Performs redundancy analysis (A.I. Glaser).
Options
PRINT = string tokens |
What to print (variance, loadings, roots, evalues, evectors, speciesscores, sitescores, fitsitescores, correlations, fitcorrelations, weights); default vari, root |
|---|---|
NROOTS = scalar |
Number of eigenvalues and eigenvectors to include in output; default * takes all the non-zero eigenvalues |
NORMALIZE = string tokens |
Whether to normalize the Y, X and/or Z variates to have unit sums-of-squares before the analysis (x, y, z); default x, z |
SCALING = string token |
Scaling for species and site scores (none, both); default none |
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 or factors to use for each pointer of y-variates |
Z = pointers |
Conditioning variates or factors to remove (“partial out”) before the analysis |
LRV = LRVs |
LRV structure from each analysis, storing the eigenvectors, eigenvalues and total variance |
SPECIESSCORES = matrices |
Saves the “species scores” from each analysis |
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 |
SAVE = pointers |
Save structure which provides information for use in CRBIPLOT and CRTRIPLOT |
Description
Redundancy analysis is the direct extension of multiple regression to the modelling of multivariate response data (see e.g. Legendre & Legendre 1998). The response data are a set of y-variates, specified in a pointer using the Y parameter. The explanatory variables, which may be either variates or factors, are specified in a pointer by the X parameter. Similarly, the Z parameter can be used to specify conditioning variables, which again may be either variates or factors; this gives partial RDA, in which the effect of the z-variables is removed before performing RDA. This may be useful in cases where the effects of the elements of Z on Y are well known, or we may wish to isolate the effect of an individual explanatory variable (in which case we would place all but one of the explanatory variables in Z). When all elements of a variable are equal to zero, CCA removes the variable.
The PRINT option controls printed output, with settings:
roots |
the eigenvalues of the fitted values; |
|---|---|
evalues |
synonym of roots; |
loadings |
the eigenvectors associated with each eigenvalue, also known as the “species scores”; |
evectors |
synonym of loadings; |
speciesscores |
the “species scores” from the analysis (synonym of loadings and evectors); |
variance |
the fraction of the variance of the y-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-variables; |
fitcorrelations |
the correlation between the fitted site scores and the x-variables; |
weights |
the weights of the x-variables in the formation of the site scores. |
By default PRINT=roots,variance. The LRV, SPECIESSCORES, 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 variates, or X or Z variables to have unit sums-of-squares before the analysis. The default is to normalize the x- and z-variables but not the y-variates. (Note: this normalization of the x’s and z’s does not affect the variances accounted for in the y-variates.) The SCALING option controls scaling for species and site scores. If both is selected, both species and site scores are multiplied by the square root of their corresponding eigenvalues. For RDA choosing none is equivalent to Scaling type 1 in Legendre & Legendre (1998), whilst both is equivalent to Scaling type 2 in the same book. 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 SAVE parameter lets you save a pointer containing full details of the analysis. This can then be used to generate plots using the CRBIPLOT or CRTRIPLOT procedures. The most recent save structure is kept automatically inside Genstat to use as a default for the SAVE options of CRBIPLOT and CRTRIPLOT. So, you need save the pointer explicitly only if you want to display output from more than one analysis at a time.
Options: PRINT, NROOTS, NORMALIZE, SCALING, TOLERANCE.
Parameters: Y, X, Z, LRV, SPECIESSCORES, SITESCORES, FITSITESCORES, CORRELATIONS, FITCORRELATIONS, WEIGHTS, SAVE.
Method
RDA and partial RDA are explained in Sections 11.1 and 11.3 of Legendre & Legendre (1998).
Action with RESTRICT
If any of the variate or factors in the Y, X or Z pointers are restricted, only the defined subset of the units will be used in the analysis.
Reference
Legendre, P. & Legendre, L. (1998). Numerical Ecology, Second English Edition. Elsevier, Amsterdam.
See also
Procedures: CRBIPLOT, CRTRIPLOT, CANCORRELATION, CCA, PLS.
Commands for: Multivariate and cluster analysis.
Example
CAPTION 'RDA example','Example from Legendre & Legendre (1998)'; STYLE=meta
" The data for this example come from Table 11.3 on page 590 of Legendre &
Legendre. The data simulate fish observations at 10 sites from a beach at
different water depths and substrates."
POINTER [VALUES=Depth_m,Coral,Sand] X1
VARIATE [NVALUES=10] Species1[1...6],X1[]; VALUES=\
!(1, 0, 0, 11, 11, 9, 9, 7, 7, 5),\
!(0, 0, 1, 4, 5, 6, 7, 8, 9, 10),\
!(0, 0, 0, 0, 17, 0, 13, 0, 10, 0),\
!(0, 0, 0, 0, 7, 0, 10, 0, 13, 0),\
!(0, 0, 0, 8, 0, 6, 0, 4, 0, 2),\
!(0, 0, 0, 1, 0, 2, 0, 3, 0, 4),\
!(1, 2, 3, 4, 5, 6, 7, 8, 9, 10),\
!(0, 0, 0, 0, 1, 0, 1, 0, 1, 0),\
!(1, 1, 1, 0, 0, 0, 0, 0, 0, 0)
RDA Species1; X1
CRBIPLOT
CRTRIPLOT