This page shows an example of canonical correlation analysis with footnotes explaining the output in Stata. 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.
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. The researcher is interested in the relationship between the psychological variables and the academic variables, with gender considered as well. Canonical correlation analysis aims to find pairs of linear combinations of each group of variables that are highly correlated. These linear combinations are called canonical variates. Each canonical variate is orthogonal to the other canonical variates except for the one with which its correlation has been maximized. The possible number of such pairs is limited to the number of variables in the smallest group. In our example, there are three psychological variables and more than three academic variables. Thus, a canonical correlation analysis on these sets of variables will generate three pairs of canonical variates.
To begin, let’s read in the dataset.
use https://stats.idre.ucla.edu/stat/stata/dae/mmreg, clear
We can now proceed with our analysis. In Stata, canonical correlation analysis is conducted using the canon command. Each group of variables is enclosed in parenthesis. We are also interested in multivariate tests for dimensionality, so we will add a test option that will allow us to determine how many of the three generated canonical dimensions are needed to describe the relationship between our sets of variables. Using the stderr option will give us the standard errors and tests of significance for the raw coefficients of the canonical correlations.
canon (locus_of_control self_concept motivation)(read write math science female), test(1 2 3) stderr
Linear combinations for canonical correlations Number of obs = 600
----------------------------------------------------------------------------------
| Coef. Std. Err. t P>|t| [95% Conf. Interval]
-----------------+----------------------------------------------------------------
u1 |
locus_of_control | 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_control | -.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_control | -.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
----------------------------------------------------------------------------
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
Linear Combination Output
Linear combinations for canonical correlations Number of obsa = 600 ------------------------------------------------------------------------------ | Coef.b Std. Err.c td P>|t|e [95% Conf. Interval]f -------------+---------------------------------------------------------------- u1g | 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 -------------+---------------------------------------------------------------- v1h | 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 -------------+---------------------------------------------------------------- u2i | 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 -------------+---------------------------------------------------------------- v2j | 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 -------------+---------------------------------------------------------------- u3k | 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 -------------+---------------------------------------------------------------- v3l | 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 correlationsm: 0.4641 0.1675 0.1040
a. Number of obs – This is the number of observations in the dataset with valid data in all of the variables listed in the canon command. In this example, as we saw in the summary, we do not have any missing data in this dataset.
b. Coef. – These are the raw canonical coefficients. They define the linear relationship between the variables in a given group and the canonical variates (v1, u1, etc.). They can be interpreted in the same way you would interpret regression coefficients, assuming the canonical variate as the outcome variable. For example, a one unit increase in locus_of_control leads to a 1.253834 unit increase in u1.
c. Std. Err. – These are the standard errors associated with the raw canonical coefficients. The standard error is used for testing whether the coefficient is significantly different from 0 by dividing the coefficient estimate by the standard error to obtain a t-value (see superscript d). The standard errors can also be used to form a confidence interval for the coefficient (see superscript f).
d. t – These are the t-values used in testing the null hypothesis that the coefficient is 0. The t-values follow a t-distribution which is used to test against a two-sided alternative hypothesis that the Coef. is not equal to zero.
e. P>|t| – This is the probability the t test statistic (or a more extreme test statistic) would be observed under the null hypothesis that a particular variable’s canonical coefficient is zero, given that the rest of the variables in the set. For a given alpha level, P>|t| determines whether or not the null hypothesis can be rejected. If P>|t| is less than alpha, then the null hypothesis can be rejected and the canonical coefficient estimate is considered statistically significant at that alpha level.
f. [95% Conf. Interval] – This shows a 95% confidence interval for the canonical coefficient. This is very useful as it helps you understand how high and how low the actual population value of the canonical coefficient might be. The confidence intervals are related to the p-values such that the coefficient will not be statistically significant if the confidence interval includes 0.
g. u1 – This is the first canonical variate, or first dimension, for variable set 1. It is a linear combination of the psychological variables: locus of control, self-concept and motivation. It is calculated to be maximally correlated with v1 and orthogonal to all of the other canonical variates.
h. v1 – This is the first canonical variate, or first dimension, for variable set 2. It is a linear combination of the academic variables (reading, writing, math and science) and female. It is calculated to be maximally correlated with u1 and orthogonal to all of the other canonical variates.
i. u2 – This is the second canonical variate, or second dimension, for variable set 1. It is a linear combination of the psychological variables: locus of control, self-concept and motivation. It is calculated to be maximally correlated with v2 and orthogonal to all of the other canonical variates.
j. v2 – This is the second canonical variate, or second dimension, for variable set 2. It is a linear combination of the academic variables (reading, writing, math and science) and female. It is calculated to be maximally correlated with u2 and orthogonal to all of the other canonical variates.
k. u3 – This is the third canonical variate, or third dimension, for variable set 1. It is a linear combination of the psychological variables: locus of control, self-concept and motivation. It is calculated to be maximally correlated with v3 and orthogonal to all of the other canonical variates.
l. v3 – This is the third canonical variate, or third dimension, for variable set 2. It is a linear combination of the academic variables (reading, writing, math and science) and female. It is calculated to be maximally correlated with u3 and orthogonal to all of the other canonical variates.
m. Canonical correlation – These are the Pearson correlation coefficients of u1 and v1, u2 and v2, and u3 and v3, respectively. We can use the predict command after running canon to generate the canonical variates, and then examine the correlation matrix of these six variables to see these correlations.
predict u1, u corr(1)
predict v1, v corr(1)
predict u2, u corr(2)
predict v2, v corr(2)
predict u3, u corr(3)
predict v3, v corr(3)
corr u1 v1 u2 v2 u3 v3
| u1 v1 u2 v2 u3 v3
-------------+------------------------------------------------------
u1 | 1.0000
v1 | 0.4641 1.0000
u2 | -0.0000 0.0000 1.0000
v2 | -0.0000 0.0000 0.1675 1.0000
u3 | -0.0000 0.0000 -0.0000 -0.0000 1.0000
v3 | -0.0000 0.0000 -0.0000 0.0000 0.1040 1.0000
From this, we can see the non-zero correlations of the canonical variate pairs (u1,v1), (u2,v2) and (u3,v3) and the zero correlations of all other combinations of the canonical variates.
Significance Test Output
Tests of significance of all canonical correlationsn Statistico df1 df2p Fq Prob>Fr Wilks' lambdas .754361 15 1634.65 11.7157 0.0000 a Pillai's tracet .254249 15 1782 11.0006 0.0000 a Lawley-Hotelling traceu .314297 15 1772 12.3763 0.0000 a Roy's largest rootv .274496 5 594 32.6101 0.0000 u ---------------------------------------------------------------------------- Test of significance of canonical correlations 1-3w Statistic df1 df2 F Prob>F Wilks' lambda .754361 15 1634.65 11.7157 0.0000 a ---------------------------------------------------------------------------- Test of significance of canonical correlations 2-3x Statistic df1 df2 F Prob>F Wilks' lambda .96143 8 1186 2.9445 0.0029 e ---------------------------------------------------------------------------- Test of significance of canonical correlation 3y Statistic df1 df2 F Prob>F Wilks' lambda .989186 3 594 2.1646 0.0911 e ---------------------------------------------------------------------------- e = exact, a = approximate, u = upper bound on Fz
n. Tests of significance of all canonical correlations – By default, Stata tests all of the canonical dimensions together, listing four multivariate test statistics and their significance levels. Our null hypothesis is that our two sets of variables are not linearly related. We evaluate this hypothesis based on the p-values for the multivariate tests.
o. Statistic – This is the test statistic based on the multivariate statistic indicated in the prior column. We wish to test the hypothesis that our two sets of variables are not linearly related. Since our canonical variates are linear combinations of the sets, correlations between our canonical variates imply linear relationships between our sets of variables. Thus, these statistics are calculated from the canonical correlations.
p. df1 & df2 – These are the degrees of freedom used in determining the F statistic. Note that there are instances in manova when the degrees of freedom may be a non-integer (here, the df2 associated with Wilks’ lambda is a non-integer) because these degrees of freedom are calculated using the mean squared errors, which are often non-integers.
q. F – This is the F statistic for the given multivariate test. Note that the F statistic for Roy’s largest root is quite different that the other F statistics. For more details about each of the tests, see superscripts s, t, u, and v.
r. Prob>F – This is the p-value associated with the F statistic of a given test statistic. The null hypothesis that our two sets of variables are not linearly related is evaluated with regard to this p-value. For a given alpha level, if the p-value is less than alpha, the null hypothesis is rejected. If not, then we fail to reject the null hypothesis. In this example, we reject the null hypothesis that our two sets of variables are not linearly related at alpha level .05 because the p-values are all less than .05.
s. Wilks’ lambda – This is one of the four multivariate statistics calculated by Stata. Wilks’ lambda is the product of the values of (1-canonical correlation2). In this example, our canonical correlations are 0.4641, 0.1675, and 0.1040 so the Wilks’ Lambda testing all three of the correlations is (1- 0.46412)*(1-0.16752)*(1-0.10402) = 0.754361. For the dimensions 2-3, Wilks’ lambda is calculated using just 0.1675 and 0.1040: (1-0.16752)*(1-0.10402) = 0.96143. For dimension 3, Wilks’ lambda is (1-0.10402) = 0.989186.
t. Pillai’s trace – Pillai’s trace is another of the four multivariate statistics calculated by Stata. Pillai’s trace is the sum of the squared canonical correlations. It is only presented in the test of all three correlations: 0.46412 + 0.16752 + 0.10402 = .254249.
u. Lawley-Hotelling trace – This is very similar to Pillai’s trace. It is the sum of the values of (canonical correlation2/(1-canonical correlation2)). We can calculate 0.46412 /(1- 0.46412) + 16752/(1-0.16752) + 0.10402/(1-0.10402) = 0.314297.
v. Roy’s largest root – This is the square of the largest canonical correlation. Because it is based on a maximum, it can behave differently from the other three test statistics. In instances where the other three are not significant and Roy’s is significant, the effect should be considered not significant.
w. Test of significance of canonical correlations 1-3 – In addition to the tests given in the output produced without options, we specified additional tests in our command. The first test of dimensions again tests whether all three dimensions are significant. We see that, for this test, only one multivariate test statistic is listed (Wilks’ lambda), and it is redundant given the default test printed above. In this example, we reject the null hypothesis that our two sets of variables are not linearly related at alpha level .05 because the p-values are all less than .05.
x. Test of significance of canonical correlations 2-3 – Here, we test whether dimensions 2 and 3 combined are significant. We reject the null hypothesis at alpha level 0.05 because the p-value is less than .05.
y. Test of significance of canonical correlation 3 – Here, we test whether dimension 3, by itself, is significant. We fail to reject the null hypothesis at alpha level 0.05 because the p-value is greater than 0.05. Based on the results of the three tests, we can conclude that dimensions 1 and 2 must each be significant. This answers the researcher’s original question of how many dimensions are needed to describe the relationship between the two sets of variables.
z. e = exact, a = approximate, u = upper bound on F – This indicates how the F statistic was calculated (whether it was an exact calculation, an approximation, or an upper bound) for each of the multivariate tests.
For a further exploration of the options available in canon, see the corresponding Data Analysis Example page.