As in the case of PCA, this output requires some explanations. Some of the results
are similar to those of a PCA, but additional elements are provided.
• Partitioning of variance: the overall variance is partitioned into constrained and
unconstrained fractions. The constrained fraction is the amount of variance of the
Y matrix explained by the explanatory variables. Expressed as a proportion, it is
equivalent to an R
2 in multiple regression; in RDA this quantity is also called the
bimultivariate redundancy statistic. However, this R
2 is biased, like the
unadjusted R
2 of multiple regression, as shown by Peres-Neto et al. (2006). We
will present the computation of an adjusted, unbiased R
2 below.
• Eigenvalues and their contribution to the variance: this analysis yielded
12 canonical axes (with eigenvalues labelled RDA1 to RDA12) and 16 additional,
unconstrained axes for the residuals (with eigenvalues labelled PC1 to PC16).
The results give the eigenvalues themselves, as well as the cumulative proportion
of variance explained (for the RDA axes) or represented (for the residual axes).
The last cumulative value is therefore 1. The cumulative contribution to the
variance obtained by the 12 canonical axes is the (biased) proportion of the
total variance of the response data explained by the RDA. It is the same value
as the “Proportion constrained” presented above; it is 0.7271 in this example.
• One feature of the eigenvalues is worth mentioning. Observe that the canonical
eigenvalues RDA1 to RDA12 are (of course) decreasing in values; however, the
first residual eigenvalue (PC1) is larger than the last canonical eigenvalue (in this
example it is actually larger than most RDA eigenvalues). This means that the
first residual structure (axis) of the data has more variance than some of the
structures that can be explained by the explanatory variables in X. It is up to the
user to exploit this information, for example by plotting the first pair of residual
axes and devising hypotheses about the causes of the features revealed. These
causes should not, however, involve variables that have already been used in the
model, except if one suspects that products (interactions) or higher-order combinations (e.g. squared variables) may be required.
• An important distinction must be made: the canonical (RDAx) eigenvalues
measure amounts of variance explained by the RDA model, whereas the residual
(PCx) eigenvalues measures amounts of variance represented by the residual
axes, but not explained by any model.
• Accumulated constrained eigenvalues: these are cumulative amounts of variance expressed as proportions of the total explained variance, as opposed to their
contributions to the total variance described above.
• Species scores are the coordinates of the tips of the vectors representing the
response variables in the bi- or triplots. As in PCA, they depend on the scaling
chosen: scaling 1 or scaling 2.
• Site scores (weighted sums of species scores): coordinates of the sites as
expressed in the space of the response variables Y.
• Site constraints (linear combinations of constraining variables): coordinates
of the sites in the space of the explanatory variables X. These are the fitted site
scores.
6.3 Redundancy Analysis (RDA)
211
are similar to those of a PCA, but additional elements are provided.
• Partitioning of variance: the overall variance is partitioned into constrained and
unconstrained fractions. The constrained fraction is the amount of variance of the
Y matrix explained by the explanatory variables. Expressed as a proportion, it is
equivalent to an R
2 in multiple regression; in RDA this quantity is also called the
bimultivariate redundancy statistic. However, this R
2 is biased, like the
unadjusted R
2 of multiple regression, as shown by Peres-Neto et al. (2006). We
will present the computation of an adjusted, unbiased R
2 below.
• Eigenvalues and their contribution to the variance: this analysis yielded
12 canonical axes (with eigenvalues labelled RDA1 to RDA12) and 16 additional,
unconstrained axes for the residuals (with eigenvalues labelled PC1 to PC16).
The results give the eigenvalues themselves, as well as the cumulative proportion
of variance explained (for the RDA axes) or represented (for the residual axes).
The last cumulative value is therefore 1. The cumulative contribution to the
variance obtained by the 12 canonical axes is the (biased) proportion of the
total variance of the response data explained by the RDA. It is the same value
as the “Proportion constrained” presented above; it is 0.7271 in this example.
• One feature of the eigenvalues is worth mentioning. Observe that the canonical
eigenvalues RDA1 to RDA12 are (of course) decreasing in values; however, the
first residual eigenvalue (PC1) is larger than the last canonical eigenvalue (in this
example it is actually larger than most RDA eigenvalues). This means that the
first residual structure (axis) of the data has more variance than some of the
structures that can be explained by the explanatory variables in X. It is up to the
user to exploit this information, for example by plotting the first pair of residual
axes and devising hypotheses about the causes of the features revealed. These
causes should not, however, involve variables that have already been used in the
model, except if one suspects that products (interactions) or higher-order combinations (e.g. squared variables) may be required.
• An important distinction must be made: the canonical (RDAx) eigenvalues
measure amounts of variance explained by the RDA model, whereas the residual
(PCx) eigenvalues measures amounts of variance represented by the residual
axes, but not explained by any model.
• Accumulated constrained eigenvalues: these are cumulative amounts of variance expressed as proportions of the total explained variance, as opposed to their
contributions to the total variance described above.
• Species scores are the coordinates of the tips of the vectors representing the
response variables in the bi- or triplots. As in PCA, they depend on the scaling
chosen: scaling 1 or scaling 2.
• Site scores (weighted sums of species scores): coordinates of the sites as
expressed in the space of the response variables Y.
• Site constraints (linear combinations of constraining variables): coordinates
of the sites in the space of the explanatory variables X. These are the fitted site
scores.
6.3 Redundancy Analysis (RDA)
211