CANCORRELATION procedure

Does canonical correlation analysis (P.G.N. Digby).

Option

PRINT = strings

Printed output from the analysis (correlations, pcoeff, qcoeff, pscores, qscores); default * i.e. no output

Parameters

PVARIATES = pointers

Pointer to P-set of variates to be analysed

QVARIATES = pointers

Pointer to Q-set of variates to be analysed

CORRELATIONS = diagonal matrices

Stores the canonical correlations from each analysis

PCOEFF = matrices

Stores the coefficients for the P-set of variates

QCOEFF = matrices

Stores the coefficients for the Q-set of variates

PSCORES = matrices

Stores the unit scores from the P-set of variates

QSCORES = matrices

Stores the unit scores from the Q-set of variates

Description

CANCORRELATION does canonical correlation analysis; see, for example, Mardia, Kent & Bibby (1979) or Digby & Kempton (1987).

The data for the procedure are two pointers specified by the PVARIATES and QVARIATES parameters; these must point directly to two sets of variates. The variates may have missing values, or be restricted: any units with any values missing will be excluded from the analysis; any restrictions on the variates must be consistent (the rules here are exactly as used by the FSSPM directive).

Printed output is controlled by the option PRINT with settings: correlations to print the canonical correlations (also expressed as percentages, and cumulative percentages, of their total); pcoeff to print the canonical correlation coefficients for the P-set of variates; qcoeff to print the canonical correlation coefficients for the Q-set of variates; pscores to print the canonical correlation scores for the units calculated from the P-set of variates; qscores to print the canonical correlation scores for the units calculated from the Q-set of variates.

Results from the analysis can be saved using the parameters CORRELATIONS, PCOEFF, QCOEFF, PSCORES and QSCORES. The structures specified for these parameters need not be declared in advance.

Option: PRINT.

Parameters: PVARIATES, QVARIATES, CORRELATIONS, PCOEFF, QCOEFF, PSCORES, QSCORES.

Method

The method used is as described in Digby & Kempton (1987). Spectral decompositions (LRL′) of the SSPMs for the P-set and Q-set are used to form the inverse square root matrices, F and G (as L R^-½) ). The singular value decomposition ( U S V ′ ) of ( F ′ C G )is then formed, where C is the matrix of sums of products between the two sets of variates. The diagonal matrix S contains the canonical correlations; the canonical correlation coefficients for the two sets of variates are ( F U ) and ( G V ). The scores for the units from the two sets of variates are formed by subtracting the variate means and applying the matrices of coefficients as loadings.

References

Digby, P.G.N. & Kempton, R.A. (1987). Multivariate Analysis of Ecological Communities. Chapman & Hall, London.

Mardia, K.V., Kent, J.T. & Bibby, J.M. (1979). Multivariate Analysis. Academic Press, London.