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