dissimilarities themselves (Cailliez correction) (Gower and Legendre 1986). For
hypothesis testing in two-way MANOVA (Sect. 6.3.3), the Lingoes correction is
preferable because it produces a test with correct type I error, whereas the Cailliez
correction produces a test with slightly inflated rate of type I error (Legendre and
Anderson 1999). In the function cmdscale() presented below, the Cailliez
correction is obtained with the argument add ¼ TRUE. Note also that many
similarity or dissimilarity coefficients are non-Euclidean, but the square root of the
dissimilarity form (
ffiffiffiffiffiffiffiffiffiffiffi
1 À S
p
or
ffiffiffi ffi
D
p
) of some of them is Euclidean. The simple
matching (S 1 ), Jaccard (S 7 ), Sørensen (S 8 ) and percentage difference (aka BrayCurtis, D 14 ) coefficients are examples. See Legendre and Legendre (2012) Tables 7.2
and 7.3. The function dist.ldc() of adespatial tells users whether a
dissimilarity coefficient is Euclidean or not, and if its square root would be
Euclidean.
Advanced note – One can avoid complex axes by keeping the eigenvectors with
their original Euclidean norm (vector length ¼ 1) instead of dividing each one by the
square root of its eigenvalue, as is usual in the PCoA procedure. This workaround is
used in the MEM spatial analysis presented in Chap. 7. It should not be used for
routine ordination by PCoA since eigenvectors that have not been rescaled to
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
eigenvalue
p
cannot be used to produce plots that preserve the original dissimilarities
among the objects.
The ordination axes of a PCoA can be interpreted like those of a PCA or CA:
proximity of objects in the ordination represents their similarity in the sense of the
association measure used.
Finally, vegan offers a weighted version of PCoA with function
wcmdscale(). This function is based on cmdscale() with the added possibility to provide a vector of site weights. Weights are real positive numbers that can be
larger than 1. A 0 weight returns NA for the corresponding object. Lingoes and
Cailliez corrections are available through argument add ¼ "lingoes" (default)
or "cailliez".
5.5.2 Application of PCoA to the Doubs Data Set Using
cmdscale() and vegan
As an example, let us compute a matrix of percentage difference (aka Bray-Curtis)
dissimilarities among sites, and subject this matrix to PCoA. In vegan, there is a
way to project species abundances as weighted averages on a PCoA plot, by means
of function wascores() (Fig. 5.10). Since species are projected as weighted
averages of their contributions to the sites, their interpretation with respect to the
sites is akin to that of a CA with scaling 2. It is also possible to project other (here,
environmental) variables into the PCoA biplot using envfit().
188
5 Unconstrained Ordination
hypothesis testing in two-way MANOVA (Sect. 6.3.3), the Lingoes correction is
preferable because it produces a test with correct type I error, whereas the Cailliez
correction produces a test with slightly inflated rate of type I error (Legendre and
Anderson 1999). In the function cmdscale() presented below, the Cailliez
correction is obtained with the argument add ¼ TRUE. Note also that many
similarity or dissimilarity coefficients are non-Euclidean, but the square root of the
dissimilarity form (
ffiffiffiffiffiffiffiffiffiffiffi
1 À S
p
or
ffiffiffi ffi
D
p
) of some of them is Euclidean. The simple
matching (S 1 ), Jaccard (S 7 ), Sørensen (S 8 ) and percentage difference (aka BrayCurtis, D 14 ) coefficients are examples. See Legendre and Legendre (2012) Tables 7.2
and 7.3. The function dist.ldc() of adespatial tells users whether a
dissimilarity coefficient is Euclidean or not, and if its square root would be
Euclidean.
Advanced note – One can avoid complex axes by keeping the eigenvectors with
their original Euclidean norm (vector length ¼ 1) instead of dividing each one by the
square root of its eigenvalue, as is usual in the PCoA procedure. This workaround is
used in the MEM spatial analysis presented in Chap. 7. It should not be used for
routine ordination by PCoA since eigenvectors that have not been rescaled to
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
eigenvalue
p
cannot be used to produce plots that preserve the original dissimilarities
among the objects.
The ordination axes of a PCoA can be interpreted like those of a PCA or CA:
proximity of objects in the ordination represents their similarity in the sense of the
association measure used.
Finally, vegan offers a weighted version of PCoA with function
wcmdscale(). This function is based on cmdscale() with the added possibility to provide a vector of site weights. Weights are real positive numbers that can be
larger than 1. A 0 weight returns NA for the corresponding object. Lingoes and
Cailliez corrections are available through argument add ¼ "lingoes" (default)
or "cailliez".
5.5.2 Application of PCoA to the Doubs Data Set Using
cmdscale() and vegan
As an example, let us compute a matrix of percentage difference (aka Bray-Curtis)
dissimilarities among sites, and subject this matrix to PCoA. In vegan, there is a
way to project species abundances as weighted averages on a PCoA plot, by means
of function wascores() (Fig. 5.10). Since species are projected as weighted
averages of their contributions to the sites, their interpretation with respect to the
sites is akin to that of a CA with scaling 2. It is also possible to project other (here,
environmental) variables into the PCoA biplot using envfit().
188
5 Unconstrained Ordination