Canonical Structures

The understanding of canonical analysis is facilitated by awareness of the basic relationships between the component canonical structures. Initially, let us consider the canonical weights C for predictor and D for criterion variables obtained from a computer analysis of the data for our example. The canonical weights are analogous to the beta weights of regression analysis. The notational convention using C to denote the set of canonical standardized eigenvectors for the predictor set and D to denote the set of standardized canonical eigenvectors for the criterion set of variables uses an alphabetic order to stress the progression from the multiple regression to the canonical analysis. For our example, the canonical weights are presented in the following table.

 

 

In the previous discussion we have shown only how to obtain the first normalized eigenvector [ .50 .62] from the above sets of predictor and criterion normalized eigenvectors. The remaining standardized eigenvectors could have been obtained in the similar way. Using these canonical weights, we can obtain the canonical variates U and V as

 

and

 

For the example, the predictor and criterion sets of canonical variates were calculated as presented in the following table.

 

      

 

The complete solution thus can be schematized as presented in the following table.

 

 

The structures, described by submatrices of the supermatrix of correlations in the above table can be classified into several categories.

Intra-correlations

Submatrices  and  contain correlations between variables within the predictor set and correlations between variables within the criterion set. These matrices can be computed as

 

and

 

For the example, the matrix of inter-correlations between the predictor variables equals

 

 

and the matrix of inter-correlations between criterion variables is

 

 

 

The above matrices, together with the matrices in the following set are incipient to the canonical analysis proper.

Inter-correlations

Submatrices  and  contain correlations between variables of predictor and criterion sets. In formal matrix notation

 

and

 

The  matrix is the transpose of the  matrix. For the example, the matrixes of inter-correlations between predictor and criterion sets of variables were computed as

 

 

for the  matrix and as

 

 

for its transpose.

 

Identity Matrices

To facilitate discussion within this and the following sections, remember that matrix multiplication of symmetric matrices is commutative. As discussed earlier, ipsative correlations of canonical variates equal unity. Since canonical variates are orthogonal, ipsative products of matrices of canonical variates divided by n are identity matrices, i.e.,

 

and

 

For the example used, these matrices will be

 

 

In general, the order of the  matrix will be equal to the number of canonical variates in the predictor set and the order of the  matrix will equal the number of canonical variates of the criterion set.

Matrices of Canonical Correlations

Canonical analysis is a procedure extracting dimensions of the predictor and criterion sets of variables and maximizing canonical correlations between corresponding dimensions of the predictor and criterion sets. The dimensions of each set are extracted as orthogonal with respect to each other. Thus, the matrix of canonical correlations between dimensions of the predictor and criterion sets

 

and the  matrix

 

 

are identical, containing canonical correlations o in its principal diagonal and zeroes in the off-diagonal elements. For the example,

 

 

the canonical correlations corresponding to the first () and second () eigenvalue are located along the principal diagonal.

Inter-variate Structure

The  and  matrices contain correlations between canonical variates and the original variables from the opposite set. This inter-variate structure can be computed from matrices of cross-correlations and matrices of canonical weights as

 

and

 

 

Intra-variate Structure

Intra-variate structure contains correlations between predictor variables and canonical variates extracted from the predictor set

 

 

and the criterion variables and the canonical variates extracted from the criterion set

 

Canonical  Structure

The global canonical structure can be obtained by inter-correlating variables within the X, Y, U and V sets of variables as presented in the following table.

 

 

or, in the formal notation

 

                                                                     

 

 

The submatrices of the above supermatrix can be compared with the canonical structures discussed in the preceding section. The understanding of the global structure of the canonical analysis is necessary for the interpretation of results.