• PCA must not be computed on the transposed data matrix. The reason is that
covariances or correlations among objects are meaningless.
• Covariances and correlations are defined for quantitative variables. However,
PCA is very robust to variations in the precision of data. Since a Pearson
correlation coefficient on semi-quantitative data is equivalent to a Spearman’s
correlation, a PCA on such variables yields an ordination where the relationship
among variables is estimated using that measure.
• PCA can be applied to binary data. Gower (1966, in Legendre and Legendre
2012) has shown that with binary descriptors PCA positions the objects in the
multidimensional space at distances that are the square roots of complements of
simple matching coefficients S 1 (i.e.,
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À S 1
p
) times a constant which is the
square root of the number of binary variables.
• Species presence-absence data can be subjected to a chord, Hellinger or log-chord
transformation prior to PCA. The justification is that the chord, Hellinger and logchord distances computed on presence-absence data are equal to
ffiffi ffi
2
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À Ochiai similarity
p
, so PCA after Hellinger or chord transformation preserves the Ochiai distance among objects in scaling type 1 plots. We also know
that
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À Ochiai similarity
p
is a metric and Euclidean distance (Legendre and
Legendre 2012 Table 7.2), which is appropriate for ordination analysis of community composition presence-absence data.
• Avoid the mistake of interpreting the relationships among variables based on the
proximities of the apices (tips) of the vector arrows instead of their angles in
biplots.
5.3.5 PCA Using Function PCA.newr()
For someone who wants a quick assessment of the structure of his or her data, we
provide functions PCA.newr() and biplot.PCA.newr() in file PCA.
newr.R. Here is an example of how they work using the Doubs environmental data.
# PCA; scaling 1 is the default for biplots in this function
env.PCA.PL <- PCA.newr(env, stand = TRUE)
biplot.PCA.newr(env.PCA.PL)
# PCA; scaling 2 in the biplot
biplot.PCA.newr(env.PCA.PL, scaling = 2)
170
5 Unconstrained Ordination
covariances or correlations among objects are meaningless.
• Covariances and correlations are defined for quantitative variables. However,
PCA is very robust to variations in the precision of data. Since a Pearson
correlation coefficient on semi-quantitative data is equivalent to a Spearman’s
correlation, a PCA on such variables yields an ordination where the relationship
among variables is estimated using that measure.
• PCA can be applied to binary data. Gower (1966, in Legendre and Legendre
2012) has shown that with binary descriptors PCA positions the objects in the
multidimensional space at distances that are the square roots of complements of
simple matching coefficients S 1 (i.e.,
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À S 1
p
) times a constant which is the
square root of the number of binary variables.
• Species presence-absence data can be subjected to a chord, Hellinger or log-chord
transformation prior to PCA. The justification is that the chord, Hellinger and logchord distances computed on presence-absence data are equal to
ffiffi ffi
2
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À Ochiai similarity
p
, so PCA after Hellinger or chord transformation preserves the Ochiai distance among objects in scaling type 1 plots. We also know
that
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À Ochiai similarity
p
is a metric and Euclidean distance (Legendre and
Legendre 2012 Table 7.2), which is appropriate for ordination analysis of community composition presence-absence data.
• Avoid the mistake of interpreting the relationships among variables based on the
proximities of the apices (tips) of the vector arrows instead of their angles in
biplots.
5.3.5 PCA Using Function PCA.newr()
For someone who wants a quick assessment of the structure of his or her data, we
provide functions PCA.newr() and biplot.PCA.newr() in file PCA.
newr.R. Here is an example of how they work using the Doubs environmental data.
# PCA; scaling 1 is the default for biplots in this function
env.PCA.PL <- PCA.newr(env, stand = TRUE)
biplot.PCA.newr(env.PCA.PL)
# PCA; scaling 2 in the biplot
biplot.PCA.newr(env.PCA.PL, scaling = 2)
170
5 Unconstrained Ordination