Example of Canonical Correlation
Women's Health Study

Adapted from the work of Tabachnick & Fidell

You are interested in the relationship between a set of attitudinal variables and set of health variables. The attitudinal variables include attitudes towards locus of control (control), toward current martial status (lattmar), toward self-worth (esteem), and toward the role of women (attrole). The health variables are mental health (menheal), physical health (lphysheal), visits to health professionals (ltimedrs), attitude toward use of medication (attdrug), and use of psychotropic drugs (ldruguse). Larger scores indicate more conservative attitudes toward women's role, external locus of control, dissatisfaction with marital status, low self-esteem, poor mental health, poor physical health, more numerous health visits, favorable attitudes toward drug use, and more drug use.

The dataset below contains the results of interviews from 465 women.

[Download Data]

Note. I did a logarithmic transformation of some of the variables due to non-normal distributions.


SPSS (Syntax)

include file 'c:\Program files\spss\canonical correlation.sps'.
cancorr set1=CONTROL LATTMAR ESTEEM ATTROLE
/set2=MENHEAL LPHYHEAL LTIMEDRS ATTDRUG LDRUGUSE.

Note. The first line in the syntax may change according to where the 'canonical correlation.sps'  file is located on your computer.


SPSS Output

Run MATRIX procedure:

Correlations for Set-1
control lattmar esteem attrole
control 1.0000 .1605 .3431 .0007
lattmar .1605 1.0000 .2055 -.0077
esteem .3431 .2055 1.0000 .1881
attrole .0007 -.0077 .1881 1.0000


Correlations for Set-2
menheal lphyheal ltimedrs attdrug ldruguse
menheal 1.0000 .5115 .3592 .0753 .3765
lphyheal .5115 1.0000 .6077 .1229 .4601
ltimedrs .3592 .6077 1.0000 .1008 .4203
attdrug .0753 .1229 .1008 1.0000 .2855
ldruguse .3765 .4601 .4203 .2855 1.0000

Correlations Between Set-1 and Set-2
menheal lphyheal ltimedrs attdrug ldruguse
control .2785 .1194 .0477 .0548 .0651
lattmar .1708 .0486 .0274 -.0679 .0928
esteem .2127 .1131 .0008 .1062 -.0316
attrole -.0709 -.0619 -.1194 .1052 -.1280

Canonical Correlations
1 .346
2 .277
3 .078
4 .032



Test that remaining correlations are zero:
Wilk's Chi-SQ DF Sig.
C

1 .807 98.264 20.000 .000
2 .917 39.885 12.000 .000
3 .993 3.220 6.000 .781
4 .999 .460 2.000 .795


Standardized Canonical Coefficients for Set-1
1 2 3 4
control -.620 -.144 -.127 .854
lattmar -.201 -.587 -.672 -.467
esteem -.527 .482 .491 -.680
attrole .107 .645 -.769 .156


Raw Canonical Coefficients for Set-1
1 2 3 4
control -.490 -.114 -.100 .674
lattmar -.977 -2.849 -3.262 -2.268
esteem -.133 .122 .125 -.172
attrole .016 .095 -.114 .023


Standardized Canonical Coefficients for Set-2
1 2 3 4
menheal -1.037 -.168 -.329 .132
lphyheal -.132 .435 .541 -.888
ltimedrs .229 -.298 .791 .545
attdrug -.187 .766 .100 .653
ldruguse .246 -.813 -.328 .377


Raw Canonical Coefficients for Set-2
1 2 3 4
menheal -.247 -.040 -.078 .032
lphyheal -.641 2.107 2.620 -4.300
ltimedrs .495 -.645 1.709 1.177
attdrug -.161 .662 .087 .564
ldruguse .469 -1.549 -.624 .718


Canonical Loadings for Set-1
1 2 3 4
control -.833 -.072 -.066 .545
lattmar -.410 -.516 -.586 -.472
esteem -.761 .433 .165 -.454
attrole .009 .740 -.672 .032


Cross Loadings for Set-1
1 2 3 4
control -.288 -.020 -.005 .017
lattmar -.142 -.143 -.045 -.015
esteem -.263 .120 .013 -.014
C



attrole .003 .205 -.052 .001


Canonical Loadings for Set-2
1 2 3 4
menheal -.944 -.301 .116 .065
lphyheal -.433 -.112 .715 -.235
ltimedrs -.140 -.359 .874 .277
attdrug -.188 .544 .128 .717
ldruguse -.162 -.583 .158 .434


Cross Loadings for Set-2
1 2 3 4
menheal -.327 -.084 .009 .002
lphyheal -.150 -.031 .055 -.007
ltimedrs -.048 -.100 .068 .009
attdrug -.065 .151 .010 .023
ldruguse -.056 -.162 .012 .014


Redundancy Analysis:


Proportion of Variance of Set-1 Explained by Its Own Can. Var.
Prop Var
CV1-1 .360
CV1-2 .252
CV1-3 .206
CV1-4 .182


Proportion of Variance of Set-1 Explained by Opposite Can.Var.
Prop Var
CV2-1 .043
CV2-2 .019
CV2-3 .001
CV2-4 .000


Proportion of Variance of Set-2 Explained by Its Own Can. Var.
Prop Var
CV2-1 .232
CV2-2 .174
CV2-3 .266
CV2-4 .168


Proportion of Variance of Set-2 Explained by Opposite Can. Var.
Prop Var
CV1-1 .028
CV1-2 .013
CV1-3 .002
C



CV1-4 .000

------ END MATRIX -----


  Results

 To improve linearity of relationship between variables and normality of their distribution, logarithmic transformations were applied to attitude toward marital status, visits to health professionals, physical health, and drug use (note--I did this already). Assumptions regarding within-set multicollinearity were met. The bivariate correlations among all the variables are reported in Table 1.

Table 1
Correlation Coefficients Among the Measures of Attitudinal and Heath Variables

 

 

 

 

 

 

 

 

 

 

Contol

Attmar

Esteem

Attrole

MentHea

PhyHeal

TimeDr

AttDrugs

Attmar

.160

 

 

 

 

 

 

 

Self esteem

.343

.206

 

 

 

 

 

 

Attitudes toward role of women

.001

-.008

.188

 

 

 

 

 

Mental health symptoms

.279

.171

.213

-.071

 

 

 

 

lphyheal

.119

.049

.113

-.062

.511

 

 

 

ltimedrs

.048

.027

.001

-.119

.359

.608

 

 

Attitude toward use of drugs

.055

-.068

.106

.105

.075

.123

.101

 

ldruguse

.065

.093

-.032

-.128

.376

.460

.420

.286

The first canonical correlation was .35 (12% overlapping variance); the second was .28 (7% overlapping variance). The remaining two canonical correlations were effectively zero. With all four canonical correlations included, c2 (20)=98.26, p<.001, and with the first canonical correlation removed, c2 (12)=39.88 p<.001. Subsequent c2 tests were not statistically significant.

Correlations, standardized canonical coefficients are reported in Table 2. The total percent of variance (59%) [add the first proportion of variance explained by its own can var for both set 1 and set 2] and total redundancy (7%) [add the first proportion of variance explained by opposite can var for both set 1 and set 2] suggesting that the first pair of canonical variates was moderately related. However, the second pair, which had a total percent of variance of only 42% and redundancy of 4%, suggesting the second pair are only minimally related suggestions questionable interpretation.

Using a cutoff correlation of .30, the variables in the attitudinal set that correlated with the first canonical variate were locus of control, attitude toward marriage, and self-esteem. Among the health set, mental health and physical health  correlated to the first canonical variate. The first canonical variates indicate that with less external locus of contol (-.83), more feeling of satisfaction towards marriage (-.41), and higher self-esteem (-.76) are associated with fewer mental (-.94) and physical (-.43) health symptoms. [Note--you have to remember how the variables were coded to interpret the results.]

The second canonical variate in the attitudinal set was composed of attitude toward role of women, self-esteem, and attitude toward marital status, and corresponding canonical variate from the health set was composed of drug use, attitude toward drugs, visits to health professionals, and physical health.

Table 2

Correlations, Standardized Canonical Coefficients for the First and Second Canonical Variates

 

 

 

First Canonical Variate

 

Second Canonical Variate

 

 

 

 

Attitudinal

 

 

Correlation

Coefficient

 

Correlation

Coefficient

 

CONTROL

 

-.83

-.62

 

-.07

-.14

 

LATTMAR

 

-.41

-.20

 

-.52

-.59

 

ESTEEM

 

-.76

-.53

 

.43

.48

 

ATTROLE

 

.01

.11

 

.74

.65

 

 

 

 

 

 

 

 

Health

 

 

 

 

 

 

 

 

MENHEAL

 

-.94

-1.04

 

-.30

-.17

 

LPHYHEAL

 

-.43

-.13

 

-.11

.44

 

LTIMEDRS

 

-.14

.23

 

-.36

-.30

 

ATTDURG

 

-.19

-.19

 

.54

.77

 

LDRUGUSE

 

-.16

.25

 

-.58

-.81