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Canonical Correlation Analysis

**Version info: **Code for this page was tested in SAS 9.3.

**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.

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.

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 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 -----------------------------------------------------------------------------------------------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.

- Canonical correlation analysis, the focus of this page.
- 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.

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 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 VAR Variables and the WITH Variables READ WRITE MATH SCIENCE FEMALE 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 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 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 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

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 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

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 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 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 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

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

- 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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