Canonical Correlation Example -- Timm -- pg 314
Same data as Assignment 7 and Multiple Regression - Example 2
data timm;
input apt ppvt rpmt n s ns na ss;
cards;
49 48 8 1 2 6 12 16
47 76 13 5 14 14 30 27
11 40 13 0 10 21 16 16
9 52 9 0 2 5 17 8
69 63 15 2 7 11 26 17
35 82 14 2 15 21 34 25
6 71 21 0 1 20 23 18
8 68 8 0 0 10 19 14
49 74 11 0 0 7 16 13
8 70 15 3 2 21 26 25
47 70 15 8 16 15 35 24
6 61 11 5 4 7 15 14
14 54 12 1 12 13 27 21
30 55 13 2 1 12 20 17
4 54 10 3 12 20 26 22
24 40 14 0 2 5 14 8
19 66 13 7 12 21 35 27
45 54 10 0 6 6 14 16
22 64 14 12 8 19 27 26
16 47 16 3 9 15 18 10
32 48 16 0 7 9 14 18
37 52 14 4 6 20 26 26
47 74 19 4 9 14 23 23
5 57 12 0 2 4 11 8
6 57 10 0 1 16 15 17
60 80 11 3 8 18 28 21
58 78 13 1 18 19 34 23
6 70 16 2 11 9 23 11
16 47 14 0 10 7 12 8
45 94 19 8 10 28 32 32
9 63 11 2 12 5 25 14
69 76 16 7 11 18 29 21
35 59 11 2 5 10 23 24
19 55 8 0 1 14 19 12
58 74 14 1 0 10 18 18
58 71 17 6 4 23 31 26
79 54 14 0 6 6 15 14
;
proc cancorr all;
var apt ppvt rpmt;
with n s ns na ss;
run;
-----------------------------------------------------------------------------
Means and Standard Deviations
3 'VAR' Variables
5 'WITH' Variables
37 Observations
Variable Mean Std Dev
APT 31.270270 22.100338
PPVT 62.648649 12.526009
RPMT 13.243243 3.094847
N 2.540541 2.949525
S 6.918919 5.068302
NS 13.486486 6.353571
NA 22.378378 7.243201
SS 18.378378 6.369857
Correlations Among the Original Variables
Correlations Among the 'VAR' Variables
APT PPVT RPMT
APT 1.0000 0.3703 0.2114
PPVT 0.3703 1.0000 0.3548
RPMT 0.2114 0.3548 1.0000
Correlations Among the 'WITH' Variables
N S NS NA SS
N 1.0000 0.4007 0.5370 0.6481 0.6704
S 0.4007 1.0000 0.3523 0.6478 0.4252
NS 0.5370 0.3523 1.0000 0.7136 0.7695
NA 0.6481 0.6478 0.7136 1.0000 0.7951
SS 0.6704 0.4252 0.7695 0.7951 1.0000
Correlations Among the Original Variables
Correlations Between the 'VAR' Variables and the 'WITH' Variables
N S NS NA SS
APT 0.1860 0.1609 0.0685 0.2617 0.3341
PPVT 0.4444 0.2682 0.4692 0.6720 0.5876
RPMT 0.3504 0.2386 0.4388 0.3390 0.3404
Canonical Correlation Analysis
Adjusted Approx Squared
Canonical Canonical Standard Canonical
Correlation Correlation Error Correlation
1 0.716526 0.655198 0.081098 0.513409
2 0.490621 0.414578 0.126549 0.240709
3 0.266778 0.211906 0.154805 0.071170
Eigenvalues of INV(E)*H
= CanRsq/(1-CanRsq)
Eigenvalue Difference Proportion Cumulative
1 1.0551 0.7381 0.7283 0.7283
2 0.3170 0.2404 0.2188 0.9471
3 0.0766 . 0.0529 1.0000
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 0.34316907 2.5381 15 80.45763 0.0039
2 0.70525204 1.4308 8 60 0.2025
3 0.92882959 0.7918 3 31 0.5078
Canonical Correlation Analysis
Multivariate Statistics and F Approximations
S=3 M=0.5 N=13.5
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.34316907 2.5381 15 80.45763 0.0039
Pillai's Trace 0.82528864 2.3529 15 93 0.0066
Hotelling-Lawley Trace 1.44875712 2.6722 15 83 0.0023
Roy's Greatest Root 1.05511542 6.5417 5 31 0.0003
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
Canonical Correlation Analysis
Raw Canonical Coefficients for the 'VAR' Variables
V1 V2 V3
APT 0.0032263523 -0.033176025 0.0358057012
PPVT 0.076224788 -0.001671498 -0.048255281
RPMT 0.0141322568 0.2756473326 0.2104352766
Raw Canonical Coefficients for the 'WITH' Variables
W1 W2 W3
N -0.007150868 0.1599910602 0.0992870525
S -0.075658454 0.0421240468 0.1746238789
NS -0.035321752 0.2381604803 -0.010080608
NA 0.1579154702 -0.059418808 -0.229030304
SS 0.0435562566 -0.182391116 0.2019492827
Canonical Correlation Analysis
Standardized Canonical Coefficients for the 'VAR' Variables
V1 V2 V3
APT 0.0713 -0.7332 0.7913
PPVT 0.9548 -0.0209 -0.6044
RPMT 0.0437 0.8531 0.6513
Standardized Canonical Coefficients for the 'WITH' Variables
W1 W2 W3
N -0.0211 0.4719 0.2928
S -0.3835 0.2135 0.8850
NS -0.2244 1.5132 -0.0640
NA 1.1438 -0.4304 -1.6589
SS 0.2774 -1.1618 1.2864
Canonical Structure
Correlations Between the 'VAR' Variables and Their Canonical Variables
V1 V2 V3
APT 0.4341 -0.5606 0.7052
PPVT 0.9967 0.0102 -0.0803
RPMT 0.3976 0.6906 0.6041
Correlations Between the 'WITH' Variables and Their Canonical Variables
W1 W2 W3
N 0.6320 0.3122 0.4004
S 0.3879 0.1630 0.4521
NS 0.6588 0.6406 0.2112
NA 0.9422 0.1697 0.0814
SS 0.8371 0.0675 0.4906
Canonical Structure
Correlations Between the 'VAR' Variables and the
Canonical Variables of the 'WITH' Variables
W1 W2 W3
APT 0.3111 -0.2750 0.1881
PPVT 0.7142 0.0050 -0.0214
RPMT 0.2849 0.3388 0.1612
Canonical Structure
Correlations Between the 'WITH' Variables and
the Canonical Variables of the 'VAR' Variables
V1 V2 V3
N 0.4529 0.1532 0.1068
S 0.2780 0.0800 0.1206
NS 0.4721 0.3143 0.0563
NA 0.6751 0.0833 0.0217
SS 0.5998 0.0331 0.1309
Canonical Redundancy Analysis
Raw Variance of the 'VAR' Variables
Explained by
Their Own The Opposite
Canonical Variables Canonical Variables
Cumulative Canonical Cumulative
Proportion Proportion R-Squared Proportion Proportion
1 0.3809 0.3809 0.5134 0.1955 0.1955
2 0.2414 0.6223 0.2407 0.0581 0.2536
3 0.3777 1.0000 0.0712 0.0269 0.2805
Raw Variance of the 'WITH' Variables
Explained by
Their Own The Opposite
Canonical Variables Canonical Variables
Cumulative Canonical Cumulative
Proportion Proportion R-Squared Proportion Proportion
1 0.5952 0.5952 0.5134 0.3056 0.3056
2 0.1180 0.7132 0.2407 0.0284 0.3340
3 0.1106 0.8238 0.0712 0.0079 0.3418
Standardized Variance of the 'VAR' Variables
Explained by
Their Own The Opposite
Canonical Variables Canonical Variables
Cumulative Canonical Cumulative
Proportion Proportion R-Squared Proportion Proportion
1 0.4467 0.4467 0.5134 0.2293 0.2293
2 0.2638 0.7104 0.2407 0.0635 0.2928
3 0.2896 1.0000 0.0712 0.0206 0.3134
Standardized Variance of the 'WITH' Variables
Explained by
Their Own The Opposite
Canonical Variables Canonical Variables
Cumulative Canonical Cumulative
Proportion Proportion R-Squared Proportion Proportion
1 0.5145 0.5145 0.5134 0.2641 0.2641
2 0.1136 0.6281 0.2407 0.0273 0.2915
3 0.1313 0.7594 0.0712 0.0093 0.3008
Squared Multiple Correlations Between the 'VAR' Variables and
the First 'M' Canonical Variables of the 'WITH' Variables
M 1 2 3
APT 0.0968 0.1724 0.2078
PPVT 0.5100 0.5101 0.5105
RPMT 0.0812 0.1960 0.2219
Squared Multiple Correlations Between the 'WITH' Variables and
the First 'M' Canonical Variables of the 'VAR' Variables
M 1 2 3
N 0.2051 0.2285 0.2400
S 0.0773 0.0837 0.0982
NS 0.2229 0.3216 0.3248
NA 0.4558 0.4627 0.4632
SS 0.3597 0.3608 0.3780