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. However, the amount of computational work involved is substantial and was relegated to a computer program for canonical analysis.
The matrix analyzed was the standardized data set for our example, presented in the following table.
with its corresponding matrix of Pearson’s product-moment correlations presented in the following table.
Let us return once again to our schematic outlay of the canonical analysis. Our inceptive structure is the supermatrix of inter-correlations between predictor and criterion variables, presented in the following table.
Using the canonical weights listed at the beginning of this section we can obtain two more sets of variables using equations
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.
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.
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.
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.
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.
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
For the example, the inter-variate structure is presented in the following table.
together with its associated row and column marginal referents and communalities.
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
For the example, the intra-variate structure is presented in the following table.
together with its row and column marginal referents and associated communalities.
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.
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
