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

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```

Richard Lockhart
Wed Mar 18 08:41:12 PST 1998