Example 26.1 Canonical Correlation Analysis of Fitness Club Data

Three physiological and three exercise variables are measured on 20 middle-aged men in a fitness club. You can use the CANCORR procedure to determine whether the physiological variables are related in any way to the exercise variables. The following statements create the SAS data set Fit and produce Output 26.1.1 through Output 26.1.5:

data Fit;
   input Weight Waist Pulse Chins Situps Jumps;
   datalines;
191  36  50   5  162   60
189  37  52   2  110   60
193  38  58  12  101  101
162  35  62  12  105   37
189  35  46  13  155   58
182  36  56   4  101   42
211  38  56   8  101   38
167  34  60   6  125   40
176  31  74  15  200   40
154  33  56  17  251  250
169  34  50  17  120   38
166  33  52  13  210  115
154  34  64  14  215  105
247  46  50   1   50   50
193  36  46   6   70   31
202  37  62  12  210  120
176  37  54   4   60   25
157  32  52  11  230   80
156  33  54  15  225   73
138  33  68   2  110   43
;

proc cancorr data=Fit all
     vprefix=Physiological vname='Physiological Measurements'
     wprefix=Exercises wname='Exercises';
   var Weight Waist Pulse;
   with Chins Situps Jumps;
   title 'Middle-Aged Men in a Health Fitness Club';
   title2 'Data Courtesy of Dr. A. C. Linnerud, NC State Univ';
run;

Output 26.1.1 displays the correlations among the original variables. The correlations between the physiological and exercise variables are moderate, the largest being between Waist and Situps. There are larger within-set correlations: 0.8702 between Weight and Waist, 0.6957 between Chins and Situps, and 0.6692 between Situps and Jumps.

As Output 26.1.2 shows, the first canonical correlation is 0.7956, which would appear to be substantially larger than any of the between-set correlations. The probability level for the null hypothesis that all the canonical correlations are zero in the population is only 0.0635, so no firm conclusions can be drawn. The remaining canonical correlations are not worthy of consideration, as can be seen from the probability levels and especially from the negative adjusted canonical correlations.

Because the variables are not measured in the same units, the standardized coefficients rather than the raw coefficients should be interpreted. The correlations given in the canonical structure matrices should also be examined.

The first canonical variable for the exercise variables also shows a mixture of signs, subtracting Situps () and Chins () from Jumps (0.7164), with the most weight on Situps. All the correlations are negative, indicating that Jumps is also a suppressor variable.

It might seem contradictory that a variable should have a coefficient of opposite sign from that of its correlation with the canonical variable. In order to understand how this can happen, consider a simplified situation: predicting Situps from Waist and Weight by multiple regression. In informal terms, it seems plausible that obese people should do fewer sit-ups than skinny people. Assume that the men in the sample do not vary much in height, so there is a strong correlation between Waist and Weight (0.8702). Examine the relationships between obesity and the independent variables:

• People with large waists tend to be more obese than people with small waists. Hence, the correlation between Waist and Situps should be negative.

• People with high weights tend to be more obese than people with low weights. Therefore, Weight should correlate negatively with Situps.

• For a fixed value of Weight, people with large waists tend to be shorter and more obese. Thus, the multiple regression coefficient for Waist should be negative.

• For a fixed value of Waist, people with higher weights tend to be taller and skinnier. The multiple regression coefficient for Weight should therefore be positive, of opposite sign from the correlation between Weight and Situps.

Therefore, the general interpretation of the first canonical correlation is that Weight and Jumps act as suppressor variables to enhance the correlation between Waist and Situps. This canonical correlation might be strong enough to be of practical interest, but the sample size is not large enough to draw definite conclusions.

The canonical redundancy analysis (Output 26.1.4) shows that neither of the first pair of canonical variables is a good overall predictor of the opposite set of variables, the proportions of variance explained being 0.2854 and 0.2584. The second and third canonical variables add virtually nothing, with cumulative proportions for all three canonical variables being 0.2969 and 0.2767.

The squared multiple correlations (Output 26.1.5) indicate that the first canonical variable of the physiological measurements has some predictive power for Chins (0.3351) and Situps (0.4233) but almost none for Jumps (0.0167). The first canonical variable of the exercises is a fairly good predictor of Waist (0.5421), a poorer predictor of Weight (0.2438), and nearly useless for predicting Pulse (0.0701).