**Version info**: Code for this page was tested in Stata 12.

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

For our analysis example, we are going to expand example 1 about investigating the associations between psychological measures and academic achievement measures.

We have a data file, **mmreg.dta**, with 600 observations on eight variables.
The psychological variables are **locus of control**, **self-concept** and
**motivation**. The academic variables are standardized tests in
reading** **(**read**), writing (**write**),** **math (**math**) and science
(**science**). Additionally,
the variable **female** is a zero-one indicator variable
with the one indicating a female student.

Let’s look at the data.

use http://www.ats.ucla.edu/stat/stata/dae/mmreg, clear summarize locus_of_control self_concept motivationVariable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- locus_of_c~l | 600 .0965333 .6702799 -2.23 1.36 self_concept | 600 .0049167 .7055125 -2.62 1.19 motivation | 600 .6608333 .3427294 0 1summarize read write math science femaleVariable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- read | 600 51.90183 10.10298 28.3 76 write | 600 52.38483 9.726455 25.5 67.1 math | 600 51.849 9.414736 31.8 75.5 science | 600 51.76333 9.706179 26 74.2 female | 600 .545 .4983864 0 1

## Analysis methods you might consider

Below is a list of some analysis methods you may have encountered. Some of the methods listed are quite reasonable while others have either fallen out of favor or have limitations.

- 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.

## Canonical correlation analysis

Below we use the **canon** command to conduct a canonical correlation
analysis. It requires two sets of variables enclosed with a pair of
parentheses. We specify our psychological variables as the first set of
variables and our academic variables plus gender as the second set. For
convenience, the variables in the first set are called "u" variables and the
variables in the second set are called "v" variables.

canon (locus_of_control self_concept motivation)(read write math science female)Canonical correlation analysis Number of obs = 600 Raw coefficients for the first variable set | 1 2 3 -------------+------------------------------ locus_of_c~l | 1.2538 -0.6215 -0.6617 self_concept | -0.3513 -1.1877 0.8267 motivation | 1.2624 2.0273 2.0002 -------------------------------------------- Raw coefficients for the second variable set | 1 2 3 -------------+------------------------------ read | 0.0446 -0.0049 0.0214 write | 0.0359 0.0421 0.0913 math | 0.0234 0.0042 0.0094 science | 0.0050 -0.0852 -0.1098 female | 0.6321 1.0846 -1.7946 -------------------------------------------- ---------------------------------------------------------------------------- Canonical correlations: 0.4641 0.1675 0.1040 ---------------------------------------------------------------------------- Tests of significance of all canonical correlations Statistic df1 df2 F Prob>F Wilks' lambda .754361 15 1634.65 11.7157 0.0000 a Pillai's trace .254249 15 1782 11.0006 0.0000 a Lawley-Hotelling trace .314297 15 1772 12.3763 0.0000 a Roy's largest root .274496 5 594 32.6101 0.0000 u ---------------------------------------------------------------------------- e = exact, a = approximate, u = upper bound on F

The output for canonical correlation analysis is made up of two parts. First is the raw canonical coefficients. The second part begins with the canonical correlations and includes the overall multivariate tests for dimensionality.

The raw canonical coefficients can be used to generate the canonical variates,
represented by the columns (1 2 3) in the coefficient tables,
for each set. They 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 the "v" set when all of the
other variables are held constant. Here is another example: being female leads
to a .6321 increase in the dimension 1 for the "v" set with the other predictors held constant.

The number of possible canonical variates, also known as canonical dimensions, is
equal to the number of variables in the smaller set. In our example, the "u"
set (the first set) has three variables and the "v" set (the second set)
has five. This leads to three possible canonical variates for each set,
which corresponds to the three columns for each set and three canonical
correlation coefficients in the output. Canonical dimensions are latent variables that are analogous to factors obtained in factor analysis,
except that canonical variates also maximize the correlation between the two
sets of variables. In general,
not all the canonical dimensions would be statistically significant. A
significant dimension corresponds to a significant canonical correlation and
vice versa. To test if a canonical correlation is statistically different from zero, we
can use the **test** option in **canon** command as shown below. We don’t
need to rerun the model, instead we just ask Stata to redisplay the model with
additional information on the requested tests. In order to test all the
canonical dimensions, we need to specify **test(1 2 3)**. Essentially **
test(1)** is the overall test on three dimensions, **test(2)** will test
the significance of canonical correlations 2 and 3, and **test(3)** will test
the significance of the third canonical correlation alone.

canon, test(1 2 3)(some output is omitted) ---------------------------------------------------------------------------- Tests of significance of all canonical correlations Statistic df1 df2 F Prob>F Wilks' lambda .754361 15 1634.65 11.7157 0.0000 a Pillai's trace .254249 15 1782 11.0006 0.0000 a Lawley-Hotelling trace .314297 15 1772 12.3763 0.0000 a Roy's largest root .274496 5 594 32.6101 0.0000 u ---------------------------------------------------------------------------- Test of significance of canonical correlations 1-3 Statistic df1 df2 F Prob>F Wilks' lambda .754361 15 1634.65 11.7157 0.0000 a ---------------------------------------------------------------------------- Test of significance of canonical correlations 2-3 Statistic df1 df2 F Prob>F Wilks' lambda .96143 8 1186 2.9445 0.0029 e ---------------------------------------------------------------------------- Test of significance of canonical correlation 3 Statistic df1 df2 F Prob>F Wilks' lambda .989186 3 594 2.1646 0.0911 e ---------------------------------------------------------------------------- e = exact, a = approximate, u = upper bound on F

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 combined are significant (they are), the next test tests whether dimensions 2 and 3 combined are significant (they are). Finally, the last test tests whether dimension 3, by itself, is significant (it is not). Therefore dimensions 1 and 2 must each be significant.

Now, we might want to inspect what raw coefficients for each of the canonical variates are
significant. We can request the standard errors and significant tests via **
stderr** option.

canon, stderrLinear combinations for canonical correlations Number of obs = 600 ------------------------------------------------------------------------------ | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- u1 | locus_of_c~l | 1.253834 .1210229 10.36 0.000 1.016153 1.491515 self_concept | -.3513499 .116424 -3.02 0.003 -.5799987 -.1227012 motivation | 1.26242 .2435532 5.18 0.000 .7840983 1.740742 -------------+---------------------------------------------------------------- v1 | read | .0446206 .0122741 3.64 0.000 .0205152 .068726 write | .0358771 .0122944 2.92 0.004 .0117318 .0600224 math | .0234172 .0127339 1.84 0.066 -.0015914 .0484258 science | .0050252 .0122762 0.41 0.682 -.0190845 .0291348 female | .6321192 .1747222 3.62 0.000 .2889767 .9752618 -------------+---------------------------------------------------------------- u2 | locus_of_c~l | -.6214775 .3731786 -1.67 0.096 -1.354375 .11142 self_concept | -1.187687 .3589975 -3.31 0.001 -1.892733 -.4826399 motivation | 2.027264 .7510053 2.70 0.007 .5523406 3.502187 -------------+---------------------------------------------------------------- v2 | read | -.00491 .0378475 -0.13 0.897 -.07924 .0694199 write | .0420715 .0379101 1.11 0.268 -.0323814 .1165244 math | .0042295 .0392656 0.11 0.914 -.0728854 .0813444 science | -.0851622 .0378541 -2.25 0.025 -.1595052 -.0108192 female | 1.084642 .5387622 2.01 0.045 .02655 2.142735 -------------+---------------------------------------------------------------- u3 | locus_of_c~l | -.6616896 .6064262 -1.09 0.276 -1.85267 .5292904 self_concept | .8267209 .5833814 1.42 0.157 -.3190007 1.972443 motivation | 2.000228 1.220406 1.64 0.102 -.3965655 4.397022 -------------+---------------------------------------------------------------- v3 | read | .0213806 .0615033 0.35 0.728 -.0994078 .1421689 write | .0913073 .0616051 1.48 0.139 -.0296808 .2122955 math | .0093982 .0638077 0.15 0.883 -.1159158 .1347122 science | -.109835 .0615141 -1.79 0.075 -.2306445 .0109745 female | -1.794647 .8755045 -2.05 0.041 -3.514078 -.0752155 ------------------------------------------------------------------------------ (Standard errors estimated conditionally) Canonical correlations: 0.4641 0.1675 0.1040 ---------------------------------------------------------------------------- Tests of significance of all canonical correlations Statistic df1 df2 F Prob>F Wilks' lambda .754361 15 1634.65 11.7157 0.0000 a Pillai's trace .254249 15 1782 11.0006 0.0000 a Lawley-Hotelling trace .314297 15 1772 12.3763 0.0000 a Roy's largest root .274496 5 594 32.6101 0.0000 u ---------------------------------------------------------------------------- e = exact, a = approximate, u = upper bound on F

Note that for the first dimension all of the variables except for **math** and
**science**
are statistically significant along with the dimension as a whole. Thus, l**ocus
of control**, **self-** **concept**, and **motivation** share some
variability with one another, as well as with **read**, **write**, and **
female**, which also share variablity among each other. For the second
dimension only **self-concept**, **motivation**, **math** and **female** are significant. The third
dimension is not significant and no attention will be paid to its coefficients or
to the Wald tests.

When the variables in the model have very different standard deviations, the standardized coefficients allow for easier comparisons among the variables. Next we’ll display the standardized canonical coefficients for the first two (significant) dimensions.

canon (locus_of_control self_concept motivation)(read write math science female), first(2) stdcoef notestCanonical correlation analysis Number of obs = 600 Standardized coefficients for the first variable set | 1 2 -------------+-------------------- locus_of_c~l | 0.8404 -0.4166 self_concept | -0.2479 -0.8379 motivation | 0.4327 0.6948 ---------------------------------- Standardized coefficients for the second variable set | 1 2 -------------+-------------------- read | 0.4508 -0.0496 write | 0.3490 0.4092 math | 0.2205 0.0398 science | 0.0488 -0.8266 female | 0.3150 0.5406 ---------------------------------- Canonical correlations: 0.4641 0.1675 0.1040

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.

Next, we’ll use the **estat correlations** command to look at all of the correlations
within and between sets of variables.

estat correlationsCorrelations for variable list 1 | locus_~l self_c~t motiva~n -------------+------------------------------ locus_of_c~l | 1.0000 self_concept | 0.1712 1.0000 motivation | 0.2451 0.2886 1.0000 -------------------------------------------- Correlations for variable list 2 | read write math sci female -------------+-------------------------------------------------- read | 1.0000 write | 0.6286 1.0000 math | 0.6793 0.6327 1.0000 science | 0.6907 0.5691 0.6495 1.0000 female | -0.0417 0.2443 -0.0482 -0.1382 1.0000 ---------------------------------------------------------------- Correlations between variable lists 1 and 2 | locus_~l self_c~t motiva~n -------------+------------------------------ read | 0.3736 0.0607 0.2106 write | 0.3589 0.0194 0.2542 math | 0.3373 0.0536 0.1950 science | 0.3246 0.0698 0.1157 female | 0.1134 -0.1260 0.0981 --------------------------------------------

Finally, we’ll use the **estat loadings** command to display the
loadings of the variables on the canonical dimensions (variates). These
loadings are correlations between variables and the canonical variates.

estat loadingsCanonical loadings for variable list 1 | 1 2 -------------+-------------------- locus_of_c~l | 0.9040 -0.3897 self_concept | 0.0208 -0.7087 motivation | 0.5672 0.3509 ---------------------------------- Canonical loadings for variable list 2 | 1 2 -------------+-------------------- read | 0.8404 -0.3588 write | 0.8765 0.0648 math | 0.7639 -0.2979 science | 0.6584 -0.6768 female | 0.3641 0.7549 ---------------------------------- Correlation between variable list 1 and canonical variates from list 2 | 1 2 -------------+-------------------- locus_of_c~l | 0.4196 -0.0653 self_concept | 0.0097 -0.1187 motivation | 0.2632 0.0588 ---------------------------------- Correlation between variable list 2 and canonical variates from list 1 | 1 2 -------------+-------------------- read | 0.3900 -0.0601 write | 0.4068 0.0109 math | 0.3545 -0.0499 science | 0.3056 -0.1134 female | 0.1690 0.1265 ----------------------------------

## 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.

## See also

- Stata 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.