Example
Example 20.1. Canonical Correlation Analysis of Fitness Club Data
Three physiological and three exercise variables are measured on twenty middle-aged meninafitness 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 :
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;
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Middle-Aged Men in a Health Fitness Club Data Courtesy of Dr. A. C. Linnerud, NC State Univ The CANCORR Procedure Correlations Among the Original Variables Correlations Among the Physiological Measurements Weight Waist Pulse Weight 1.0000 0.8702 0.3658 Waist 0.8702 1.0000 0.3529 Pulse 0.3658 0.3529 1.0000 Correlations Among the Exercises Chins Situps Jumps Chins 1.0000 0.6957 0.4958 Situps 0.6957 1.0000 0.6692 Jumps 0.4958 0.6692 1.0000 Correlations Between the Physiological Measurements and the Exercises Chins Situps Jumps Weight 0.3897 0.4931 0.2263 Waist 0.5522 0.6456 0.1915 Pulse 0.1506 0.2250 0.0349
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Output 20.1.1 displays the correlations among the original variables. The correlations between the physiological and exercise variables are moderate, the largest being ˆ’ . 6456 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 .
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Middle-Aged Men in a Health Fitness Club Data Courtesy of Dr. A. C. Linnerud, NC State Univ The CANCORR Procedure Canonical Correlation Analysis Adjusted Approximate Squared Canonical Canonical Standard Canonical Correlation Correlation Error Correlation 1 0.795608 0.754056 0.084197 0.632992 2 0.200556 .076399 0.220188 0.040223 3 0.072570 . 0.228208 0.005266 Eigenvalues of Inv(E)*H = CanRsq/(1 CanRsq) Eigenvalue Difference Proportion Cumulative 1 1.7247 1.6828 0.9734 0.9734 2 0.0419 0.0366 0.0237 0.9970 3 0.0053 0.0030 1.0000 Test of H0: The canonical correlations in the current row and all that follow are zero Likelihood Approximate Ratio F Value Num DF Den DF Pr > F 1 0.35039053 2.05 9 34.223 0.0635 2 0.95472266 0.18 4 30 0.9491 3 0.99473355 0.08 1 16 0.7748 Multivariate Statistics and F Approximations S=3 M=-0.5 N=6 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.35039053 2.05 9 34.223 0.0635 Pillai's Trace 0.67848151 1.56 9 48 0.1551 Hotelling-Lawley Trace 1.77194146 2.64 9 19.053 0.0357 Roy's Greatest Root 1.72473874 9.20 3 16 0.0009 NOTE: F Statistic for Roy's Greatest Root is an upper bound.
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As Output 20.1.2 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 0 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.
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Middle-Aged Men in a Health Fitness Club Data Courtesy of Dr. A. C. Linnerud, NC State Univ The CANCORR Procedure Canonical Correlation Analysis Raw Canonical Coefficients for the Physiological Measurements Physiological1 Physiological2 Physiological3 Weight 0.031404688 0.076319506 0.007735047 Waist 0.4932416756 0.3687229894 0.1580336471 Pulse 0.008199315 0.032051994 0.1457322421 Raw Canonical Coefficients for the Exercises Exercises1 Exercises2 Exercises3 Chins 0.066113986 0.071041211 0.245275347 Situps 0.016846231 0.0019737454 0.0197676373 Jumps 0.0139715689 0.0207141063 0.008167472 Middle-Aged Men in a Health Fitness Club Data Courtesy of Dr. A. C. Linnerud, NC State Univ The CANCORR Procedure Canonical Correlation Analysis Standardized Canonical Coefficients for the Physiological Measurements Physiological1 Physiological2 Physiological3 Weight 0.7754 1.8844 0.1910 Waist 1.5793 1.1806 0.5060 Pulse 0.0591 0.2311 1.0508 Standardized Canonical Coefficients for the Exercises Exercises1 Exercises2 Exercises3 Chins 0.3495 0.3755 1.2966 Situps 1.0540 0.1235 1.2368 Jumps 0.7164 1.0622 0.4188
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The first canonical variable for the physiological variables, displayed in Output 20.1.3, is a weighted difference of Waist (1.5793) and Weight ( ˆ’ . 7754), with more emphasis on Waist . The coefficient for Pulse is near 0. The correlations between Waist and Weight and the first canonical variable are both positive, 0.9254 for Waist and 0.6206 for Weight . Weight is therefore a suppressor variable, meaning that its coefficient and its correlation have opposite signs.
The first canonical variable for the exercise variables also shows a mixture of signs, subtracting Situps ( ˆ’ 1 . 0540) and Chins ( ˆ’ . 3495) from Jumps (0.7164), with the most weight on Situps . All the correlations are negative, indicating that Jumps is also a suppressor variable.
It may 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 fat 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 fatness and the independent variables:
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People with large waists tend to be fatter than people with small waists. Hence, the correlation between Waist and Situps should be negative.
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People with high weights tend to be fatter than people with low weights. Therefore, Weight should correlate negatively with Situps .
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For a fixed value of Weight , people with large waists tend to be shorter and fatter. Thus, the multiple regression coefficient for Waist should be negative.
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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 may 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 20.1.4) 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.
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Middle-Aged Men in a Health Fitness Club Data Courtesy of Dr. A. C. Linnerud, NC State Univ The CANCORR Procedure Canonical Redundancy Analysis Standardized Variance of the Physiological Measurements Explained by Their Own The Opposite Canonical Variables Canonical Variables Canonical Variable Cumulative Canonical Cumulative Number Proportion Proportion R-Square Proportion Proportion 1 0.4508 0.4508 0.6330 0.2854 0.2854 2 0.2470 0.6978 0.0402 0.0099 0.2953 3 0.3022 1.0000 0.0053 0.0016 0.2969 Standardized Variance of the Exercises Explained by Their Own The Opposite Canonical Variables Canonical Variables Canonical Variable Cumulative Canonical Cumulative Number Proportion Proportion R-Square Proportion Proportion 1 0.4081 0.4081 0.6330 0.2584 0.2584 2 0.4345 0.8426 0.0402 0.0175 0.2758 3 0.1574 1.0000 0.0053 0.0008 0.2767
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Middle-Aged Men in a Health Fitness Club Data Courtesy of Dr. A. C. Linnerud, NC State Univ The CANCORR Procedure Canonical Redundancy Analysis Squared Multiple Correlations Between the Physiological Measurements and the First M Canonical Variables of the Exercises M 1 2 3 Weight 0.2438 0.2678 0.2679 Waist 0.5421 0.5478 0.5478 Pulse 0.0701 0.0702 0.0749 Squared Multiple Correlations Between the Exercises and the First M Canonical Variables of the Physiological Measurements M 1 2 3 Chins 0.3351 0.3374 0.3396 Situps 0.4233 0.4365 0.4365 Jumps 0.0167 0.0536 0.0539
The squared multiple correlations 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).
EAN: 2147483647
Pages: 156