Canonical Correlation

What is canonical Correlation?

Canonical correlation analysis is a multivariate statistical model that facilitates the study of linear interrelationships between two sets of variables. One set of variables is referred to as independent variables and the others are considered dependent variables a canonical variate is formed for each set.

It may be helpful to think of a canonical variate as being like the variate (i.e., linear composite) formed from the set of independent variables in a multiple regression analysis. But in canonical correlation there is also a variate formed from several dependent variables whereas multiple regression can accommodate only one dependent variable. 

Canonical correlation analysis develops a canonical function that maximizes the canonical correlation coefficient between the two canonical variates. The canonical correlation coefficient measures the strength of the relationship between the two canonical variates. Each canonical variate is interpreted with canonical loadings, the

correlation of the individual variables and their respective variates. Canonical loadings are similar to the factor loadings of each variable and the factors that were described in factor analysis (see Chapter 3 in the text). So in a sense this is analogous to estimating a separate factor for each set of variables in order to maximize the correlation between factors. 

Another unique characteristic of canonical correlation is that it develops multiple canonical functions. Each canonical function is independent (orthogonal) from the other canonical functions so that they represent different relationships found among the sets of dependent and independent variables. The canonical loadings of the individual variables differ in each canonical function and represent that variable’s contribution to the specific relationship being depicted. 

Extending the simple example above, each canonical function consists of a different pair of variates (one for the independent variables and the other for the dependent variables), each function representing a different relationship between the sets of variables. The researcher retains and interprets all of the statistically significant canonical functions. Here we see the development of different canonical functions as somewhat analogous to the discriminant functions in discriminant analysis in which each represents a different dimension of discrimination in the dependent variable (see text for more discussion of discriminant analysis).

(Pre published - under editing process)