROWTH 0.9799 0.0006 -0.1996 PROFIT 0.9464 0.3229 0.0075 NEW 0.9519 -0.1863 0.2434 Correlations Between the 'WITH' Variables and Their Canonical Variables W1 W2 W3 CREATE 0.6383 -0.2157 0.6514 MECH 0.7212 0.2376 -0.0677 ABST 0.6472 -0.5013 -0.5742 MATH 0.9441 0.1975 -0.0942 The SAS System 6 17:49 Monday, March 9, 1998 Canonical Structure Correlations Between the 'VAR' Variables and the Canonical Variables of the 'WITH' Variables W1 W2 W3 GROWTH 0.9745 0.0006 -0.0766 PROFIT 0.9412 0.2835 0.0029 NEW 0.9466 -0.1636 0.0934 Correlations Between the 'WITH' Variables and the Canonical Variables of the 'VAR' Variables V1 V2 V3 CREATE 0.6348 -0.1894 0.2499 MECH 0.7172 0.2086 -0.0260 ABST 0.6437 -0.4402 -0.2203 MATH 0.9389 0.1735 -0.0361

Wed Mar 18 08:41:12 PST 1998