An RDA produces min[p, m, nÀ1] canonical axes, where n is the number of
objects, p is the number of response variables and m is the number of degrees of
freedom of the model (rank of cov(X), which is number of numeric explanatory
variables, including levels of factors if qualitative explanatory variables are
included; a factor with k classes bears (k–1) degrees of freedom; if needed, it requires
(k–1) dummy variables for coding). Each of the canonical axes is a linear combination (like a multiple regression model) of all explanatory variables. RDA may be
computed, for convenience, using standardized explanatory variables; the fitted
values of the regressions, as well as the canonical analysis results, are unchanged
by standardization of the X variables. In vegan’s rda() function, the variation of
the data matrix that cannot be explained by the environmental variables (i.e., the
residuals of the regressions) is expressed by unconstrained PCA eigenvectors, which
are given after the canonical eigenvectors.
For the reasons explained in Chap. 5 about PCA, an RDA can be computed on a
covariance or a correlation response matrix. To obtain an analysis on the correlation
response matrix, standardization of the response data is done by the option
scale ¼ TRUE in vegan’s rda(). This option should not be used with community composition data.
The statistical significance of an RDA (global model) and that of individual
canonical axes can be tested by permutations. These tests will be introduced in
due course.
6.3.2 RDA of the Doubs River Data
You will now explore various aspects of RDA. To achieve this, you will first prepare
the data sets as usual, but also transform a variable and divide the explanatory
variables into two subsets.
6.3.2.1 Preparation of the Data
# Load the required packages
library(ade4)
library(adegraphics)
library(adespatial)
library(vegan)
library(vegan3d)
library(MASS)
library(ellipse)
library(FactoMineR)
library(rrcov)
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6 Canonical Ordination
objects, p is the number of response variables and m is the number of degrees of
freedom of the model (rank of cov(X), which is number of numeric explanatory
variables, including levels of factors if qualitative explanatory variables are
included; a factor with k classes bears (k–1) degrees of freedom; if needed, it requires
(k–1) dummy variables for coding). Each of the canonical axes is a linear combination (like a multiple regression model) of all explanatory variables. RDA may be
computed, for convenience, using standardized explanatory variables; the fitted
values of the regressions, as well as the canonical analysis results, are unchanged
by standardization of the X variables. In vegan’s rda() function, the variation of
the data matrix that cannot be explained by the environmental variables (i.e., the
residuals of the regressions) is expressed by unconstrained PCA eigenvectors, which
are given after the canonical eigenvectors.
For the reasons explained in Chap. 5 about PCA, an RDA can be computed on a
covariance or a correlation response matrix. To obtain an analysis on the correlation
response matrix, standardization of the response data is done by the option
scale ¼ TRUE in vegan’s rda(). This option should not be used with community composition data.
The statistical significance of an RDA (global model) and that of individual
canonical axes can be tested by permutations. These tests will be introduced in
due course.
6.3.2 RDA of the Doubs River Data
You will now explore various aspects of RDA. To achieve this, you will first prepare
the data sets as usual, but also transform a variable and divide the explanatory
variables into two subsets.
6.3.2.1 Preparation of the Data
# Load the required packages
library(ade4)
library(adegraphics)
library(adespatial)
library(vegan)
library(vegan3d)
library(MASS)
library(ellipse)
library(FactoMineR)
library(rrcov)
206
6 Canonical Ordination