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, http://stats.idre.ucla.edu/wp-content/uploads/2016/02/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 READ reading score 600 51.9018334 10.1029830 28.2999992 76.0000000 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' wprefix=Academic wname='Academic 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 Correlations Among the Academic Measurements READ WRITE MATH SCIENCE FEMALE 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 and the Academic Measurements READ WRITE MATH 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 and the Academic Measurements SCIENCE FEMALE LOCUS_OF_CONTROL 0.3246 0.1134 SELF_CONCEPT 0.0698 -0.1260 MOTIVATION 0.1157 0.0981 -------------------------------------------------------------------------------- Canonical Correlation Analysis Adjusted Approximate Squared 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 Academic1 Academic2 Academic3 READ reading score 0.0446205959 0.0049100176 0.0213805581 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 Academic1 Academic2 Academic3 READ reading score 0.4508 0.0496 0.2160 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 Academic1 Academic2 Academic3 READ reading score 0.8404 0.3588 0.1354 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 Academic1 Academic2 Academic3 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 READ reading score 0.3900 0.0601 0.0141 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 READ reading score 0.1521 0.1557 0.1559 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 MeasurementsLOCUS_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 Correlations Among the Academic Measurements^{a}^{b}READ WRITE MATH SCIENCE FEMALE 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 and the Academic Measurements^{c}READ WRITE MATH 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 and the Academic 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 3×3.
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 5×5. 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 CorrelationCorrelation^{d}^{e}Error^{f}Correlation^{g}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^{h}Difference^{i}Proportion^{j}Cumulative^{k}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^{l}F Value^{m}Num DF Den DF^{n}Pr > F^{o}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^{m}Num DF Den DF^{n}Pr > F^{o}Wilks' Lambda^{p}0.75436113 11.72 15 1634.7 <.0001 Pillai's Trace^{q}0.25424936 11.00 15 1782 <.0001 Hotelling-Lawley Trace^{r}0.31429738 12.38 15 1113 <.0001 Roy's Greatest Root^{s}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 correlation^{2}). 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- 0.4641^{2})*(1-0.1675^{2})*(1-0.1040^{2})
= 0.754361. To test that the two smaller canonical correlations, 0.1675
and 0.1040, are zero in the population, the likelihood is (1-0.1675^{2})*(1-0.1040^{2})
= 0.96143. The likelihood that the smallest canonical correlation is zero is (1-0.1040^{2}) = 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 correlation^{2}). 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- 0.4641^{2})*(1-0.1675^{2})*(1-0.1040^{2})
= 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: 0.4641^{2} + 0.1675^{2} + 0.1040^{2} =
0.25424936.

r. **
Hotelling-Lawley Trace** – This is very similar to Pillai’s trace. It is the sum
of the values of (canonical correlation^{2}/(1-canonical correlation^{2})). We can calculate 0.4641^{2
}/(1- 0.4641^{2}) + 0.1675^{2}/(1-0.1675^{2})
+ 0.1040^{2}/(1-0.1040^{2}) = 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 MeasurementsPsych1 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^{t}

Raw Canonical Coefficients for the Academic Measurements^{t}Academic1 Academic2 Academic3 READ reading score 0.0446205959 0.0049100176 0.0213805581 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 Measurements^{u}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^{u}Academic1 Academic2 Academic3 READ reading score 0.4508 0.0496 0.2160 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 VariablesPsych1 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^{v}

Correlations Between the Academic Measurements and Their Canonical Variables^{v}Academic1 Academic2 Academic3 READ reading score 0.8404 0.3588 0.1354 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^{w}Academic1 Academic2 Academic3 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^{x}Psych1 Psych2 Psych3 READ reading score 0.3900 0.0601 0.0141 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 3×3=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 5×3=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 VariablesCanonical Variables^{y}^{z}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^{y}Canonical Variables^{z}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 Variables^{aa}Canonical Variables^{bb}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^{aa}Canonical Variables^{bb}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 MeasurementsM 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^{cc}^{cc}M 1 2 3 READ reading score 0.1521 0.1557 0.1559 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.