### SAS Annotated Output: Canonical Correlation Analysis

This page shows an example of a canonical correlation analysis in SAS with footnotes explaining the output.  A researcher has collected data on three psychological variables, four academic variables (standardized test scores) and gender for 600 college freshman.  She is interested in how the set of psychological variables relates to the academic variables and gender.  In particular, the researcher is interested in how many dimensions are necessary to understand the association between the two sets of variables.

We have a data file, mmr.sas7bdat, with 600 observations on eight variables.  The psychological variables are locus of control, self-concept and motivation.  The academic variables are standardized test scores in reading, writing, math and science.  Additionally, the variable female is a zero-one indicator variable with the one indicating a female student.  The researcher is interested in the relationship between the psychological variables and the academic variables, with gender considered as well.  Canonical correlation analysis aims to find pairs of linear combinations of each group of variables that are highly correlated.  These linear combinations are called canonical variates.  Each canonical variate is orthogonal to the other canonical variates except for the one with which its correlation has been maximized.  The possible number of such pairs is limited to the number of variables in the smallest group.  In our example, there are three psychological variables and more than three academic variables.  Thus, a canonical correlation analysis on these sets of variables will generate three pairs of canonical variates.

To begin, let's read in and explore the dataset.

proc means data = mmr; run;
The SAS System

The MEANS Procedure

Variable          Label               N          Mean       Std Dev       Minimum       Maximum
ID                                  600   300.5000000   173.3493582     1.0000000   600.0000000
LOCUS_OF_CONTROL  locus of control  600     0.0965333     0.6702799    -2.2300000     1.3600000
SELF_CONCEPT      self-concept      600     0.0049167     0.7055125    -2.6199999     1.1900001
MOTIVATION        motivation        600     0.6608333     0.3427294             0     1.0000000
WRITE             writing score     600    52.3848333     9.7264550    25.5000000    67.0999985
MATH              math score        600    51.8490000     9.4147363    31.7999992    75.5000000
SCIENCE           science score     600    51.7633332     9.7061789    26.0000000    74.1999969
FEMALE                              600     0.5450000     0.4983864             0     1.0000000


To run our canonical correlation, we will use the cancorr procedure in SAS.  We list the set of variables in our first group in the var statement and the set of variables in our second group in the with statement.  We include the optional commands vprefix, wprefix, vname and wname in the proc cancor statement to give understandable prefixes to our sets of variables and make the output easier to interpret.

proc cancorr data=mmr
vprefix=Psych vname='Psychological Measurements'
var locus_of_control self_concept motivation;
with read write math science female;
run;
...[additional output omitted]...
Correlations Among the Original Variables

Correlations Among the Psychological Measurements

LOCUS_OF_
CONTROL      SELF_CONCEPT        MOTIVATION

LOCUS_OF_CONTROL            1.0000            0.1712            0.2451
SELF_CONCEPT                0.1712            1.0000            0.2886
MOTIVATION                  0.2451            0.2886            1.0000

READ           1.0000        0.6286        0.6793        0.6907       -0.0417
WRITE          0.6286        1.0000        0.6327        0.5691        0.2443
MATH           0.6793        0.6327        1.0000        0.6495       -0.0482
SCIENCE        0.6907        0.5691        0.6495        1.0000       -0.1382
FEMALE        -0.0417        0.2443       -0.0482       -0.1382        1.0000

Correlations Between the Psychological Measurements

LOCUS_OF_CONTROL            0.3736            0.3589            0.3373
SELF_CONCEPT                0.0607            0.0194            0.0536
MOTIVATION                  0.2106            0.2542            0.1950

Correlations Between the Psychological Measurements

SCIENCE            FEMALE

LOCUS_OF_CONTROL            0.3246            0.1134
SELF_CONCEPT                0.0698           -0.1260
MOTIVATION                  0.1157            0.0981

--------------------------------------------------------------------------------

Canonical Correlation Analysis

Canonical      Canonical       Standard      Canonical
Correlation    Correlation          Error    Correlation

1       0.464086       0.455474       0.032059       0.215376
2       0.167509        .             0.039712       0.028059
3       0.103991        .             0.040417       0.010814

Eigenvalues of Inv(E)*H
= CanRsq/(1-CanRsq)

Eigenvalue    Difference    Proportion    Cumulative

1        0.2745        0.2456        0.8734        0.8734
2        0.0289        0.0179        0.0919        0.9652
3        0.0109                      0.0348        1.0000

Test of H0: The canonical correlations in the
current row and all that follow are zero

Likelihood    Approximate
Ratio        F Value    Num DF    Den DF    Pr > F

1    0.75436113          11.72        15    1634.7    <.0001
2    0.96142996           2.94         8      1186    0.0029
3    0.98918584           2.16         3       594    0.0911

Multivariate Statistics and F Approximations

S=3    M=0.5    N=295

Statistic                        Value    F Value    Num DF    Den DF    Pr > F

Wilks' Lambda               0.75436113      11.72        15    1634.7    <.0001
Pillai's Trace              0.25424936      11.00        15      1782    <.0001
Hotelling-Lawley Trace      0.31429738      12.38        15      1113    <.0001
Roy's Greatest Root         0.27449563      32.61         5       594    <.0001

NOTE: F Statistic for Roy's Greatest Root is an upper bound.

--------------------------------------------------------------------------------

Canonical Correlation Analysis

Raw Canonical Coefficients for the Psychological Measurements

Psych1         Psych2         Psych3

LOCUS_OF_CONTROL   locus of control   1.2538339076   0.6214775237   -0.661689607
SELF_CONCEPT       self-concept        -0.35134993   1.1876866562   0.8267209411
MOTIVATION         motivation         1.2624203286   -2.027264053   2.0002284379

Raw Canonical Coefficients for the Academic Measurements

WRITE        writing score      0.0358771125      -0.042071471      0.0913073288
MATH         math score         0.0234171847      -0.004229472      0.0093982096
SCIENCE      science score      0.0050251567      0.0851621751      -0.109835018
FEMALE                          0.6321192387      -1.084642482      -1.794646917

--------------------------------------------------------------------------------

Canonical Correlation Analysis

Standardized Canonical Coefficients for the Psychological Measurements

Psych1        Psych2        Psych3

LOCUS_OF_CONTROL      locus of control        0.8404        0.4166       -0.4435
SELF_CONCEPT          self-concept           -0.2479        0.8379        0.5833
MOTIVATION            motivation              0.4327       -0.6948        0.6855

Standardized Canonical Coefficients for the Academic Measurements

WRITE        writing score         0.3490        -0.4092         0.8881
MATH         math score            0.2205        -0.0398         0.0885
SCIENCE      science score         0.0488         0.8266        -1.0661
FEMALE                             0.3150        -0.5406        -0.8944

--------------------------------------------------------------------------------

Canonical Structure

Correlations Between the Psychological Measurements
and Their Canonical Variables

Psych1        Psych2        Psych3

LOCUS_OF_CONTROL      locus of control        0.9040        0.3897       -0.1756
SELF_CONCEPT          self-concept            0.0208        0.7087        0.7052
MOTIVATION            motivation              0.5672       -0.3509        0.7451

Correlations Between the Academic Measurements and Their Canonical Variables

WRITE        writing score         0.8765        -0.0648         0.2546
MATH         math score            0.7639         0.2979         0.1478
SCIENCE      science score         0.6584         0.6768        -0.2304
FEMALE                             0.3641        -0.7549        -0.5434

Correlations Between the Psychological Measurements and
the Canonical Variables of the Academic Measurements

LOCUS_OF_CONTROL     locus of control        0.4196        0.0653       -0.0183
SELF_CONCEPT         self-concept            0.0097        0.1187        0.0733
MOTIVATION           motivation              0.2632       -0.0588        0.0775

Correlations Between the Academic Measurements and the
Canonical Variables of the Psychological Measurements

Psych1        Psych2        Psych3

WRITE        writing score        0.4068       -0.0109        0.0265
MATH         math score           0.3545        0.0499        0.0154
SCIENCE      science score        0.3056        0.1134       -0.0240
FEMALE                            0.1690       -0.1265       -0.0565

--------------------------------------------------------------------------------

Canonical Redundancy Analysis

Raw Variance of the Psychological Measurements Explained by
Their Own                               The Opposite
Canonical Variables                       Canonical Variables
Canonical
Variable                  Cumulative     Canonical                  Cumulative
Number    Proportion    Proportion      R-Square    Proportion    Proportion

1        0.3806        0.3806        0.2154        0.0820        0.0820
2        0.3126        0.6932        0.0281        0.0088        0.0908
3        0.3068        1.0000        0.0108        0.0033        0.0941

Raw Variance of the Academic Measurements Explained by
Their Own                               The Opposite
Canonical Variables                       Canonical Variables
Canonical
Variable                  Cumulative     Canonical                  Cumulative
Number    Proportion    Proportion      R-Square    Proportion    Proportion

1        0.6251        0.6251        0.2154        0.1346        0.1346
2        0.1704        0.7955        0.0281        0.0048        0.1394
3        0.0395        0.8350        0.0108        0.0004        0.1398

--------------------------------------------------------------------------------

Canonical Redundancy Analysis

Standardized Variance of the Psychological Measurements Explained by
Their Own                               The Opposite
Canonical Variables                       Canonical Variables
Canonical
Variable                  Cumulative     Canonical                  Cumulative
Number    Proportion    Proportion      R-Square    Proportion    Proportion

1        0.3798        0.3798        0.2154        0.0818        0.0818
2        0.2591        0.6389        0.0281        0.0073        0.0891
3        0.3611        1.0000        0.0108        0.0039        0.0930

Standardized Variance of the Academic Measurements Explained by
Their Own                               The Opposite
Canonical Variables                       Canonical Variables
Canonical
Variable                  Cumulative     Canonical                  Cumulative
Number    Proportion    Proportion      R-Square    Proportion    Proportion

1        0.5249        0.5249        0.2154        0.1130        0.1130
2        0.2499        0.7748        0.0281        0.0070        0.1201
3        0.0907        0.8655        0.0108        0.0010        0.1210

--------------------------------------------------------------------------------

Canonical Redundancy Analysis

Squared Multiple Correlations Between the Psychological Measurements
and the First M Canonical Variables of the Academic Measurements

M                                                  1             2             3

LOCUS_OF_CONTROL      locus of control        0.1760        0.1803        0.1806
SELF_CONCEPT          self-concept            0.0001        0.0142        0.0196
MOTIVATION            motivation              0.0693        0.0727        0.0787

Squared Multiple Correlations Between the Academic Measurements and
the First M Canonical Variables of the Psychological Measurements

M                                      1             2             3

WRITE        writing score        0.1655        0.1656        0.1663
MATH         math score           0.1257        0.1282        0.1284
SCIENCE      science score        0.0934        0.1062        0.1068
FEMALE                            0.0286        0.0445        0.0477

#### Correlations Among the Original Variables

The SAS System

The CANCORR Procedure

Correlations Among the Original Variables

Correlations Among the Psychological Measurementsa

LOCUS_OF_
CONTROL      SELF_CONCEPT        MOTIVATION

LOCUS_OF_CONTROL            1.0000            0.1712            0.2451
SELF_CONCEPT                0.1712            1.0000            0.2886
MOTIVATION                  0.2451            0.2886            1.0000

READ           1.0000        0.6286        0.6793        0.6907       -0.0417
WRITE          0.6286        1.0000        0.6327        0.5691        0.2443
MATH           0.6793        0.6327        1.0000        0.6495       -0.0482
SCIENCE        0.6907        0.5691        0.6495        1.0000       -0.1382
FEMALE        -0.0417        0.2443       -0.0482       -0.1382        1.0000

Correlations Between the Psychological Measurements

LOCUS_OF_CONTROL            0.3736            0.3589            0.3373
SELF_CONCEPT                0.0607            0.0194            0.0536
MOTIVATION                  0.2106            0.2542            0.1950

Correlations Between the Psychological Measurements

SCIENCE            FEMALE

LOCUS_OF_CONTROL            0.3246            0.1134
SELF_CONCEPT                0.0698           -0.1260
MOTIVATION                  0.1157            0.0981

a. Correlations Among the Psychological Measurements - This is the Pearson correlation matrix for the three psychological variables.  This gives us a sense of the relationships between the variables within this group.  Because there are three variables in this group, the correlation matrix is 3x3.  The psychological variables are not highly correlated.  This suggests that knowing the values in one of the psychological variables does not provide much information about the other psychological variables.  These relationships between the variables will effect the way in which the group is summarized as a linear combination of these variables.

b. Correlations Among the Academic Measurements - This is the Pearson correlation matrix for the four academic variables and female.  This gives us a sense of the relationships between the variables within this group.  Because there are three variables in this group, the correlation matrix is 5x5.  We can see that the four standardized test variables (read, write, math, and science) are much more highly correlated than the psychological variables.

c. Correlations Between the Psychological Measurements and the Academic Measurements - This matrix presents the psychological variables in rows and the academic variables in columns.  The correlations in the matrix are between all combinations of variables in different groups.  Because we have 3 variables in one group and 5 in the other, a total of 15 such correlations exist.  In this table, we can see that all of the correlations are less than 0.4.

#### Canonical Correlations

                              Adjusted    Approximate        Squared
Canonical      Canonical       Standard      Canonical
Correlationd   Correlatione         Errorf   Correlationg

1       0.464086       0.455474       0.032059       0.215376
2       0.167509        .             0.039712       0.028059
3       0.103991        .             0.040417       0.010814

Eigenvalues of Inv(E)*H
= CanRsq/(1-CanRsq)

Eigenvalueh   Differencei   Proportionj   Cumulativek

1        0.2745        0.2456        0.8734        0.8734
2        0.0289        0.0179        0.0919        0.9652
3        0.0109                      0.0348        1.0000

Test of H0: The canonical correlations in the
current row and all that follow are zero

Likelihood    Approximate
Ratiol       F Valuem   Num DF    Den DFn   Pr > Fo

1    0.75436113          11.72        15    1634.7    <.0001
2    0.96142996           2.94         8      1186    0.0029
3    0.98918584           2.16         3       594    0.0911

Multivariate Statistics and F Approximations

S=3    M=0.5    N=295

Statistic                         Value    F Valuem    Num DF   Den DFn   Pr > Fo

Wilks' Lambdap               0.75436113      11.72        15    1634.7    <.0001
Pillai's Traceq              0.25424936      11.00        15      1782    <.0001
Hotelling-Lawley Tracer      0.31429738      12.38        15      1113    <.0001
Roy's Greatest Roots         0.27449563      32.61         5       594    <.0001

NOTE: F Statistic for Roy's Greatest Root is an upper bound.

d. Canonical Correlation - These are the Pearson correlations of the pairs of canonical variates.  The first pair of variates, a linear combination of the psychological measurements and a linear combination of the academic measurements, has a correlation coefficient of 0.464086.  The second pair has a correlation coefficient of 0.167509, and the third pair 0.103991.

e. Adjusted Canonical Correlation - These are adjusted canonical correlations which are less biased than the raw correlations.  These adjusted values may be negative.  If an adjusted canonical correlation is close to zero or if it is greater than the previous adjusted canonical correlation, then it is reported as missing.

f. Approximate Standard Error - These are the approximate standard errors for the canonical correlations.

g. Squared Canonical Correlation - These are the squares of the canonical correlations.  For example, (0.464086*0.464086) = 0.215376. These values can be interpreted similarly to R-squared values in OLS regression: they are the proportion of the variance in the canonical variate of one set of variables explained by the canonical variate of the other set of variables.

h. Eigenvalue - These are the eigenvalues of the product of the model matrix and the inverse of the error matrix.  These eigenvalues can also be calculated using the squared canonical correlations.  The largest eigenvalue is equal to largest squared correlation /(1- largest squared correlation).  So 0.215376/(1-0.215376) = 0.2745.  These calculations can be completed for each correlation to find the corresponding eigenvalue. The magnitudes of the eigenvalues are related to the tests of the correlations.  The larger the eigenvalues are associated with lower p-values. If we think about the relationship between the canonical correlations and the eigenvalues, it makes sense that the larger correlations are more likely to be significantly different from zero.

i. Difference - This is the difference between the given eigenvalue and the next-largest eigenvalue: 0.2745-0.0289 = 0.2456 and 0.0289-0.0109 = 0.0179 (with rounding).

j. Proportion - This is the proportion of the sum of the eigenvalues represented by a given eigenvalue.  The sum of the three eigenvalues is (0.2745+0.0289+0.0109) = 0.3143.  Then, the proportions can be calculated: 0.2745/0.3143 = 0.8734, 0.0289/0.3143 = 0.0919, and 0.0109/0.3143 = 0.0348.

k. Cumulative - This is the cumulative sum of the proportions.

l. Likelihood Ratio - This is the likelihood ratio for testing the hypothesis that the given canonical correlation and all smaller ones are equal to zero in the population.  It is equivalent to Wilks' lambda (see superscript p) and can be calculated as the product of the values of (1-canonical correlation2).  In this example, our canonical correlations are 0.4641, 0.1675, and 0.1040.  Hence the likelihood ratio for testing that all three of the correlations are zero is (1- 0.46412)*(1-0.16752)*(1-0.10402) = 0.754361.  To test that the two smaller canonical correlations, 0.1675 and 0.1040, are zero in the population, the likelihood is (1-0.16752)*(1-0.10402) = 0.96143.  The likelihood that the smallest canonical correlation is zero is (1-0.10402) = 0.989186.

m. (Approximate) F Value - These are the F values associated with the various tests (likelihood ratio or one of the four multivariate tests) that are included in SAS's cancorr procedure.  For the likelihood ratio tests, the F values are approximate.  For Roy's Greatest Root, the F value is an upper bound.  For the likelihood tests, the F values are testing the hypotheses that the given canonical correlation and all smaller ones are equal to zero in the population.  For the multivariate tests, the F values are testing the hypothesis that all three canonical correlations are equal to zero in the population.

n. Num DF, Den DF - These are the degrees of freedom used in determining the F values.  Note that there are instances in which the degrees of freedom may be a non-integer (here, the Den DF associated with Wilks' lambda is a non-integer) because these degrees of freedom are calculated using the mean squared errors, which are often non-integers.

o. Pr > F - This is the p-value associated with the F value of a given test statistic.  The null hypothesis that our two sets of variables are not linearly related is evaluated with regard to this p-value.  The null hypothesis is rejected if the p-value is less than our specified alpha level (often 0.05).  If not, then we fail to reject the null hypothesis.  In this example, we reject the null hypothesis that all three canonical correlations are equal to zero at alpha level 0.05 because the p-values for all tests of this hypothesis are less than 0.05 (Wilks' Lambda, Pillai's Trace, Hotelling-Lawley Trace, Roy's Greatest Root and the first Likelihood Ratio).  The p-value associated with the likelihood ratio test of the second and third canonical correlations suggest that they we can also reject the hypothesis that both the second and third canonical correlations are zero, but the p-value associated with the likelihood ratio test of the third canonical correlation alone is 0.0911.  Because this is greater than 0.05, we fail to reject the hypothesis that the third canonical correlation is zero.

p. Wilks' Lambda - This is one of the four multivariate statistics calculated by SAS to test the null hypothesis that the canonical correlations are zero (which, in turn, means that there is no linear relationship between the two specified groups of variables).  Wilks' lambda is the product of the values of (1-canonical correlation2).  In this example, our canonical correlations are 0.4641, 0.1675, and 0.1040 so the Wilks' Lambda testing all three of the correlations is (1- 0.46412)*(1-0.16752)*(1-0.10402) = 0.75436113.  This test statistic is equal to the likelihood ratio (see superscript l).

q. Pillai's Trace - Pillai's trace is another of the four multivariate statistics calculated by SAS.  Pillai's trace is the sum of the squared canonical correlations: 0.46412 + 0.16752 + 0.10402 =  0.25424936.

r. Hotelling-Lawley Trace -  This is very similar to Pillai's trace.  It is the sum of the values of (canonical correlation2/(1-canonical correlation2)).  We can calculate 0.46412 /(1- 0.46412) + 0.16752/(1-0.16752) + 0.10402/(1-0.10402) = 0.31429738.

s. Roy's Greatest Root - This is the largest eigenvalue.  Because it is based on a maximum, it can behave differently from the other three test statistics.  In instances where the other three are not significant and Roy's is significant, the effect should be considered not significant.

#### Canonical Coefficients

               Raw Canonical Coefficients for the Psychological Measurementst
Psych1            Psych2            Psych3
LOCUS_OF_CONTROL      locus of control      1.2538339076      0.6214775237      -0.661689607
SELF_CONCEPT          self-concept           -0.35134993      1.1876866562      0.8267209411
MOTIVATION            motivation            1.2624203286      -2.027264053      2.0002284379


            Raw Canonical Coefficients for the Academic Measurementst
WRITE        writing score      0.0358771125      -0.042071471      0.0913073288
MATH         math score         0.0234171847      -0.004229472      0.0093982096
SCIENCE      science score      0.0050251567      0.0851621751      -0.109835018
FEMALE                          0.6321192387      -1.084642482      -1.794646917


     Standardized Canonical Coefficients for the Psychological Measurementsu
Psych1        Psych2        Psych3
LOCUS_OF_CONTROL      locus of control        0.8404        0.4166       -0.4435
SELF_CONCEPT          self-concept           -0.2479        0.8379        0.5833
MOTIVATION            motivation              0.4327       -0.6948        0.6855


   Standardized Canonical Coefficients for the Academic Measurementsu
WRITE        writing score         0.3490        -0.4092         0.8881
MATH         math score            0.2205        -0.0398         0.0885
SCIENCE      science score         0.0488         0.8266        -1.0661
FEMALE                             0.3150        -0.5406        -0.8944

t. Raw Canonical Coefficients for the Psychological/Academic Measurements - These are the raw canonical coefficients.  They define the linear relationship between the variables in a given group and the canonical variates.  They can be interpreted in the same way you would interpret regression coefficients, assuming the canonical variate as the outcome variable.  For example, a one unit increase in locus_of_control leads to a 1.253834 unit increase in the first variate of the psychological measurements ("Psych1"), and a one unit increase in read score leads to a 0.0446206 unit increase in the first variate of the academic measurements ("Academic1").

u. Standardized Canonical Coefficients for the Psychological/Academic Measurements - These are the standardized canonical coefficients.  This means that, if all of the variables in the analysis are rescaled to have a mean of zero and a standard deviation of 1, the coefficients generating the canonical variates would indicate how a one standard deviation increase in the variable would change the variate. For example, an increase of one standard deviation in locus_of_control would lead to a 0.8404 unit increase in the first variate of the psychological measurements ("Psych1"), and an increase of one standard deviation in read would lead to a 0.4508 unit increase in the first variate of the academic measurements ("Academic1").

#### Correlations Among Original Variables and Canonical Variates

Correlations Between the Psychological Measurements and Their Canonical Variablesv
Psych1        Psych2        Psych3
LOCUS_OF_CONTROL      locus of control        0.9040        0.3897       -0.1756
SELF_CONCEPT          self-concept            0.0208        0.7087        0.7052
MOTIVATION            motivation              0.5672       -0.3509        0.7451


Correlations Between the Academic Measurements and Their Canonical Variablesv
WRITE        writing score         0.8765        -0.0648         0.2546
MATH         math score            0.7639         0.2979         0.1478
SCIENCE      science score         0.6584         0.6768        -0.2304
FEMALE                             0.3641        -0.7549        -0.5434


                    Correlations Between the Psychological Measurements and
the Canonical Variables of the Academic Measurementsw
LOCUS_OF_CONTROL      locus of control         0.4196         0.0653        -0.0183
SELF_CONCEPT          self-concept             0.0097         0.1187         0.0733
MOTIVATION            motivation               0.2632        -0.0588         0.0775


                     Correlations Between the Academic Measurements and the
Canonical Variables of the Psychological Measurementsx
Psych1        Psych2        Psych3
WRITE        writing score        0.4068       -0.0109        0.0265
MATH         math score           0.3545        0.0499        0.0154
SCIENCE      science score        0.3056        0.1134       -0.0240
FEMALE                            0.1690       -0.1265       -0.0565

v. Correlations Between the Psychological/Academic Measurements and Their Canonical Variables - Here, SAS presents the correlations between each variable in a group and the group's canonical variates.  These can allow us to see if the variates are combining the variables in such a way that might represent a particular idea. For example, we can see that the first variate for the psychological variables, Psych1, is highly correlated with locus_of_control and motivation, but uncorrelated with self-concept.  Thus, this variate arguably captures much of the shared variance of locus_of_control and motivation. If we look at the academic variables, we can see that the first variate is highly correlated with all four of the subject variables. Those four variables were very highly correlated with each other (see superscript b), so it is not surprising that they should all be highly correlated with a variate that captures their shared variance.  The second variate is highly correlated with science and negatively correlated with female. Thus, the first variate might represent overall academic performance with an emphasis on reading and writing, while the second variate emphasizes performance in science and is possibly indicative of male students.

w. Correlations Between the Psychological Measurements and the Canonical Variables of the Academic Measurements - In addition to the correlations between the variables in a group and the group's canonical variates, SAS also presents the correlations between each variable in one group and the canonical variates of the other.  We see that the psychological variables locus_of_control, self_concept and motivation are correlated with Academic1, Academic2 and Academic3 (a total of 3x3=9 correlations).   Here, we can see that locus_of_control and motivation are correlated with the first academic variate, while self_concept is uncorrelated with the first variate but slightly correlated with the second variate. Based on our observations about these two variates in superscript v, we might interpret these correlations to mean that overall academic performance, especially reading and writing, are related to locus_of_control and motivation, while performance in science and gender may be related to self_concept.

x. Correlations Between the Academic Measurements and the Canonical Variables of the Psychological Measurements - Here, we see how the academic variables read, write, math, science and female are correlated with Psych1, Psych2 and Psych3 (a total of 5x3=15 correlations).   We see that the academic variables read, write, math and science are all correlated with Psych1, the first psychological variate strongly correlated with locus_of_control and motivation. This supports what we noted in superscript w about the possible relationship between overall academic performance and these two psychological variables.

#### Canonical Redundancy Analysis

Canonical Redundancy Analysis

Raw Variance of the Psychological Measurements Explained by
Their Own                               The Opposite
Canonical Variablesy                      Canonical Variablesz
Canonical
Variable                  Cumulative     Canonical                  Cumulative
Number    Proportion    Proportion      R-Square    Proportion    Proportion
1        0.3806        0.3806        0.2154        0.0820        0.0820
2        0.3126        0.6932        0.0281        0.0088        0.0908
3        0.3068        1.0000        0.0108        0.0033        0.0941

Raw Variance of the Academic Measurements Explained by
Their Own                               The Opposite
Canonical Variablesy                      Canonical Variablesz
Canonical
Variable                  Cumulative     Canonical                  Cumulative
Number    Proportion    Proportion      R-Square    Proportion    Proportion
1        0.6251        0.6251        0.2154        0.1346        0.1346
2        0.1704        0.7955        0.0281        0.0048        0.1394
3        0.0395        0.8350        0.0108        0.0004        0.1398

      Standardized Variance of the Psychological Measurements Explained by
Their Own                               The Opposite
Canonical Variablesaa                      Canonical Variablesbb
Canonical
Variable                  Cumulative     Canonical                  Cumulative
Number    Proportion    Proportion      R-Square    Proportion    Proportion
1        0.3798        0.3798        0.2154        0.0818        0.0818
2        0.2591        0.6389        0.0281        0.0073        0.0891
3        0.3611        1.0000        0.0108        0.0039        0.0930

Standardized Variance of the Academic Measurements Explained by
Their Own                               The Opposite
Canonical Variablesaa                      Canonical Variablesbb
Canonical
Variable                  Cumulative     Canonical                  Cumulative
Number    Proportion    Proportion      R-Square    Proportion    Proportion
1        0.5249        0.5249        0.2154        0.1130        0.1130
2        0.2499        0.7748        0.0281        0.0070        0.1201
3        0.0907        0.8655        0.0108        0.0010        0.1210

y. Raw Variance of the Psychological/Academic Measurements Explained by Their Own Canonical Variables - This is the degree to which the canonical variates of a group can explain the variability in the group's variables.  For example, we see here that the first canonical variate for the academic group explains 62.5% of the variability in the academic variables and the first canonical variate for the psychological group explains 38% of the variability in the psychological variables.

z. Raw Variance of the Psychological/Academic Measurements Explained by The Opposite Canonical Variables - This is the degree to which the canonical variates of a group can explain the variability in the other group's variables.  For example, we see here that the first canonical variate for the academic group explains 8.2% of the variability in the psychological variables and the first canonical variate for the psychological group explains 13.5% of the variability in the academic variables.

aa. Standardized Variance of the Psychological/Academic Measurements Explained by Their Own Canonical Variables - This is similar to superscript y, but performed on standardized data variables.

bb. Standardized Variance of the Psychological/Academic Measurements Explained by The Opposite Canonical Variables -This is similar to superscript z, but performed on standardized data variables.

#### Squared Multiple Correlations

              Squared Multiple Correlations Between the Psychological Measurements
and the First M Canonical Variables of the Academic Measurementscc
M                                                  1             2             3
LOCUS_OF_CONTROL      locus of control        0.1760        0.1803        0.1806
SELF_CONCEPT          self-concept            0.0001        0.0142        0.0196
MOTIVATION            motivation              0.0693        0.0727        0.0787

Squared Multiple Correlations Between the Academic Measurements and
the First M Canonical Variables of the Psychological Measurementscc
M                                      1             2             3
WRITE        writing score        0.1655        0.1656        0.1663
MATH         math score           0.1257        0.1282        0.1284
SCIENCE      science score        0.0934        0.1062        0.1068
FEMALE                            0.0286        0.0445        0.047

cc. Squared Multiple Correlations Between the Psychological/Academic Measurements and the First M Canonical Variables of the Psychological Measurements - Here, the correlations that were presented earlier between each variable in a given group and the canonical variates of the other group, are squared.  Each value is equivalent to the R-squared value in an OLS regression where we are predicting a single variable with a single variate or vice versa. For example, we saw earlier in the output that locus_of_control and Academic1 have a correlation of  0.4196.  We can calculate (0.4196*0.4196) = 0.1760, the squared correlation presented in this portion of the output. This means that 17.6% of the variability in locus_of_control can be explained by Academic1.

For more on the options available in cancorr and details on the underlying calculations, see the corresponding SAS documentation page.

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