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