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# CRBIPLOT procedure

Plots correlation or distance biplots after `RDA`, or ranking biplots after `CCA` (A.I. Glaser).

### Options

`DIMENSIONS` = scalars Two numbers specifying which axes of the ordinations to plot; default 1,2 Whether to plot site or species scores (sitescores, speciesscores); default spec Which graphical window to use; default 1 Supplies results from an ordination analysis by `CCA` or `RDA`; default uses the most recent analysis

### Parameters

`X1` = scalars, variates or texts First explanatory variable to plot; default 1 Second explanatory variable to plot; default `*` i.e. none How to label the x-variables (`identifiers`, `labels`, `none`, `numbers`); default `labe` if `LXVARIABLES` is set, otherwise `iden` How to label the species scores (`identifiers`, `labels`, `none`, `numbers`); default `labe` if `LSPECIES` is set, otherwise `numb` How to label the site scores (`labels`, `none`, `numbers`); default `labe` if `LSITES` is set, otherwise `numb` Labels for variables Labels for species scores Labels for site scores

### Description

`CRBIPLOT` provides biplot representations of the results from `CCA` or `RDA`, showing projections of species or site scores onto one or two environmental variables. By default `CRBIPLOT` plots the species scores, but you can set option `PLOT=sitescores` to plot site scores instead.

The type of biplot depends on the scaling method used in the analysis. In `RDA`, Scaling Type 1 (i.e. no scaling) produces a distance biplot, while Scaling Type 2 (which scales both species and site scores) gives a correlation biplot. Similarly, for `CCA`, Scaling Type 1 (species scaling) produces a biplot with the sites at the centroids of the species, and Scaling Type 2 (site scaling) plots the species at the centroids of the site.

A distance biplot has the following features:

●        distances among elements of `Y` show approximations of their Euclidean distances in multidimensional space;

●        when an element of `Y` is projected at right angles onto a variable this approximates the position of the object on that variable;

●        since the eigenvectors have length one, the length of a projection of an element of `Y` onto a variable shows its contribution to the formation of that space;

●        the angle amongst variables is meaningless.

A correlation biplot has the following features:

●        distances among elements of `Y` are not approximations of the Euclidean distances between objects in multidimensional space (so the distance biplot is preferable if you want to interpret relationships amongst the elements of `Y`);

●        when an element of `Y` is projected at right angles onto a variable this approximates the position of the object on that variable;

●        the length of a projection of an element of `Y` onto a variable shows its contribution to the formation of that space;

●        the angles between variables approximate their correlation.

In addition when we carry out `CCA` Scaling Type 1:

●        distances among sites show approximations in reduced space of their chi-square distances;

●        the sites are at the centroids of the species, and the centroids are calculated using weights equal to the relative frequencies of the species (see Makarenkov & Legendre 2002);

●        the position of an object on an explanatory variable can be obtained by projecting the objects at right angle on the variable. This scaling is appropriate when the primary interest is the ordination of sites.

With `CCA` Scaling Type 2:

●        it is the distances among species in reduced space that are approximations of their chi-square distances;

●        the species are at the centroids of the sites in the graph;

●        any species scores that lie close to the point representing an explanatory variable are more likely to be found with higher frequency at that site than others further away (or more likely to be in State ‘1’ with binary data).

This scaling is appropriate when the primary interest is the relationship between species.

The explanatory variables to display can be specified using the `X1` and `X2` parameters. If the variable is a variate, you can set them to its identifier. Alternatively, if it is either a variate or a variable representing one of the levels of a factor, you can set them to the position of the variable in the list of variables involved in the analysis. Finally, if the variable represents the level of a factor, you can set them to a text containing the label used for the variable in the analysis (you can see the labels by looking at the row labels of the matrix showing the correlations between the environmental variables and the site scores). The `DIMENSIONS` option lists the numbers of the two canonical axes to plot; default 1,2.

The labels for the species scores, site scores and x-variable(s) can be set using the `LMSPECIES`, `LMSITES` and `LMXVARIABLES` parameters respectively, by selecting one of the following settings:

    `identifiers` uses the identifiers of the `X` variates with `LMXVARIABLES`, or of the `Y` variates with `LMSPECIES` (not available with `LMSITES`), expects labels to be supplied (in a text) using the `LSPECIES`, `LSITES` or `LXVARIABLES` parameter, gives no labels, and uses the column numbers of `X` and `Y`.

The defaults are `LMSPECIES=numbers`, `LMSITES=numbers` and `LMXVARIABLES=identifiers`, unless `LSPECIES`, `LSITES` or `LXVARIABLES` is set when the corresponding default becomes `labels`.

By default `CRBIPLOT` uses the results from the most recent analysis from `RDA` or `CCA`, but you can display results from an earlier analysis by saving the information about the analysis with the `SAVE` parameter of `CCA` or `RDA`, and then providing this to `CRBIPLOT` using its own `SAVE` option.

Options: `DIMENSIONS`, `PLOT`, `WINDOW`, `SAVE`.

Parameters: `X1`, `X2`, `LMXVARIABLES`, `LMSPECIES`, `LMSITES`, `LXVARIABLES`, `LSPECIES`, `LSITES`.

### Method

`CCA` and `RDA` are explained in Chapter 11 of Legendre & Legendre (1998).

### References

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

Makarenkov, V. & Legendre, P. (2002). Nonlinear redundancy analysis and canonical correspondence analysis based on polynomial regression. Ecology, 83, 1146-1161.

Procedures: `CCA`, `RDA`, `CRTRIPLOT`.

Commands for: Multivariate and cluster analysis, Graphics.

### Example

```CAPTION 'CRBIPLOT example','Example from Legendre & Legendre (1998)';\
STYLE=meta,plain
" 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,Other] X
VARIATE  [NVALUES=10] Species[1...9],X[]; 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),\
!(2, 5, 0, 6, 6, 10, 4, 6, 6, 0),\
!(4, 6, 2, 2, 6, 1, 5, 6, 2, 1),\
!(4, 1, 3, 0, 2, 4, 4, 4, 0, 3),\
!(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),\
!(0, 0, 0, 1, 0, 1, 0, 1, 0, 1)
CCA      Species; X
CRBIPLOT
```
Updated on March 8, 2019