CORANALYSIS procedure

Does correspondence analysis, or reciprocal averaging; synonym CORRESP (P.G.N. Digby).

Options

PRINT = strings

Printed output from the analysis (roots, rowscores, rowinertias, colscores, colinertias); default * i.e. no output

METHOD = string

Type of analysis required (correspondence, biplot, reciprocal); default corr

Parameters

DATA = matrices or pointers

Matrices to be analysed

ROOTS = diagonal matrices

Used to store the squared singular values from each analysis

ROWSCORES = matrices

Used to store the scores for the rows of the data matrix

COLSCORES = matrices

Used to store the scores for the columns of the data matrix

ROWINERTIAS = matrices

Used to store the inertias for the rows of the data matrix

COLINERTIAS = matrices

Used to store the inertias for the columns of the data matrix

Description

CORANALYSIS can do correspondence analysis, reciprocal averaging or a similar analysis of a biplot-style, as requested by the METHOD option. Greenacre (1984) describes correspondence analysis; the variations in the method are described by Digby & Kempton (1987).

The data for the procedure is specified by the DATA parameter as either a

matrix or a datamatrix (i.e. a pointer to variates, all with the same length). The matrix must not contain any missing values; it is unchanged on exit from the procedure.

Printed output is controlled by the PRINT option with settings: roots to print the roots, (also the roots expressed as percentages and cumulative percentages); rowscores to print the scores for the rows of the data matrix; rowinertias to print the inertias for the rows of the data matrix; colscores to print the scores for the columns of the data matrix; colinertias to print the inertias for the columns of the data matrix.

Results from the analysis can be saved using the parameters ROOTS, ROWSCORES, COLSCORES, ROWINERTIAS and COLINERTIAS. The structures specified for these parameters need not be declared in advance.

The results correspond to p dimensions, where p is one less than the smaller of the numbers of rows and columns of the input data matrix.

Options: PRINT, METHOD.

Parameters: DATA, ROOTS, ROWSCORES, COLSCORES, ROWINERTIAS, COLINERTIAS.

Method

Procedure CORANALYSIS provides correspondence analysis, and two related variants, Full details of correspondence are given in Greenacre (1984); the differences between this and reciprocal averaging, and a further method which allows a biplot-style interpretation, are described by Digby & Kempton (1987). The scores from correspondence analysis for the rows and columns are A = ( R^-½ U S ) and B = ( C^-½ V S ) where R and C are diagonal matrices of row and column totals of the data matrix X, scaled to have sum one, and U, S and V are from the singular value decomposition of Y = ( R^-½ X C^-½ ). With the other two methods X is not scaled to total one, and the scores are given by A = ( R^-½ U S^m ) and B = ( C^-½ V S^m ): the parameter m is zero for reciprocal averaging, and 0.5 for the third method. The inertia values for the rows and columns are given by ( R A A ′ ) and ( C B B ′ ): see Greenacre (1984) for further information. The roots are the squares of the singular values. Note that the first singular value will always be one, and corresponds to a trivial solution given in the first column of A and B above; this trivial solution is automatically removed from the results printed and saved from CORANALYSIS.

References

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

Greenacre, M.J. (1984). Theory and Applications of Correspondence Analysis. Academic Press, London.