Performs canonical correlation analysis from a variance‑covariance matrix or a correlation matrix.

Routine CANVC computes the canonical correlations, the canonical coefficients, Wilks’ lambda (for testing the independence of two sets of variates), and a series of tests due to Bartlett for testing that all canonical correlations greater than or equal to the k‑th largest are simultaneously zero. The covariance matrix is used in these computations.

The group 1 variables covariance matrix is first extracted from COV and placed in the matrix S11. Similarly, the group 2 variables covariance matrix is placed in S22. The “standardized” cross covariance matrix is then computed as:

where S12 is the NVAR1 × NVAR2 matrix of covariances between the group 1 and group 2 variables, and S1∕2denotes the upper triangular Cholesky (RT R) factorization of S. In the computation of C and in the following, it is assumed that NVAR1 is greater than NVAR2. The group 1 and group 2 variables should be interchanged in the following if this is not the case.

The canonical correlations are computed as the singular values of the matrix C. The canonical coefficients are obtained from the left and right orthogonal matrices resulting from the singular value decomposition of C. In particular, for Γ1 = COEF1.

where L is the left orthogonal matrix from the singular value decomposition.

Similarly, the correlations between the original variables and the canonical variables, R1 = COEFR1, are obtained for the group 1 variables as:

where Δ11 is a diagonal matrix containing the diagonal of S11 along its diagonal.

Wilks’ lambda, the Bartlett’s tests, Rao’s F corresponding to these tests, the numerator and denominator degrees of freedom of F , and the significance level of F are computed as in Rao (1973, page 556). Bartlett’s tests are computed as

where q = NVAR2 is the number of canonical correlations, the canonical correlations are ordered from largest to smallest, and ρj denotes the j‑th largest canonical correlation. Wilks’ lambda is given as Λ1. The degrees of freedom in the numerator of the corresponding Rao’s F statistic is given as

where p = v1 ‑ i + 1, u = v2 ‑ i + 1, v1 = NVAR2, and v2 = NVAR1. Let

where t is the degrees of freedom in COV, and let

if p2 + u2 ‑ 5 ≠ 0, and let s = 2 otherwise. Then, Rao’s F corresponding to Bartlett’s test is computed as

Rao’s F has numerator degrees of freedom d2 = ms ‑ pu/2 + 1. The significance level of F is obtained from the standard F distribution

The following example is taken from Van de Geer (1971). There are six group 1 variables and two group 2 variables. The maximum correlation turns out to be 0.609.