Although PCA has a long history as a method devoted to tables of physical and
chemical variables, the introduction of species data pre-transformations has opened
up this powerful technique to the analysis of community data. Although PCA itself is
not modified and remains a linear ordination model, the pre-transformations ensure
that the species data are treated according to their specificity, i.e. without undue
importance being given to double zeros. A scaling 1 PCA biplot thus reveals the
underlying gradients structuring the community; the sites are ordered along the axes
according to their positions along these gradients. The circle of equilibrium contribution allows the identification of the species contributing most to the plotted pair of
axes. A scaling 2 biplot reveals the relationships among species in a correlation-like
fashion; since the data have been transformed, the correlations are not equivalent to
Pearson’s r computed from the raw data.
Technical note: the chi-square transformation can also be applied to species data
prior to PCA. In that case, the PCA solution is very similar, but not identical to a
correspondence analysis (CA) of the species data (Sect. 5.4). Although the two
methods preserve the chi-square distance among the sites, the calculation of the
eigen-decomposition is not done in exactly the same way and leads to different sets
of eigenvalues and eigenvectors.
5.3.3.2 Passive (post hoc) Explanation of Axes Using Environmental
Variables
Although there are means of incorporating explanatory variables directly in the
ordination process (canonical ordination, see Chap. 6), one may be interested in
interpreting a simple ordination by means of external variables. This can be done in
vegan by means of the function envfit(), which also works with CA (Sect. 5.4),
PCoA (Sect. 5.5) and NMDS (Sect. 5.6). According to its author, Jari Oksanen,
“envfit finds vectors or factor averages of environmental variables. [...] The projections of points onto vectors have maximum correlation with corresponding
environmental variables, and the factors show the averages of factor levels.”
The result is an object containing coordinates of factor levels (points) or arrowheads (quantitative variables) that can be used to project these variables into the
ordination diagram. Furthermore, envfit() computes a permutation test of the
environmental variables and the plot() function allows users to draw only the
variables with p-values equal to or smaller than a given level.
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5 Unconstrained Ordination
chemical variables, the introduction of species data pre-transformations has opened
up this powerful technique to the analysis of community data. Although PCA itself is
not modified and remains a linear ordination model, the pre-transformations ensure
that the species data are treated according to their specificity, i.e. without undue
importance being given to double zeros. A scaling 1 PCA biplot thus reveals the
underlying gradients structuring the community; the sites are ordered along the axes
according to their positions along these gradients. The circle of equilibrium contribution allows the identification of the species contributing most to the plotted pair of
axes. A scaling 2 biplot reveals the relationships among species in a correlation-like
fashion; since the data have been transformed, the correlations are not equivalent to
Pearson’s r computed from the raw data.
Technical note: the chi-square transformation can also be applied to species data
prior to PCA. In that case, the PCA solution is very similar, but not identical to a
correspondence analysis (CA) of the species data (Sect. 5.4). Although the two
methods preserve the chi-square distance among the sites, the calculation of the
eigen-decomposition is not done in exactly the same way and leads to different sets
of eigenvalues and eigenvectors.
5.3.3.2 Passive (post hoc) Explanation of Axes Using Environmental
Variables
Although there are means of incorporating explanatory variables directly in the
ordination process (canonical ordination, see Chap. 6), one may be interested in
interpreting a simple ordination by means of external variables. This can be done in
vegan by means of the function envfit(), which also works with CA (Sect. 5.4),
PCoA (Sect. 5.5) and NMDS (Sect. 5.6). According to its author, Jari Oksanen,
“envfit finds vectors or factor averages of environmental variables. [...] The projections of points onto vectors have maximum correlation with corresponding
environmental variables, and the factors show the averages of factor levels.”
The result is an object containing coordinates of factor levels (points) or arrowheads (quantitative variables) that can be used to project these variables into the
ordination diagram. Furthermore, envfit() computes a permutation test of the
environmental variables and the plot() function allows users to draw only the
variables with p-values equal to or smaller than a given level.
168
5 Unconstrained Ordination