In vegan, partial RDA can be run in two ways, depending on whether one uses
the simple syntax or the formula interface. In the first case, explanatory variables and
covariables may or may not be assembled in separate objects, which can be vectors,
matrices or data frames; factor variables cannot be used with that notation, however.
With the formula interface, the X and W variables must be in the same object, which
must be a data frame and may contain factor variables.
# Simple syntax; X and W may be in separate tables of quantitative
# variables
(spechem.physio <- rda(spe.hel, envchem, envtopo))
summary(spechem.physio)
# Formula interface; the X and W variables must be in the same
# data frame
(spechem.physio2
Hint The formula interface may seem cumbersome, but it allows a better control of the
model and the use of factors and interactions among the constraints (X) or the
conditions (W). For example, one could have used the factor-transformed slo
variable in the second analysis, but not in the first one.
The results of the two analyses are identical.
Here again, some additional explanations are needed about the summary output.
• Partitioning of variance: This item now shows four components. The first one
(Total) is, as usual, the total inertia (variance in this case) of the response data.
The second line (Conditioned) gives the amount of variance that has been
explained by the covariables and removed
3 . The third line (Constrained)
gives the amount of variance uniquely explained by the explanatory variables.
The fourth line (Unconstrained) gives the residual variance. Beware: the
values given as proportions (right-hand column) are unadjusted and are therefore
biased. For a proper computation of unbiased, adjusted R
2 and partial R
2 , see
below (variation partitioning).
• Eigenvalues, and their contribution to the variance after removing the
contributions of conditioning variables: these values and proportions are partial in the sense that the effects of the covariables have been removed. The sum of
all these eigenvalues corresponds therefore to the sum of the constrained and
3 Mathematically, to partial out the effect of a matrix W from a canonical ordination of Y by X, one
computes the residuals of a multivariate multiple regression of X on W and uses these residuals as
the explanatory variables.
222
6 Canonical Ordination
the simple syntax or the formula interface. In the first case, explanatory variables and
covariables may or may not be assembled in separate objects, which can be vectors,
matrices or data frames; factor variables cannot be used with that notation, however.
With the formula interface, the X and W variables must be in the same object, which
must be a data frame and may contain factor variables.
# Simple syntax; X and W may be in separate tables of quantitative
# variables
(spechem.physio <- rda(spe.hel, envchem, envtopo))
summary(spechem.physio)
# Formula interface; the X and W variables must be in the same
# data frame
(spechem.physio2
Hint The formula interface may seem cumbersome, but it allows a better control of the
model and the use of factors and interactions among the constraints (X) or the
conditions (W). For example, one could have used the factor-transformed slo
variable in the second analysis, but not in the first one.
The results of the two analyses are identical.
Here again, some additional explanations are needed about the summary output.
• Partitioning of variance: This item now shows four components. The first one
(Total) is, as usual, the total inertia (variance in this case) of the response data.
The second line (Conditioned) gives the amount of variance that has been
explained by the covariables and removed
3 . The third line (Constrained)
gives the amount of variance uniquely explained by the explanatory variables.
The fourth line (Unconstrained) gives the residual variance. Beware: the
values given as proportions (right-hand column) are unadjusted and are therefore
biased. For a proper computation of unbiased, adjusted R
2 and partial R
2 , see
below (variation partitioning).
• Eigenvalues, and their contribution to the variance after removing the
contributions of conditioning variables: these values and proportions are partial in the sense that the effects of the covariables have been removed. The sum of
all these eigenvalues corresponds therefore to the sum of the constrained and
3 Mathematically, to partial out the effect of a matrix W from a canonical ordination of Y by X, one
computes the residuals of a multivariate multiple regression of X on W and uses these residuals as
the explanatory variables.
222
6 Canonical Ordination