# A posteriori projection of environmental variables in a PCA
# A PCA scaling 2 plot is produced in a new graphic window.
biplot(spe.h.pca, main = "PCA fish abundances-scaling 2")
# Scaling 2 is default
(spe.h.pca.env <- envfit(spe.h.pca, env, scaling = 2))
# Plot significant variables with a user -selected colour
plot(spe.h.pca.env, p.max = 0.05 , col = 3)
# This has added the significant environmental variables to the
# last biplot drawn by R.
# BEWARE: envfit() must be given the same scaling as the plot to
# which its result is added!
Hints See how the plot of the environmental variables has been restricted to the
significant variables by argument p.max = 0.05.
This is a post hoc interpretation of ordination axes. Compare with Chap. 6.
Does this new information help interpret the biplot?
envfit() also proposes permutation tests to assess the significance of the R
2 of
each explanatory variable regressed on the two axes of the biplot. But this is not, by
far, the best way to test the effect of explanatory variables on a table of response
variables. We will explore this topic in Chap. 6.
5.3.4 Domain of Application of PCA
Principal component analysis is a very powerful technique, but it has its limits. The
main application of PCA in ecology is the ordination of sites on the basis of
quantitative environmental variables or, after an appropriate transformation, of
community composition data. PCA has originally been defined for data with
multinormal distributions. In its applications in ecology, however, PCA is not very
sensitive to departures from multinormality, as long as the distributions are not
exaggeratedly skewed. The main computational step of PCA is the eigendecomposition of a dispersion matrix (linear covariances or correlations). Covariances must in turn be computed on quantitative data — but see below for binary data.
Here are, in more detail, the conditions of application of PCA:
• PCA must be computed on a table of dimensionally homogeneous variables. The
reason is that it is the sum of the variances of the variables that is partitioned into
eigenvalues. Variables must be in the same physical units to produce a meaningful sum of variances (the unit of a variance is the square of the unit of the variable
from which it was computed), or they must be dimensionless, which is the case
for standardized or log-transformed variables.
5.3 Principal Component Analysis (PCA)
169
# A PCA scaling 2 plot is produced in a new graphic window.
biplot(spe.h.pca, main = "PCA fish abundances-scaling 2")
# Scaling 2 is default
(spe.h.pca.env <- envfit(spe.h.pca, env, scaling = 2))
# Plot significant variables with a user -selected colour
plot(spe.h.pca.env, p.max = 0.05 , col = 3)
# This has added the significant environmental variables to the
# last biplot drawn by R.
# BEWARE: envfit() must be given the same scaling as the plot to
# which its result is added!
Hints See how the plot of the environmental variables has been restricted to the
significant variables by argument p.max = 0.05.
This is a post hoc interpretation of ordination axes. Compare with Chap. 6.
Does this new information help interpret the biplot?
envfit() also proposes permutation tests to assess the significance of the R
2 of
each explanatory variable regressed on the two axes of the biplot. But this is not, by
far, the best way to test the effect of explanatory variables on a table of response
variables. We will explore this topic in Chap. 6.
5.3.4 Domain of Application of PCA
Principal component analysis is a very powerful technique, but it has its limits. The
main application of PCA in ecology is the ordination of sites on the basis of
quantitative environmental variables or, after an appropriate transformation, of
community composition data. PCA has originally been defined for data with
multinormal distributions. In its applications in ecology, however, PCA is not very
sensitive to departures from multinormality, as long as the distributions are not
exaggeratedly skewed. The main computational step of PCA is the eigendecomposition of a dispersion matrix (linear covariances or correlations). Covariances must in turn be computed on quantitative data — but see below for binary data.
Here are, in more detail, the conditions of application of PCA:
• PCA must be computed on a table of dimensionally homogeneous variables. The
reason is that it is the sum of the variances of the variables that is partitioned into
eigenvalues. Variables must be in the same physical units to produce a meaningful sum of variances (the unit of a variance is the square of the unit of the variable
from which it was computed), or they must be dimensionless, which is the case
for standardized or log-transformed variables.
5.3 Principal Component Analysis (PCA)
169