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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 https://stats.idre.ucla.edu/stat/stata/dae/mmreg, clear

summarize locus_of_control self_concept motivation

    Variable |       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          1

summarize read write math science female

    Variable |       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

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, stderr

Linear 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, locus
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, science 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 notest
Canonical 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 correlations

Correlations 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 loadings

Canonical 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

See also

References