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Stata Data Analysis Examples
Canonical Correlation Analysis

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 are necessary to understand the association between the two sets of variables.

Description of the Data

Let's pursue Example 1 from above.

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, 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.

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

Some Strategies You Might Be Tempted To Try

Before we show how you can analyze this with a canonical correlation analysis, let's consider some other methods that you might use.

Stata Canonical Correlation Analysis

Note: Users of Stata 10.1 will need to add the stderr option to get output that looks the same as above. Here is the complete Stata 10.1 command:
canon (locus_of_control self_concept motivation)(read write math science female), test(1 2 3) stderr

The output for canonical correlation analysis is long and complex, and it is made up of two parts. First is the raw canonical coefficients with standard errors, Wald t-tests, p-values and confidence intervals for the raw coefficients. The second part begins with the canonical correlations and includes the multivariate tests for dimensionality. Stata is fairly unique among statistics packages in giving standard errors and Wald tests for the raw canonical coefficients.

The output for the coefficients is further divided into three sections, one for each of the canonical dimensions. Within each of the dimensions there are the "u" variables, variables in the first set, which for this example are the psychological variables. There are also the "v" variables, variables in the second set, in this case, the academic variables plus gender.

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 latent variables that are analogous to factors obtained in factor analysis. 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 (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.

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 the dimension 1 for set 2 with the other predictors held constant.

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. 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 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.

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.

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.

Sample Write-Up of the Analysis

There is a lot of variation in the write-ups of canonical correlation analyses. The write-up below is fairly minimal, including only the tests of dimensionality and the standardized coefficients. Typically, one does not include raw coefficients with standard errors and Wald tests of significance.

Tests of dimensionality for the canonical correlation analysis, as shown in Table 1, indicate that two of the three canonical dimensions are statistically significant at the .05 level. Dimension 1 had a canonical correlation of 0.46 between the sets of variables, while for dimension 2 the canonical correlation was much lower at 0.17.

Table 2 presents the standardized canonical coefficients for the first two dimensions across both sets of variables. For the psychological variables, the first canonical dimension is most strongly influenced by locus of control (.84) and for the second dimension self-concept (-.84) and motivation (.69). For the academic variables plus gender, the first dimension was comprised of reading (.45), writing (.35) and gender (.32). For the second dimension writing (.41), science (-.83) and gender (.54) were the dominating variables.

Cautions, Flies in the Ointment

  • Multivatiate normal distribution assumptions are required for both sets of variables.
  • Canonical correlation analysis is not recommended for small samples.

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