Canonical Correlation Example 3
data canman3;
input group y1 y2 y3 x1 x2;
cards;
1 19.6 5.15 9.5 1 1
1 15.4 5.75 9.1 1 1
1 22.3 4.35 3.3 1 1
1 24.3 7.55 5.0 1 1
1 22.5 8.50 6.0 1 1
1 20.5 10.25 5.0 1 1
1 14.1 5.95 18.8 1 1
1 13.0 6.30 16.5 1 1
1 14.1 5.45 8.9 1 1
1 16.7 3.75 6.0 1 1
1 16.8 5.10 7.4 1 1
2 17.1 9.00 7.5 -1 1
2 15.7 5.30 8.5 -1 1
2 14.9 9.85 6.0 -1 1
2 19.7 3.60 2.9 -1 1
2 17.2 4.05 0.2 -1 1
2 16.0 4.40 2.6 -1 1
2 12.8 7.15 7.0 -1 1
2 13.6 7.25 3.2 -1 1
2 14.2 5.30 6.2 -1 1
2 13.1 3.10 5.5 -1 1
2 16.5 2.40 6.6 -1 1
3 16.0 4.55 2.9 0 -2
3 12.5 2.65 0.7 0 -2
3 18.5 6.50 5.3 0 -2
3 19.2 4.85 8.3 0 -2
3 12.0 8.75 9.0 0 -2
3 13.0 5.20 10.3 0 -2
3 11.9 4.75 8.5 0 -2
3 12.0 5.85 9.5 0 -2
3 19.8 2.85 2.3 0 -2
3 16.5 6.55 3.3 0 -2
3 17.4 6.60 1.9 0 -2
;
proc cancorr all
vprefix=dv vname='dependent variables'
wprefix=iv wname='indpendent variables';
var y1 y2 y3;
with x1 x2;
run;
---------------------------------------------------------------------------------
Means and Standard Deviations
3 dependent variables
2 indpendent variables
33 Observations
Variable Mean Std Dev
Y1 16.330303 3.292462
Y2 5.715152 2.017598
Y3 6.475758 3.985131
X1 0 0.829156
X2 0 1.436141
Correlations Among the Original Variables
Correlations Among the dependent variables
Y1 Y2 Y3
Y1 1.0000 0.0978 -0.3411
Y2 0.0978 1.0000 0.1978
Y3 -0.3411 0.1978 1.0000
Correlations Among the indpendent variables
X1 X2
X1 1.0000 0.0000
X2 0.0000 1.0000
Correlations Among the Original Variables
Correlations Between the dependent variables and the indpendent variables
X1 X2
Y1 0.3262 0.2148
Y2 0.1252 0.1219
Y3 0.3717 0.1512
Adjusted Approx Squared
Canonical Canonical Standard Canonical
Correlation Correlation Error Correlation
1 0.686626 0.656071 0.093434 0.471455
2 0.072213 -.139365 0.175855 0.005215
Eigenvalues of INV(E)*H
= CanRsq/(1-CanRsq)
Eigenvalue Difference Proportion Cumulative
1 0.8920 0.8867 0.9942 0.9942
2 0.0052 . 0.0058 1.0000
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 0.52578837 3.5382 6 56 0.0049
2 0.99478527 0.0760 2 29 0.9270
Multivariate Statistics and F Approximations
S=2 M=0 N=13
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.52578837 3.5382 6 56 0.0049
Pillai's Trace 0.47667015 3.0248 6 58 0.0122
Hotelling-Lawley Trace 0.89723004 4.0375 6 54 0.0021
Roy's Greatest Root 0.89198797 8.6226 3 29 0.0003
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.
Raw Canonical Coefficients for the dependent variables
DV1 DV2
Y1 0.2636265875 0.0813957795
Y2 -0.007656089 0.3886764475
Y3 0.2210547696 -0.14785517
Raw Canonical Coefficients for the indpendent variables
IV1 IV2
X1 1.0690954306 -0.558193886
X2 0.3222733906 0.6172425347
Standardized Canonical Coefficients for the dependent variables
DV1 DV2
Y1 0.8680 0.2680
Y2 -0.0154 0.7842
Y3 0.8809 -0.5892
Standardized Canonical Coefficients for the indpendent variables
IV1 IV2
X1 0.8864 -0.4628
X2 0.4628 0.8864
Canonical Structure
Correlations Between the dependent variables and Their Canonical Variables
DV1 DV2
Y1 0.5660 0.5457
Y2 0.2437 0.6939
Y3 0.5818 -0.5255
Correlations Between the indpendent variables and Their Canonical Variables
IV1 IV2
X1 0.8864 -0.4628
X2 0.4628 0.8864
Canonical Structure
Correlations Between the dependent variables and the
Canonical Variables of the indpendent variables
IV1 IV2
Y1 0.3886 0.0394
Y2 0.1673 0.0501
Y3 0.3995 -0.0379
Correlations Between the indpendent variables and
the Canonical Variables of the dependent variables
DV1 DV2
X1 0.6087 -0.0334
X2 0.3178 0.0640
Canonical Redundancy Analysis
Raw Variance of the dependent variables
Explained by
Their Own The Opposite
Canonical Variables Canonical Variables
Cumulative Canonical Cumulative
Proportion Proportion R-Squared Proportion Proportion
1 0.2952 0.2952 0.4715 0.1392 0.1392
2 0.3109 0.6061 0.0052 0.0016 0.1408
The SAS System 19
17:11 Monday, May 6, 1996
Canonical Redundancy Analysis
Raw Variance of the indpendent variables
Explained by
Their Own The Opposite
Canonical Variables Canonical Variables
Cumulative Canonical Cumulative
Proportion Proportion R-Squared Proportion Proportion
1 0.3571 0.3571 0.4715 0.1684 0.1684
2 0.6429 1.0000 0.0052 0.0034 0.1717
The SAS System 20
17:11 Monday, May 6, 1996
Canonical Redundancy Analysis
Standardized Variance of the dependent variables
Explained by
Their Own The Opposite
Canonical Variables Canonical Variables
Cumulative Canonical Cumulative
Proportion Proportion R-Squared Proportion Proportion
1 0.2394 0.2394 0.4715 0.1129 0.1129
2 0.3518 0.5912 0.0052 0.0018 0.1147
Canonical Redundancy Analysis
Standardized Variance of the indpendent variables
Explained by
Their Own The Opposite
Canonical Variables Canonical Variables
Cumulative Canonical Cumulative
Proportion Proportion R-Squared Proportion Proportion
1 0.5000 0.5000 0.4715 0.2357 0.2357
2 0.5000 1.0000 0.0052 0.0026 0.2383
The SAS System 22
17:11 Monday, May 6, 1996
Canonical Redundancy Analysis
Squared Multiple Correlations Between the dependent variables and
the First 'M' Canonical Variables of the indpendent variables
M 1 2
Y1 0.1510 0.1526
Y2 0.0280 0.0305
Y3 0.1596 0.1610
Squared Multiple Correlations Between the indpendent variables and
the First 'M' Canonical Variables of the dependent variables
M 1 2
X1 0.3705 0.3716
X2 0.1010 0.1051