This function implements two methods for correcting for negative
values in principal coordinate analysis (PCoA). Negative eigenvalues
can be produced in PCoA when decomposing distance matrices produced by
coefficients that are not Euclidean (Gower and Legendre 1986,Legendre
and Legendre 1998).
In pcoa
, when negative eigenvalues are present in the
decomposition results, the distance matrix D can be modified using
either the Lingoes or the Cailliez procedure to produce results
without negative eigenvalues.
In the Lingoes (1971) procedure, a constant c1, equal to twice
absolute value of the largest negative value of the original principal
coordinate analysis, is added to each original squared distance in the
distance matrix, except the diagonal values. A newe principal
coordinate analysis, performed on the modified distances, has at most
(n-2) positive eigenvalues, at least 2 null eigenvalues, and no
negative eigenvalue.
In the Cailliez (1983) procedure, a constant c2 is added to the
original distances in the distance matrix, except the diagonal
values. The calculation of c2 is described in Legendre and Legendre
(1998). A new principal coordinate analysis, performed on the modified
distances, has at most (n-2) positive eigenvalues, at least 2 null
eigenvalues, and no negative eigenvalue.
In all cases, only the eigenvectors corresponding to positive
eigenvalues are shown in the output list. The eigenvectors are scaled
to the square root of the corresponding eigenvalues. Gower (1966) has
shown that eigenvectors scaled in that way preserve the original
distance (in the D matrix) among the objects. These eigenvectors can
be used to plot ordination graphs of the objects.
We recommend not to use PCoA to produce ordinations from the chord,
chi-square, abundance profile, or Hellinger distances. It is easier to
first transform the community composition data using the following
transformations, available in the decostand
function of the
vegan
package, and then carry out a principal component
analysis (PCA) on the transformed data:
Chord transformation: decostand(spiders,"normalize")
Transformation to relative abundance profiles:
decostand(spiders,"total")
Hellinger transformation: decostand(spiders,"hellinger")
Chi-square transformation: decostand(spiders,"chi.square")
The ordination results will be identical and the calculations
shorter. This two-step ordination method, called transformation-based
PCA (tb-PCA), was described by Legendre and Gallagher (2001).
The biplot.pcoa
function produces plots for any pair of
principal coordinates. The original variables can be projected onto
the ordination plot.