Conceptually, RDA is a multivariate (meaning multiresponse) multiple linear
regression followed by a PCA of the matrix of fitted values. It works as follows,
using a matrix Y of centred response data and a matrix X of centred explanatory
variables
1 :
• Regress each (centred) y variable on explanatory matrix X and compute the fitted
(ŷ) (this is the only required matrix in most analyses) and residual (y res ) values
(if needed). Assemble all vectors ŷ into a matrix Ŷ of fitted values.
• Carry out a test of significance of the canonical relationship Y ~ X.
• If the test is significant, i.e. if the X variables explain significantly more of the
variation of Y than random data would, compute a PCA of the matrix Ŷ of fitted
values; this analysis produces a vector of canonical eigenvalues and a matrix U of
canonical eigenvectors;
• Use matrix U to compute two types of ordination site scores: use either the matrix
Ŷ of fitted values to obtain an ordination in the space of variables X (i.e., compute
ŶU, which produces fitted site scores called “Site constraints (linear combinations of constraining variables)” in the summary of the RDA output, coded “lc”
in vegan) or use the original centred matrix Y to obtain an ordination in the
space of the original variables Y (i.e., compute YU, obtaining site scores called
“Site scores (weighted sums of site scores)”, coded “wa” in vegan).
• The residual values from the multiple regressions (i.e. Y res ¼ YÀŶ) may also be
submitted to a PCA to obtain an unconstrained ordination of the residuals. This
partial PCA, which is not strictly speaking part of the RDA, is computed along
with the constrained ordination by vegan’s rda() function.
Additional information on the algebra of RDA will be presented in the Code it
yourself corner at the end of this section.
As can be seen from the conceptual steps presented above, RDA computes axes
that are linear combinations of the explanatory variables. In other words, this method
seeks, in successive order, a series of linear combinations of the explanatory variables that best explain the variation of the response data. The axes defined in the
space of the explanatory variables are orthogonal to one another. RDA is therefore a
constrained ordination procedure. The difference with unconstrained ordination is
important: the matrix of explanatory variables conditions the “weights” (eigenvalues) and the directions of the ordination axes. In RDA, one can truly say that
the axes explain or model (in the statistical sense) the variation of the dependent
matrix. Furthermore, a global hypothesis (H 0 ) of absence of linear relationship
between Y and X can be tested in RDA; this is not the case in PCA.
1 For convenience, some programs and functions standardize the X variables at the beginning of the
calculation. This does not change the RDA results because standardizing X, or not, does not change
the matrix Ŷ of fitted values of the multivariate regression, upon which the RDA statistics, tests of
significance, and PCA are computed.
6.3 Redundancy Analysis (RDA)
205
regression followed by a PCA of the matrix of fitted values. It works as follows,
using a matrix Y of centred response data and a matrix X of centred explanatory
variables
1 :
• Regress each (centred) y variable on explanatory matrix X and compute the fitted
(ŷ) (this is the only required matrix in most analyses) and residual (y res ) values
(if needed). Assemble all vectors ŷ into a matrix Ŷ of fitted values.
• Carry out a test of significance of the canonical relationship Y ~ X.
• If the test is significant, i.e. if the X variables explain significantly more of the
variation of Y than random data would, compute a PCA of the matrix Ŷ of fitted
values; this analysis produces a vector of canonical eigenvalues and a matrix U of
canonical eigenvectors;
• Use matrix U to compute two types of ordination site scores: use either the matrix
Ŷ of fitted values to obtain an ordination in the space of variables X (i.e., compute
ŶU, which produces fitted site scores called “Site constraints (linear combinations of constraining variables)” in the summary of the RDA output, coded “lc”
in vegan) or use the original centred matrix Y to obtain an ordination in the
space of the original variables Y (i.e., compute YU, obtaining site scores called
“Site scores (weighted sums of site scores)”, coded “wa” in vegan).
• The residual values from the multiple regressions (i.e. Y res ¼ YÀŶ) may also be
submitted to a PCA to obtain an unconstrained ordination of the residuals. This
partial PCA, which is not strictly speaking part of the RDA, is computed along
with the constrained ordination by vegan’s rda() function.
Additional information on the algebra of RDA will be presented in the Code it
yourself corner at the end of this section.
As can be seen from the conceptual steps presented above, RDA computes axes
that are linear combinations of the explanatory variables. In other words, this method
seeks, in successive order, a series of linear combinations of the explanatory variables that best explain the variation of the response data. The axes defined in the
space of the explanatory variables are orthogonal to one another. RDA is therefore a
constrained ordination procedure. The difference with unconstrained ordination is
important: the matrix of explanatory variables conditions the “weights” (eigenvalues) and the directions of the ordination axes. In RDA, one can truly say that
the axes explain or model (in the statistical sense) the variation of the dependent
matrix. Furthermore, a global hypothesis (H 0 ) of absence of linear relationship
between Y and X can be tested in RDA; this is not the case in PCA.
1 For convenience, some programs and functions standardize the X variables at the beginning of the
calculation. This does not change the RDA results because standardizing X, or not, does not change
the matrix Ŷ of fitted values of the multivariate regression, upon which the RDA statistics, tests of
significance, and PCA are computed.
6.3 Redundancy Analysis (RDA)
205