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 selfworth (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 selfesteem, 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.
Note. I did a logarithmic transformation of some of the variables due to nonnormal 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 Set1
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 Set2
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 Set1 and Set2
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 ChiSQ 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 Set1
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 Set1
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 Set2
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 Set2
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 Set1
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 Set1
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 Set2
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 Set2
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 Set1 Explained by Its Own Can. Var.
Prop Var
CV11 .360
CV12 .252
CV13 .206
CV14 .182
Proportion of Variance of Set1 Explained by Opposite Can.Var.
Prop Var
CV21 .043
CV22 .019
CV23 .001
CV24 .000
Proportion of Variance of Set2 Explained by Its Own Can. Var.
Prop Var
CV21 .232
CV22 .174
CV23 .266
CV24 .168
Proportion of Variance of Set2 Explained by Opposite Can. Var.
Prop Var
CV11 .028
CV12 .013
CV13 .002
C
CV14 .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 (noteI did this already). Assumptions regarding withinset 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 selfesteem. 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 selfesteem (.76) are associated with fewer mental (.94) and physical (.43) health symptoms. [Noteyou 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, selfesteem, 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 







