### SAS Data Analysis Examples Canonical Correlation Analysis

#### Examples of Canonical Correlation Analysis

Canonical correlation analysis is used to identify and measure the associations among two sets of variables.  Canonical correlation is appropriate in the same situations where multiple regression would be, but where are there are multiple intercorrelated outcome variables. Canonical correlation analysis determines a set of canonical variates, orthogonal linear combinations of the variables within each set that best explain the variability both within and between sets.

Please Note: The purpose of this page is to show how to use various data analysis commands. It does not cover all aspects of the research process which researchers are expected to do. In particular, it does not cover data cleaning and checking, verification of assumptions, model diagnostics and potential follow-up analyses.

#### Examples of canonical correlation analysis

Example 1. 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 (canonical variables) are necessary to understand the association between the two sets of variables.

Example 2. A researcher is interested in exploring associations among factors from two multidimensional personality tests, the MMPI and the NEO. She is interested in what dimensions are common between the tests and how much variance is shared between them. She is specifically interested in finding whether the neuroticism dimension from the NEO can account for a substantial amount of shared variance between the two tests.

#### Description of the Data

Let's pursue Example 1 from above.

We have included the data file, which can be obtained by clicking on mmreg.sas7bdat. The dataset has 600 observations on eight variables. The psychological variables are locus of control, self-concept and motivation. The academic variables are standardized tests in reading, writing, math and science. Additionally, the variable female is a zero-one indicator variable with the one indicating a female student.

Let's look at the data.


proc means data=mylib.mmreg;
run;

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

proc freq data=mylib.mmreg;
table female;
run;

The FREQ Procedure

Cumulative    Cumulative
FEMALE    Frequency     Percent     Frequency      Percent
-----------------------------------------------------------
0         273       45.50           273        45.50
1         327       54.50           600       100.00

We did not include correlations among the variables at this point because we will get them later as part of the canonical correlation analysis.

#### Analysis methods you might consider

Before we show how you can analyze this with a canonical correlation analysis, let's consider some other methods that you might use.
• Separate OLS Regressions - You could analyze these data using separate OLS regression analyses for each variable in one set. The OLS regressions will not produce multivariate results and does not report information concerning dimensionality.
• Multivariate multiple regression is a reasonable option if you have no interest in dimensionality.

#### Canonical correlation analysis

Due to the length of the output, we will be making comments in several places along the way.

proc cancorr corr data=mylib.mmreg;
var locus_of_control self_concept motivation;
with read write math science female;
run;


The corr option on the proc cancorr statement produces correlations within and between the two sets of variables are given below.

                                      The CANCORR Procedure

Correlations Among the Original Variables

Correlations Among the VAR Variables

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 WITH Variables

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 VAR Variables and the WITH Variables

LOCUS_OF_CONTROL          0.3736          0.3589          0.3373          0.3246          0.1134
SELF_CONCEPT              0.0607          0.0194          0.0536          0.0698         -0.1260
MOTIVATION                0.2106          0.2542          0.1950          0.1157          0.0981

The SAS System          10:43 Tuesday,


The output below gives the three canonical correlations and the multivariate tests of the dimensions. These results show that the first two of the three canonical correlations are statistically significant. The output also includes the four multivariate criteria and the F approximations.

                                      The CANCORR Procedure

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

Test of H0: The canonical correlations in
Eigenvalues of Inv(E)*H           the current row and all that follow are zero
= CanRsq/(1-CanRsq)
Likelihood Approximate
Eigenvalue Difference Proportion Cumulative      Ratio     F Value Num DF Den DF Pr > F

1     0.2745     0.2456     0.8734     0.8734 0.75436113       11.72     15 1634.7 <.0001
2     0.0289     0.0179     0.0919     0.9652 0.96142996        2.94      8   1186 0.0029
3     0.0109                0.0348     1.0000 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.

In general, the number of canonical dimensions is equal to the number of variables in the smaller set; however, the number of significant dimensions may be even smaller. Canonical dimensions, also known as canonical variates, are similar to latent variables that are found in factor analysis, except that canonical variates also maximize the correlation between the two sets of variables.  For this particular model there are three canonical dimensions of which only the first two are statistically significant. The first test of dimensions tests whether all three dimensions are significant (F = 11.72), the next test tests whether dimensions 2 and 3 combined are significant (F = 2.94). Finally, the last test tests whether dimension 3, by itself, is significant (F = 2.16). Therefore dimensions 1 and 2 are each significant while the third dimension is not.

Next, the raw canonical coefficients are shown below. The raw canonical coefficients are interpreted in a manner analogous to interpreting regression coefficients i.e., for the variable read, a one unit increase in reading leads to a .0446 increase in the first canonical variate of set 2 when all of the other variables are held constant. Here is another example: being female leads to a .6321 increase in dimension 1 for set 2 with the other predictors held constant.

                         Raw Canonical Coefficients for the VAR Variables

V1                V2                V3

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 WITH Variables

W1                W2                W3

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

The raw coefficients are followed by the standardized canonical coefficients shown below.  When the variables in the model have very different standard deviations, the standardized coefficients allow for easier comparisons among the variables.  The standardized canonical coefficients are interpreted in a manner analogous to interpreting standardized regression coefficients. For example, consider the variable read, a one standard deviation increase in reading leads to a 0.45 standard deviation increase in the score on the first canonical variate for set 2 when the other variables in the model are held constant.

                    Standardized Canonical Coefficients for the VAR Variables

V1            V2            V3

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 WITH Variables

W1            W2            W3

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

Below are correlations between observed variables and canonical variables which are known as the canonical loadings, which SAS labels as the canonical structure.

                                       Canonical Structure

Correlations Between the VAR Variables and Their Canonical Variables

V1            V2            V3

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 WITH Variables and Their Canonical Variables

W1            W2            W3

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 VAR Variables and the Canonical Variables of the WITH Variables

W1            W2            W3

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 WITH Variables and the Canonical Variables of the VAR Variables

V1            V2            V3

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


#### Things to consider

As in the case of multivariate regression, MANOVA and so on, for valid inference, canonical correlation analysis requires the multivariate normal and homogeneity of variance assumption. Canonical correlation analysis assumes a linear relationship between the canonical variates and each set of variables. Similar to multivariate regression, canonical correlation analysis requires a large sample size.

• SAS Online Manual

#### References

• Afifi, A, Clark, V and May, S. 2004. Computer-Aided Multivariate Analysis. 4th ed. Boca Raton, Fl: Chapman & Hall/CRC.
• Garson, G. David (2015). GLM Multivariate, MANOVA, and Canonical Correlation. Asheboro, NC: Statistical Associates Publishers.
• G. David Garson, Canonical Correlation in Statnotes:  Topics in Multivariate Analysis
• Pedhazur, E. 1997. Multiple Regression in Behavioral Research. 3rd ed. Orlando, Fl: Holt, Rinehart and Winston, Inc.
•

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