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;
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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.