The two RDA triplots use the fitted site scores (called ‘lc’ in vegan). The
choice between these and the site scores that are weighted sums of species scores
(called ‘wa’ in vegan because in CA/CCA they are weighted averages) for the
triplots depends on the context and the purpose of the plot. On the one hand, the
fitted site scores are strictly orthogonal linear combinations of the explanatory
variables; they represent clearly and exclusively what can be modelled using the
explanatory variables at hand. We advocate the use of these lc scores in most
situations because the “true” ordination diagram of RDA is the ordination of the Ŷ
matrix of fitted values. On the other hand, the site scores that are weighted sums of
species appear more robust to noise in the environmental variables: McCune (1997)
showed that if the latter contain much random error, the resulting lc plots may be
completely scrambled. However, weighted sums of species (“wa”) scores are “contaminated” scores, halfway between the model fitted by the RDA procedure and a
PCA of the original data, and as such it is not clear how they should be interpreted.
The weighted sums of species (“wa”) triplot is preferable in one case: when RDA
is used as a form of analysis of variance (Sect. 6.3.2.9), because in that situation all
replicate sites with the same combination of factor levels are represented on top of
one another in the fitted site scores (lc) triplot. In such an analysis, one can even plot
both types of scores, with the lc scores showing the centroid of the sites sharing a
given combination of factor levels and the wa scores showing their dispersion.
Of course, the triplots can also be drawn using the wa scores. In the plot()
function, replace argument display ¼ "lc" by "wa".
Independently of the choice of site scores, the interpretation of the constrained
triplots must be preceded by a test of statistical significance of the global canonical
relationship; see below. As in multiple regression, a nonsignificant result must not be
plotted and interpreted; it must be discarded.
For the species and sites, the interpretation of the two scalings is the same as in
PCA. However, the presence of vectors and centroids of explanatory variables calls
for additional interpretation rules. Here are the essential ones (see Legendre and
Legendre 2012 p. 640–641):
• Scaling 1 À distance triplot: (1) The angles between response and explanatory
variables in the triplot reflect their correlations (but not the angles among the
response variables). (2) The relationship between the centroid of a qualitative
explanatory variable and a response variable (species) is found by projecting the
centroid at right angle on the response variable, as for individual objects, since we
are projecting the centroid of a group of objects. (3) Distances among centroids,
and between centroids and individual objects, approximate their Euclidean
distances.
• Scaling 2 À correlation triplot: (1) Projecting an object at right angle on a
response or a quantitative explanatory variable approximates the value of the
object along that variable. (2) The angles in the triplot between response and
explanatory variables, and between response variables themselves or explanatory variables themselves, reflect their correlations. (3) The relationship
between the centroid of a qualitative explanatory variable and a response variable
216
6 Canonical Ordination
choice between these and the site scores that are weighted sums of species scores
(called ‘wa’ in vegan because in CA/CCA they are weighted averages) for the
triplots depends on the context and the purpose of the plot. On the one hand, the
fitted site scores are strictly orthogonal linear combinations of the explanatory
variables; they represent clearly and exclusively what can be modelled using the
explanatory variables at hand. We advocate the use of these lc scores in most
situations because the “true” ordination diagram of RDA is the ordination of the Ŷ
matrix of fitted values. On the other hand, the site scores that are weighted sums of
species appear more robust to noise in the environmental variables: McCune (1997)
showed that if the latter contain much random error, the resulting lc plots may be
completely scrambled. However, weighted sums of species (“wa”) scores are “contaminated” scores, halfway between the model fitted by the RDA procedure and a
PCA of the original data, and as such it is not clear how they should be interpreted.
The weighted sums of species (“wa”) triplot is preferable in one case: when RDA
is used as a form of analysis of variance (Sect. 6.3.2.9), because in that situation all
replicate sites with the same combination of factor levels are represented on top of
one another in the fitted site scores (lc) triplot. In such an analysis, one can even plot
both types of scores, with the lc scores showing the centroid of the sites sharing a
given combination of factor levels and the wa scores showing their dispersion.
Of course, the triplots can also be drawn using the wa scores. In the plot()
function, replace argument display ¼ "lc" by "wa".
Independently of the choice of site scores, the interpretation of the constrained
triplots must be preceded by a test of statistical significance of the global canonical
relationship; see below. As in multiple regression, a nonsignificant result must not be
plotted and interpreted; it must be discarded.
For the species and sites, the interpretation of the two scalings is the same as in
PCA. However, the presence of vectors and centroids of explanatory variables calls
for additional interpretation rules. Here are the essential ones (see Legendre and
Legendre 2012 p. 640–641):
• Scaling 1 À distance triplot: (1) The angles between response and explanatory
variables in the triplot reflect their correlations (but not the angles among the
response variables). (2) The relationship between the centroid of a qualitative
explanatory variable and a response variable (species) is found by projecting the
centroid at right angle on the response variable, as for individual objects, since we
are projecting the centroid of a group of objects. (3) Distances among centroids,
and between centroids and individual objects, approximate their Euclidean
distances.
• Scaling 2 À correlation triplot: (1) Projecting an object at right angle on a
response or a quantitative explanatory variable approximates the value of the
object along that variable. (2) The angles in the triplot between response and
explanatory variables, and between response variables themselves or explanatory variables themselves, reflect their correlations. (3) The relationship
between the centroid of a qualitative explanatory variable and a response variable
216
6 Canonical Ordination