# Multivariate Analysis: Canonical Correlation Analysis

## Overview of Canonical Correlation Analysis

Canonical correlation analysis is a technique for analyzing the relationship between two sets (or groups) of variables. Each set can contain multiple variables.

Given two sets of variables, canonical correlation analysis finds a linear combination from each set, called a canonical variable, such that the correlation between the two canonical variables is maximized. This correlation between the two canonical variables is the first canonical correlation. The first canonical correlation is at least as large as the multiple correlation between any variable and the opposite set of variables. The coefficients of the linear combinations are canonical coefficients or canonical weights. It is customary to normalize the canonical coefficients so that each canonical variable has a variance of 1.

Canonical correlation analysis continues by finding a second set of canonical variables, uncorrelated with the first pair, that produces the second-highest correlation coefficient. The process of constructing canonical variables continues until the number of pairs of canonical variables equals the number of variables in the smaller group.

Each canonical variable is uncorrelated with all the other canonical variables of either set except for the one corresponding canonical variable in the opposite set. However, the canonical variables do not represent jointly perpendicular directions through the space of the original variables.

You can run the Canonical Correlation analysis by selecting AnalysisMultivariate AnalysisCanonical Correlation Analysis from the main menu. The analysis is implemented by calling the CANCORR procedure in SAS/STAT software. See the CANCORR procedure documentation in the SAS/STAT User's Guide for additional details.