# Contingency table crossing variables "Shrub" and "Topo"
table(mite.env$Shrub, mite.env$Topo)
The table shows that sites where shrubs are present (modalities “Few” and
“Many”) are rather evenly represented on blankets and hummocks, but all sites
devoid of shrubs have blanket-type soil coverage (or, conversely, no hummock is
devoid of shrubs). Observe in Fig. 5.9b that modalities “Blanket” and “None” are
close.
5.5 Principal Coordinate Analysis (PCoA)
5.5.1 Introduction
PCA as well as CA impose the distance that is preserved among objects: the
Euclidean distance (and several others after pre-transformation of the species data)
for PCA and the χ
2 distance for CA. If one wishes to ordinate objects on the basis of
some other dissimilarity measure, more appropriate to the problem at hand, then
PCoA is the method of choice. It provides a Euclidean representation of a set of
objects whose relationships are measured by any dissimilarity measure chosen by the
user. For example, if the coefficient is Gower’s index D ¼ (1 – S 15 ), which can
combine descriptors of many mathematical types into a single measure of resemblance, then the ordination will represent the relationships among the objects based
upon these variables and measured through the Gower index. This would not be
possible with PCA or CA.
Like PCA and CA, PCoA produces a set of orthogonal axes whose importance is
measured by eigenvalues. Since it is based on an association matrix, it can directly
represent the relationships either among objects (if the association matrix was in Q
mode) or among variables (if the association matrix was in R mode). If it is necessary
to project variables, e.g. species, on a PCoA ordination of the objects (sites), the
variables can be related a posteriori to the ordination axes using correlations or
weighted averages and drawn on the ordination plot. In the case of association
measures that have the Euclidean property (Sect. 3.3), PCoA behaves in a Euclidean
manner. For instance, computing a Euclidean distance among sites and running a
PCoA will yield the same results as running a PCA on a covariance matrix of the
same data and looking at the scaling 1 ordination biplot. But if the association
coefficient used is non-Euclidean, then PCoA may react by producing several
negative eigenvalues in addition to the positive ones, plus a null eigenvalue
in-between. The axes corresponding to negative eigenvalues cannot be represented
on real ordination axes since they are complex. In most applications, this does not
affect the representation of the objects on the several first principal axes, but it can
lead to problems if the largest negative eigenvalues are of the same magnitude in
absolute value as the first positive ones.
There are technical solutions to this problem, which consist in adding a constant
to either the squared dissimilarities among objects (Lingoes correction) or to the
5.5 Principal Coordinate Analysis (PCoA)
187
table(mite.env$Shrub, mite.env$Topo)
The table shows that sites where shrubs are present (modalities “Few” and
“Many”) are rather evenly represented on blankets and hummocks, but all sites
devoid of shrubs have blanket-type soil coverage (or, conversely, no hummock is
devoid of shrubs). Observe in Fig. 5.9b that modalities “Blanket” and “None” are
close.
5.5 Principal Coordinate Analysis (PCoA)
5.5.1 Introduction
PCA as well as CA impose the distance that is preserved among objects: the
Euclidean distance (and several others after pre-transformation of the species data)
for PCA and the χ
2 distance for CA. If one wishes to ordinate objects on the basis of
some other dissimilarity measure, more appropriate to the problem at hand, then
PCoA is the method of choice. It provides a Euclidean representation of a set of
objects whose relationships are measured by any dissimilarity measure chosen by the
user. For example, if the coefficient is Gower’s index D ¼ (1 – S 15 ), which can
combine descriptors of many mathematical types into a single measure of resemblance, then the ordination will represent the relationships among the objects based
upon these variables and measured through the Gower index. This would not be
possible with PCA or CA.
Like PCA and CA, PCoA produces a set of orthogonal axes whose importance is
measured by eigenvalues. Since it is based on an association matrix, it can directly
represent the relationships either among objects (if the association matrix was in Q
mode) or among variables (if the association matrix was in R mode). If it is necessary
to project variables, e.g. species, on a PCoA ordination of the objects (sites), the
variables can be related a posteriori to the ordination axes using correlations or
weighted averages and drawn on the ordination plot. In the case of association
measures that have the Euclidean property (Sect. 3.3), PCoA behaves in a Euclidean
manner. For instance, computing a Euclidean distance among sites and running a
PCoA will yield the same results as running a PCA on a covariance matrix of the
same data and looking at the scaling 1 ordination biplot. But if the association
coefficient used is non-Euclidean, then PCoA may react by producing several
negative eigenvalues in addition to the positive ones, plus a null eigenvalue
in-between. The axes corresponding to negative eigenvalues cannot be represented
on real ordination axes since they are complex. In most applications, this does not
affect the representation of the objects on the several first principal axes, but it can
lead to problems if the largest negative eigenvalues are of the same magnitude in
absolute value as the first positive ones.
There are technical solutions to this problem, which consist in adding a constant
to either the squared dissimilarities among objects (Lingoes correction) or to the
5.5 Principal Coordinate Analysis (PCoA)
187