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Canonical correlation analysis (CCA) is a statistical method used to analyze relationships between two sets of variables. CCA finds linear combinations of variables from each set that have the highest correlation with each other. The first pair of linear combinations have maximum correlation, and each subsequent pair is orthogonal to previous pairs and has successively smaller correlations. CCA can be used to understand how variables from different tests relate to each other or to build models relating two sets of variables.
![Canonical correlation 2
Solution
The solution is therefore:
• is an eigenvector of
• is proportional to
Reciprocally, there is also:
• is an eigenvector of
• is proportional to
Reversing the change of coordinates, we have that
• is an eigenvector of
• is an eigenvector of
• is proportional to
• is proportional to
The canonical variables are defined by:
Implementation
CCA can be computed using singular value decomposition on a correlation matrix. It is available as a function in
• MATLAB as canoncorr
[1]
• R as cancor
[2]
or in FactoMineR
[3]
• SAS as proc cancorr
[4]
• Scikit-Learn,Python as Cross decomposition
[5]
Hypothesis testing
Each row can be tested for significance with the following method. Since the correlations are sorted, saying that row
is zero implies all further correlations are also zero. If we have independent observations in a sample and is
the estimated correlation for . For the th row, the test statistic is:
which is asymptotically distributed as a chi-squared with degrees of freedom for large
. Since all the correlations from to are logically zero (and estimated that way also) the product
for the terms after this point is irrelevant.
Practical uses
A typical use for canonical correlation in the experimental context is to take two sets of variables and see what is
common amongst the two sets. For example in psychological testing, you could take two well established
multidimensional personality tests such as the Minnesota Multiphasic Personality Inventory (MMPI-2) and the NEO.
By seeing how the MMPI-2 factors relate to the NEO factors, you could gain insight into what dimensions were
common between the tests and how much variance was shared. For example you might find that an extraversion or
neuroticism dimension accounted for a substantial amount of shared variance between the two tests.
One can also use canonical-correlation analysis to produce a model equation which relates two sets of variables, for
example a set of performance measures and a set of explanatory variables, or a set of outputs and set of inputs.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
