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 1.0000
Notice 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 0.1194
The 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