community composition data with large numbers of zeros must be transformed
before they are used in MANOVA and other Euclidean-based models of analysis.
The Legendre and Gallagher (2001) transformations (Sect. 3.5) are one way to solve
that problem: transformed species data were used in the ANOVA by RDA described
in Sect. 6.3.2.9. These transformations cover only the chord, Hellinger, chi-square,
and Ochiai distance cases, however. Ecologists may want to compute RDA based on
other dissimilarity indices that cannot be computed by a data transformation
followed by the calculation of the Euclidean distance.
Legendre and Anderson (1999) proposed the method of distance-based redundancy analysis (db-RDA) to solve that problem. They showed that RDA could be
used as a form of ANOVA that was applicable to community composition data if
these were transformed in some appropriate way, which went through the calculation
of a dissimilarity matrix of the user’s choice. This approach remains fully valid and
useful for all dissimilarity measures that cannot be obtained by a data transformation
followed by the calculation of the Euclidean distance. Among the dissimilarities
devoted to communities of living organisms, let us mention some measures for
binary data (e.g. Jaccard (
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À S 7
p
), Sørensen (
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À S 8
p
), and quantitative dissimilarity measures like percentage difference (aka Bray-Curtis, D 14 ), asymmetric
Gower (
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À S 19
p
), Whittaker (D 9 ) and Canberra (D 10 ). Dissimilarities intended
for other types of data, e.g. symmetric Gower (
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À S 15
p
), Estabrook-Rogers (
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À S 16
p
), and the generalized Mahalanobis distance for groups of observations,
can also be used in canonical ordination through db-RDA when the analysis
concerns response variables describing the physical environment. Examples of
papers involving db-RDA are Anderson (1999), Geffen et al. (2004) and Lear
et al. (2008). The method goes as follows:
• Compute a Q-mode dissimilarity matrix for the response data.
• Compute a principal coordinate analysis (PCoA) of the dissimilarity matrix,
correcting for negative eigenvalues if necessary, using the Lingoes correction.
Keep all principal coordinates in a file; they express all the variance of the data as
seen through the dissimilarity measure.
• Run and test an RDA of the principal coordinates created above (which act as the
response data) constrained by the explanatory variables available in the study.
The explanatory variables may, for example, represent the factors of a manipulative or mensurative experiment.
These steps are quite simple; they can be run one by one in R with a few lines of
code only, but more directly vegan proposes the function capscale()to that
effect. This function allows the direct plotting of the weighted average species scores
if the user provides the matrix of species data in argument comm.
There is another way of computing db-RDA. McArdle and Anderson (2001)
proposed an alternative method that runs the analysis directly on a dissimilarity
response matrix without having to go through PCoA. This alternative way is
provided in vegan by function dbrda(). Unfortunately, species scores cannot
be added directly to the plot with this function. But the main interest in this
alternative procedure is that, according to the authors’ simulations, the permutation
test computed on their method based on a dissimilarity matrix has correct type I error
250
6 Canonical Ordination
before they are used in MANOVA and other Euclidean-based models of analysis.
The Legendre and Gallagher (2001) transformations (Sect. 3.5) are one way to solve
that problem: transformed species data were used in the ANOVA by RDA described
in Sect. 6.3.2.9. These transformations cover only the chord, Hellinger, chi-square,
and Ochiai distance cases, however. Ecologists may want to compute RDA based on
other dissimilarity indices that cannot be computed by a data transformation
followed by the calculation of the Euclidean distance.
Legendre and Anderson (1999) proposed the method of distance-based redundancy analysis (db-RDA) to solve that problem. They showed that RDA could be
used as a form of ANOVA that was applicable to community composition data if
these were transformed in some appropriate way, which went through the calculation
of a dissimilarity matrix of the user’s choice. This approach remains fully valid and
useful for all dissimilarity measures that cannot be obtained by a data transformation
followed by the calculation of the Euclidean distance. Among the dissimilarities
devoted to communities of living organisms, let us mention some measures for
binary data (e.g. Jaccard (
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À S 7
p
), Sørensen (
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À S 8
p
), and quantitative dissimilarity measures like percentage difference (aka Bray-Curtis, D 14 ), asymmetric
Gower (
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À S 19
p
), Whittaker (D 9 ) and Canberra (D 10 ). Dissimilarities intended
for other types of data, e.g. symmetric Gower (
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À S 15
p
), Estabrook-Rogers (
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À S 16
p
), and the generalized Mahalanobis distance for groups of observations,
can also be used in canonical ordination through db-RDA when the analysis
concerns response variables describing the physical environment. Examples of
papers involving db-RDA are Anderson (1999), Geffen et al. (2004) and Lear
et al. (2008). The method goes as follows:
• Compute a Q-mode dissimilarity matrix for the response data.
• Compute a principal coordinate analysis (PCoA) of the dissimilarity matrix,
correcting for negative eigenvalues if necessary, using the Lingoes correction.
Keep all principal coordinates in a file; they express all the variance of the data as
seen through the dissimilarity measure.
• Run and test an RDA of the principal coordinates created above (which act as the
response data) constrained by the explanatory variables available in the study.
The explanatory variables may, for example, represent the factors of a manipulative or mensurative experiment.
These steps are quite simple; they can be run one by one in R with a few lines of
code only, but more directly vegan proposes the function capscale()to that
effect. This function allows the direct plotting of the weighted average species scores
if the user provides the matrix of species data in argument comm.
There is another way of computing db-RDA. McArdle and Anderson (2001)
proposed an alternative method that runs the analysis directly on a dissimilarity
response matrix without having to go through PCoA. This alternative way is
provided in vegan by function dbrda(). Unfortunately, species scores cannot
be added directly to the plot with this function. But the main interest in this
alternative procedure is that, according to the authors’ simulations, the permutation
test computed on their method based on a dissimilarity matrix has correct type I error
250
6 Canonical Ordination