(species) is found by projecting the centroid at right angle on the response
variable (as for individual objects). (4) Distances among centroids, and between
centroids and individual objects, do not approximate their Euclidean distances.
On these bases, it is now possible to interpret the triplots. Let us take as examples
the pair of plots representing the fitted site scores (Fig. 6.1). The numerical output
shows that the first two canonical axes explain together 56.1% of the total variance of
the response data, the first axis alone explaining 45.4%. These are unadjusted values,
however. Since the adjusted R
2 of the RDA
2 is R
2
adj ¼ 0.5224, the proportions of
accumulated constrained eigenvalues (i.e., proportions with respect to the explained
variance, not the total variance) show that the first axis alone explains
0.5224 Â 0.6242 ¼ 0.326 or 32.6% variance, and the first two axes together
0.5224 Â 0.7712 ¼ 0.4029 or 40.3% variance. We can be confident that the major
trends have been well modelled in this analysis. Because ecological data are generally quite noisy, one should not expect very high values of R
2
adj . Furthermore, the
first unconstrained eigenvalue (PC1) is comparatively small (0.04581), compared to
the first constrained eigenvalue (0.2281), which means that it does not display any
dominant residual structure of the response data.
These triplots show that oxygen (oxy), elevation (ele), nitrates (nit) and
discharge (dis), as well as slope (mainly the level slo.very_steep) play an
important role in the dispersion of the sites along the first axis. Both triplots oppose
the upper and lower parts of the river along the first axis. The scaling 2 triplot shows
three groups of fish species correlated with different sets of explanatory variables:
the brown trout (Satr), Eurasian minnow (Phph) and stone loach (Babl) are
found in the first half of the sites, and are correlated with high oxygen content and
slope as well as high elevation. The bleak (Alal), roach (Ruru) and European chub
(Sqce), on the opposite, are related to sites 23, 24 and 25 characterized by high
phosphates (pho), ammonium (amm) and biological oxygen demand (bod) levels.
Most other species are bunched together away from these extremes. They show
mostly shorter projections, indicating that they are either present over most portions
of the river or related to intermediate ecological conditions.
To avoid the need to manually retrieve the biplot scores and ask for the arrows,
we provide a homemade function called triplot.rda() to produce an RDA
triplot in one command. The function offers many choices: lc or wa scores, scaling 1
or 2, choice of a subgroup of variables (species) as well as several graphical
parameters. In the example below, we propose to draw only the species that reach
a (cumulative) goodness-of-fit of at least 0.6 (an arbitrary value) in the ordination
plane formed by axes 1 and 2. The higher the goodness-of-fit, the better the species is
fitted on the corresponding axis. Goodness-of-fit can be retrieved from an rda()
object by function goodness() of package vegan.
The function works as follows. Scaling 1: Fig. 6.2a; scaling 2: Fig. 6.2b.
2 Obtained with RsquareAdj(spe.rda).
6.3 Redundancy Analysis (RDA)
217
variable (as for individual objects). (4) Distances among centroids, and between
centroids and individual objects, do not approximate their Euclidean distances.
On these bases, it is now possible to interpret the triplots. Let us take as examples
the pair of plots representing the fitted site scores (Fig. 6.1). The numerical output
shows that the first two canonical axes explain together 56.1% of the total variance of
the response data, the first axis alone explaining 45.4%. These are unadjusted values,
however. Since the adjusted R
2 of the RDA
2 is R
2
adj ¼ 0.5224, the proportions of
accumulated constrained eigenvalues (i.e., proportions with respect to the explained
variance, not the total variance) show that the first axis alone explains
0.5224 Â 0.6242 ¼ 0.326 or 32.6% variance, and the first two axes together
0.5224 Â 0.7712 ¼ 0.4029 or 40.3% variance. We can be confident that the major
trends have been well modelled in this analysis. Because ecological data are generally quite noisy, one should not expect very high values of R
2
adj . Furthermore, the
first unconstrained eigenvalue (PC1) is comparatively small (0.04581), compared to
the first constrained eigenvalue (0.2281), which means that it does not display any
dominant residual structure of the response data.
These triplots show that oxygen (oxy), elevation (ele), nitrates (nit) and
discharge (dis), as well as slope (mainly the level slo.very_steep) play an
important role in the dispersion of the sites along the first axis. Both triplots oppose
the upper and lower parts of the river along the first axis. The scaling 2 triplot shows
three groups of fish species correlated with different sets of explanatory variables:
the brown trout (Satr), Eurasian minnow (Phph) and stone loach (Babl) are
found in the first half of the sites, and are correlated with high oxygen content and
slope as well as high elevation. The bleak (Alal), roach (Ruru) and European chub
(Sqce), on the opposite, are related to sites 23, 24 and 25 characterized by high
phosphates (pho), ammonium (amm) and biological oxygen demand (bod) levels.
Most other species are bunched together away from these extremes. They show
mostly shorter projections, indicating that they are either present over most portions
of the river or related to intermediate ecological conditions.
To avoid the need to manually retrieve the biplot scores and ask for the arrows,
we provide a homemade function called triplot.rda() to produce an RDA
triplot in one command. The function offers many choices: lc or wa scores, scaling 1
or 2, choice of a subgroup of variables (species) as well as several graphical
parameters. In the example below, we propose to draw only the species that reach
a (cumulative) goodness-of-fit of at least 0.6 (an arbitrary value) in the ordination
plane formed by axes 1 and 2. The higher the goodness-of-fit, the better the species is
fitted on the corresponding axis. Goodness-of-fit can be retrieved from an rda()
object by function goodness() of package vegan.
The function works as follows. Scaling 1: Fig. 6.2a; scaling 2: Fig. 6.2b.
2 Obtained with RsquareAdj(spe.rda).
6.3 Redundancy Analysis (RDA)
217