applied cannonical correlation analysis by Holli Tonyan - 11/24/97 The variables included in the cannonical correlation analysis: PRESCHOOL BEHAVIOR QUESTIONNAIRE (PBQ)- a screener for behavior problems; filled out by teachers where each item is marked as "does not apply" "sometimes applies" or "definately applies"; high scores represent high levels of behavior problems (Behar & Stringfield, 1974). PBQ1: the aggressive/hostile subscale PBQ2: the hyperactive/inattentive subscale PBQ3: the depressed/anxious subscale CHILD TEACHER RELATIONSHIP SCALE (CTR)- teacher ratings of the relationship she has with a particular child; each item is rated on a scale of 1 ("definately does not apply") to 5 ("definately applies") (Pianta, 1992) CTR1: warm subscale CTR2: conflicted subscale CTR3: open subscale CTR4: dependent subscale CTR5: troubled subscale SECURITY: the most global use of the Attachment Q-Sort where the results of the sort are correlated with an "ideal" sort representing the ideally attached child in terms of the security of the attachment relationship; since scores are a correlation, they range from -1 to +1 (Waters, 1990) AGE: in months SEX: 1=female; 2=male proc cancorr all; with pbq1 pbq2 pbq3 age gender; var ctr1 ctr2 ctr3 ctr4 ctr5 security; run; --------------------------------------------------------------------------- Means and Standard Deviations 5 'VAR' Variables 6 'WITH' Variables 144 Observations Variable Mean Std Dev PBQ1 1.496528 0.540675 PBQ2 1.603009 0.584538 PBQ3 1.269792 0.317344 AGE 51.791667 8.867222 SEX 1.534722 0.500534 SECURITY 0.185347 0.214003 CTR1 3.974082 0.613779 CTR2 1.932300 0.702375 CTR3 3.782407 0.930005 CTR4 2.316551 0.832259 CTR5 2.583333 0.931200 Correlations Among the Original Variables Correlations Among the 'VAR' Variables PBQ1 PBQ2 PBQ3 AGE SEX PBQ1 1.0000 0.6894 0.4267 0.0372 0.1956 PBQ2 0.6894 1.0000 0.3954 0.0152 0.2247 PBQ3 0.4267 0.3954 1.0000 0.0832 -0.0495 AGE 0.0372 0.0152 0.0832 1.0000 -0.1449 SEX 0.1956 0.2247 -0.0495 -0.1449 1.0000 Correlations Among the Original Variables Correlations Among the 'WITH' Variables SECURITY CTR1 CTR2 SECURITY 1.0000 0.1087 -0.1494 CTR1 0.1087 1.0000 0.0410 CTR2 -0.1494 0.0410 1.0000 CTR3 0.0176 0.3929 0.0764 CTR4 0.1178 0.2723 0.3417 CTR5 -0.1347 0.1854 0.2308 Correlations Among the Original Variables Correlations Among the 'WITH' Variables CTR3 CTR4 CTR5 SECURITY 0.0176 0.1178 -0.1347 CTR1 0.3929 0.2723 0.1854 CTR2 0.0764 0.3417 0.2308 CTR3 1.0000 0.1345 -0.0859 CTR4 0.1345 1.0000 0.1575 CTR5 -0.0859 0.1575 1.0000 Correlations Among the Original Variables Correlations Between the 'VAR' Variables and the 'WITH' Variables SECURITY CTR1 CTR2 PBQ1 -0.2969 0.0150 0.5954 PBQ2 -0.2114 -0.0958 0.4927 PBQ3 -0.0921 0.0423 0.3998 AGE -0.0641 0.1863 0.1426 SEX -0.0556 -0.1005 0.1109 Correlations Among the Original Variables Correlations Between the 'VAR' Variables and the 'WITH' Variables CTR3 CTR4 CTR5 PBQ1 0.1013 0.0118 0.1893 PBQ2 -0.0115 0.1212 0.1436 PBQ3 -0.0986 0.1035 0.0944 AGE 0.0314 -0.0990 -0.2401 SEX -0.0463 0.0189 0.0088 Canonical Correlation Analysis Adjusted Approx Squared Canonical Canonical Standard Canonical Correlation Correlation Error Correlation 1 0.694337 0.669543 0.043309 0.482104 2 0.441049 0.391867 0.067357 0.194524 3 0.300234 0.259383 0.076086 0.090141 4 0.142627 . 0.081923 0.020342 5 0.046331 . 0.083445 0.002147 Canonical Correlation Analysis Eigenvalues of INV(E)*H = CanRsq/(1-CanRsq) Eigenvalue Difference Proportion Cumulative 1 0.9309 0.6894 0.7192 0.7192 2 0.2415 0.1424 0.1866 0.9058 3 0.0991 0.0783 0.0765 0.9823 4 0.0208 0.0186 0.0160 0.9983 5 0.0022 . 0.0017 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 0.37103100 5.0070 30 534 0.0001 2 0.71642050 2.3556 20 445.3776 0.0009 3 0.88943752 1.3488 12 357.4679 0.1889 4 0.97755480 0.5175 6 272 0.7949 5 0.99785348 0.1474 2 137 0.8631 Canonical Correlation Analysis Multivariate Statistics and F Approximations S=5 M=0 N=65.5 Statistic Value F Num DF Den DF Pr > F Wilks' Lambda 0.37103100 5.0070 30 534 0.0001 Pillai's Trace 0.78925788 4.2799 30 685 0.0001 Hotelling-Lawley Trace 1.29437993 5.6694 30 657 0.0001 Roy's Greatest Root 0.93089119 21.2553 6 137 0.0001 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 V4 V5 PBQ1 1.5236177419 0.1010997647 -1.974500437 0.6529256474 -0.552732112 PBQ2 0.0399530647 0.8718075085 1.18896837 -1.451681269 1.2527483903 PBQ3 0.5181187692 -0.290485024 2.5175859846 2.3954632589 -0.652971373 AGE 0.0377405045 -0.09893358 0.01299824 -0.040145281 0.009691907 SEX 0.0545457644 -0.156163947 0.620652104 -0.682317127 -1.887195877 Canonical Correlation Analysis Raw Canonical Coefficients for the 'WITH' Variables W1 W2 W3 W4 W5 SECURITY -1.2331674 0.066140313 0.8060051 1.7595049571 -0.763364901 CTR1 0.3571416769 -1.377108594 0.0231285391 1.0615349577 0.4520683958 CTR2 1.4089079488 -0.06409179 0.3388352073 0.0600284761 -0.194118435 CTR3 -0.010452847 0.3389874894 -0.880054488 -0.383143189 0.4270899711 CTR4 -0.422935888 0.4710511054 0.6702792758 -0.269065212 0.9177977217 CTR5 -0.121955423 0.8376783836 -0.392171707 0.6659415433 -0.134133901 Canonical Correlation Analysis Standardized Canonical Coefficients for the 'VAR' Variables V1 V2 V3 V4 V5 PBQ1 0.8238 0.0547 -1.0676 0.3530 -0.2988 PBQ2 0.0234 0.5096 0.6950 -0.8486 0.7323 PBQ3 0.1644 -0.0922 0.7989 0.7602 -0.2072 AGE 0.3347 -0.8773 0.1153 -0.3560 0.0859 SEX 0.0273 -0.0782 0.3107 -0.3415 -0.9446 Canonical Correlation Analysis Standardized Canonical Coefficients for the 'WITH' Variables W1 W2 W3 W4 W5 SECURITY -0.2639 0.0142 0.1725 0.3765 -0.1634 CTR1 0.2192 -0.8452 0.0142 0.6515 0.2775 CTR2 0.9896 -0.0450 0.2380 0.0422 -0.1363 CTR3 -0.0097 0.3153 -0.8185 -0.3563 0.3972 CTR4 -0.3520 0.3920 0.5578 -0.2239 0.7638 CTR5 -0.1136 0.7800 -0.3652 0.6201 -0.1249 Canonical Structure Correlations Between the 'VAR' Variables and Their Canonical Variables V1 V2 V3 V4 V5 PBQ1 0.9278 0.3187 -0.1825 0.0123 -0.0640 PBQ2 0.6675 0.4799 0.3465 -0.3867 0.2333 PBQ3 0.5516 0.0635 0.6125 0.5625 0.0087 AGE 0.3754 -0.8638 0.1076 -0.2430 0.2056 SEX 0.1371 0.1787 0.2017 -0.4492 -0.8407 Canonical Structure Correlations Between the 'WITH' Variables and Their Canonical Variables W1 W2 W3 W4 W5 SECURITY -0.4142 -0.1244 0.2390 0.3249 0.0010 CTR1 0.1104 -0.4703 -0.1947 0.6082 0.5950 CTR2 0.8908 0.2563 0.2566 0.0520 0.1619 CTR3 0.1097 -0.0343 -0.6852 -0.1739 0.6064 CTR4 -0.0044 0.3134 0.4957 0.0619 0.8073 CTR5 0.1364 0.6457 -0.1727 0.6953 0.0032 Canonical Structure Correlations Between the 'VAR' Variables and the Canonical Variables of the 'WITH' Variables W1 W2 W3 W4 W5 PBQ1 0.6442 0.1406 -0.0548 0.0018 -0.0030 PBQ2 0.4635 0.2117 0.1040 -0.0552 0.0108 PBQ3 0.3830 0.0280 0.1839 0.0802 0.0004 AGE 0.2607 -0.3810 0.0323 -0.0347 0.0095 SEX 0.0952 0.0788 0.0606 -0.0641 -0.0390 Canonical Structure Correlations Between the 'WITH' Variables and the Canonical Variables of the 'VAR' Variables V1 V2 V3 V4 V5 SECURITY -0.2876 -0.0549 0.0718 0.0463 0.0000 CTR1 0.0766 -0.2074 -0.0585 0.0867 0.0276 CTR2 0.6185 0.1130 0.0771 0.0074 0.0075 CTR3 0.0762 -0.0151 -0.2057 -0.0248 0.0281 CTR4 -0.0031 0.1382 0.1488 0.0088 0.0374 CTR5 0.0947 0.2848 -0.0519 0.0992 0.0001 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 0.1447 0.1447 0.4821 0.0698 0.0698 2 0.7385 0.8832 0.1945 0.1436 0.2134 3 0.0127 0.8958 0.0901 0.0011 0.2145 4 0.0600 0.9558 0.0203 0.0012 0.2158 5 0.0442 1.0000 0.0021 0.0001 0.2159 Canonical Redundancy Analysis 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 0.1289 0.1289 0.4821 0.0621 0.0621 2 0.1638 0.2926 0.1945 0.0319 0.0940 3 0.1951 0.4877 0.0901 0.0176 0.1116 4 0.1777 0.6653 0.0203 0.0036 0.1152 5 0.2742 0.9395 0.0021 0.0006 0.1158 Canonical Redundancy Analysis 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 0.3541 0.3541 0.4821 0.1707 0.1707 2 0.2228 0.5769 0.1945 0.0433 0.2141 3 0.1162 0.6931 0.0901 0.0105 0.2245 4 0.1454 0.8385 0.0203 0.0030 0.2275 5 0.1615 1.0000 0.0021 0.0003 0.2278 Canonical Redundancy Analysis 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 0.1680 0.1680 0.4821 0.0810 0.0810 2 0.1365 0.3044 0.1945 0.0265 0.1075 3 0.1510 0.4554 0.0901 0.0136 0.1211 4 0.1659 0.6214 0.0203 0.0034 0.1245 5 0.2333 0.8547 0.0021 0.0005 0.1250 Canonical Redundancy Analysis Squared Multiple Correlations Between the 'VAR' Variables and the First 'M' Canonical Variables of the 'WITH' Variables 1 2 3 4 5 PBQ1 0.4150 0.4348 0.4378 0.4378 0.4378 PBQ2 0.2148 0.2596 0.2705 0.2735 0.2736 PBQ3 0.1467 0.1475 0.1813 0.1877 0.1877 AGE 0.0679 0.2131 0.2141 0.2153 0.2154 SEX 0.0091 0.0153 0.0189 0.0230 0.0246 Canonical Redundancy Analysis Squared Multiple Correlations Between the 'WITH' Variables and the First 'M' Canonical Variables of the 'VAR' Variables 1 2 3 4 5 SECURITY 0.0827 0.0857 0.0909 0.0930 0.0930 CTR1 0.0059 0.0489 0.0523 0.0598 0.0606 CTR2 0.3825 0.3953 0.4012 0.4013 0.4014 CTR3 0.0058 0.0060 0.0484 0.0490 0.0498 CTR4 0.0000 0.0191 0.0413 0.0414 0.0427 CTR5 0.0090 0.0901 0.0928 0.1026 0.1026 INTERPRETATION: There are two statistically significant pairs of cannonical variates. The first pair accounts for 48% of the variance (cann. corr. = .69) while the second accounts for 19% of the variance (cann. corr. = .44). Judging from the cannonical structure matrices, one of those pairs represents a relationship between high scores on primarily hostility/agression (PBQ1, very high), but also hyperactivity/inattention (PBQ2, high) and to a lesser extent depression/anxiety (PBQ3, moderately high) on the "with" side and high scores on the conflicted subscale of the Child Teacher Relationship measure (CTR2, high) and security of attachment (SECURITY, low) on the "var" side, representing the most disruptive behavior problems as related to a "workability" dimension of the child teacher relationship. The second pair of variates represents a relationship between primarily hyperactivity/inattention (PBQ2, high), but also age (low) on the "var" side with warmth (CTR1, low), dependence (CTR4, high), and troubled (CTR5, high) subscales on the Child Teacher Relationship measure. The redundancy analyses indicate that much of the variance accounted for by the pairs of variates is driven by the presence of PBQ1 and CTR2.