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.