RDA procedure

Performs redundancy analysis (A. Glaser).


Options

PRINT = strings
What to print (model, deviance, summary, estimates, correlations, fittedvalues, accumulated, monitoring, cparameter, cmonitoring, cplot); default mode, summ, esti, cpar

NROOTS = scalar
Number of eigenvalues and eigenvectors to include in output; default * takes all the non-zero eigenvalues

NORMALIZE = strings
Whether to normalize the Y and/or X variates to have unit sums-of-squares before the analysis (x, y); default x

BIPLOT = strings
Types of biplots to produce (xcorrelation, xdistance, ycorrelation, ydistance); default *

TOLERANCE = scalar
Tolerance for detecting non-zero eigenvalues; default 10-5


Parameters

Y = pointers
Each pointer defines a set of response variates to be modelled

X = pointers
Explanatory variates to use for for each pointer of x-variates

LRV = LRVs
LRV structure from each analysis, storing the eigenvectors, eigenvalues and total variance

SITESCORES = matrices
Save the "site scores" from each analysis

FITSITESCORES = matrices
Save the fitted "site scores" from each analysis

CORRELATIONS = matrices
Saves the correlations between the site scores and the x-variates

FITCORRELATIONS = matrices
Saves the correlations between the fitted site scores and the x-variates

WEIGHTS = matrices
Save the weights of the x-variates in the formation of the site scores


Description

Redundancy analysis is the direct extension of multiple regression to the modelling of multivariate response data (see e.g. Legendre, Legendre & Legendre 1998). The response data is a set of y-variates, specified in a pointer using the Y parameter, and the explanatory variates are specified in a pointer using the X parameter. The analysis forms an ordination of the y-variates, constrained so that the ordination variates that are formed are linear combinations of the x-variates.

   The PRINT option controls printed output, with settings:

    evalues
the eigenvalues of the fitted values;

    evectors
the eigenvectors associated with each eigenvalue;

    variance
the fraction of the variance of the response variates associated with each eigenvalue;

    sitescores
the "site scores" of the y-variates (i.e. the ordination of the units in the y-variate space);

    fitsitescores
the fitted "site scores" of the fitted values of the y-variates (i.e. the ordination of the units in the y-variate space);

    correlations
the correlation between the site scores and the x-variates;

    fitcorrelations
the correlation between the fitted site scores and the x-variates;

    weights
the weights of the x-variates in the formation of the site scores.

By default PRINT=evalues,variance. The LRV, SITESCORES, FITSITESCORES, CORRELATIONS, FITCORRELATIONS and WEIGHTS parameters allow this information to be saved.

   The NROOTS option specifies the number of eigenvalues and eigenvectors to include in the output. By default all the non-zero eigenvalues are included. The NORMALIZE option controls whether to normalize the Y and/or X variates to have unit sums-of-squares before the analysis. The default is to normalize the x-variates but not the y-variates. (Note: this normalization of the x's does not affect the variances accounted for in the y-variates.) The TOLERANCE option specifies a threshold for the detection of non-zero eigenvalues (default 10-5). An eigenvalue is taken to be non zero if is it greater than TOLERANCE multiplied by the total variance.

   The BIPLOT option allows you to request various types of biplot:

    xcorrelation
correlation biplot of the site scores and x-variates;

    xdistance
distance biplot of the site scores and x-variates;

    ycorrelation
correlation biplot of the site scores and y-variates;

    ydistance
distance biplot of the site scores and y-variates.

By default none are plotted.

 

Options: PRINT, NROOTS, NORMALIZE, BIPLOT, TOLERANCE.

Parameters: Y, X, LRV, SITESCORES, FITSITESCORES, CORRELATIONS, FITCORRELATIONS, WEIGHTS.


Method

RDA uses the method described in Section 11.1 of Legendre et al. (1998).


Action with RESTRICT

The X and Y variates must not be restricted.


Reference

Legendre, P., Legendre, L. & Legendre, L. (1998). Numerical Ecology, Second English Edition. Elsevier, Amsterdam.