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 The CANDISC Procedure

## Computational Details

### General Formulas

Canonical discriminant analysis is equivalent to canonical correlation analysis between the quantitative variables and a set of dummy variables coded from the class variable. In the following notation the dummy variables will be denoted by y and the quantitative variables by x. The total sample covariance matrix for the x and y variables is
When c is the number of groups, nt is the number of observations in group t, and St is the sample covariance matrix for the x variables in group t, the within-class pooled covariance matrix for the x variables is
The canonical correlations, , are the square roots of the eigenvalues, , of the following matrix. The corresponding eigenvectors are vi.
Sp-1/2SxySyy-1SyxSp-1/2
Let V be the matrix with the eigenvectors vi that correspond to nonzero eigenvalues as columns. The raw canonical coefficients are calculated as follows
R = Sp-1/2V
The pooled within-class standardized canonical coefficients are
P = diag(Sp)1/2R
And the total sample standardized canonical coefficients are
T = diag(Sxx)1/2R
Let Xc be the matrix with the centered x variables as columns. The canonical scores may be calculated by any of the following
Xc  R
Xc  diag(Sp)-1/2P
Xc  diag(Sxx)-1/2T
For the Multivariate tests based on E-1H
E = (n-1)(Syy - SyxSxx-1Sxy)
H = (n-1)SyxSxx-1Sxy
where n is the total number of observations.

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