6.2 Canonical Ordination Overview
In the methods explored in Chap. 5, the ordination procedure itself is not influenced
by external variables; these may only be considered after the computation of the
ordination. One lets the data matrix express the relationships among objects and
variables without constraint. This is an exploratory, descriptive approach. Canonical
ordination, on the contrary, explicitly explores the relationships between two matrices: a response matrix and an explanatory matrix in some cases (asymmetric
analysis), or two matrices with symmetric roles in other cases. Both matrices are
used in the production of the ordination.
The way to combine the information of two (or, in some cases, more) data
matrices depends on the method of analysis. We will first explore the two asymmetric methods that are mostly used in ecology nowadays, i.e., redundancy analysis
(RDA) and canonical correspondence analysis (CCA). Both combine multiple
regression with classical ordination (PCA or CA). Partial RDA will also be explored,
as well as a procedure of variation partitioning. The significance of canonical
ordinations will be tested by means of permutations. After that, we will devote
sections to three other asymmetric methods: linear discriminant analysis (LDA),
which looks for a linear combination of explanatory variables to explain a predefined
grouping of the objects, principal response curves (PRC), developed for the analysis
of multivariate results of designed experiments that involve repeated measurements,
and co-correspondence analysis (CoCA), which is devoted to the simultaneous
ordination of two communities sampled at the same sites. Then, we turn to three
symmetric methods that compute eigenvectors describing the common structure of
two or several data sets: canonical correlation analysis (CCorA), co-inertia analysis
(CoIA) and multiple factor analysis (MFA). We conclude the chapter by visiting two
methods devoted to the study of the relationship between species traits and the
environment: the fourth-corner method and the RLQ analysis.
6.3 Redundancy Analysis (RDA)
6.3.1 Introduction
RDA is a method combining regression and principal component analysis (PCA). It
is a direct extension of multiple regression analysis to model multivariate response
data. RDA is an extremely powerful tool in the hands of ecologists, especially since
the introduction of the Legendre and Gallagher (2001) transformations that opened
RDA to the analysis of community composition data (transformation-based RDA, or
tb-RDA).
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6 Canonical Ordination
In the methods explored in Chap. 5, the ordination procedure itself is not influenced
by external variables; these may only be considered after the computation of the
ordination. One lets the data matrix express the relationships among objects and
variables without constraint. This is an exploratory, descriptive approach. Canonical
ordination, on the contrary, explicitly explores the relationships between two matrices: a response matrix and an explanatory matrix in some cases (asymmetric
analysis), or two matrices with symmetric roles in other cases. Both matrices are
used in the production of the ordination.
The way to combine the information of two (or, in some cases, more) data
matrices depends on the method of analysis. We will first explore the two asymmetric methods that are mostly used in ecology nowadays, i.e., redundancy analysis
(RDA) and canonical correspondence analysis (CCA). Both combine multiple
regression with classical ordination (PCA or CA). Partial RDA will also be explored,
as well as a procedure of variation partitioning. The significance of canonical
ordinations will be tested by means of permutations. After that, we will devote
sections to three other asymmetric methods: linear discriminant analysis (LDA),
which looks for a linear combination of explanatory variables to explain a predefined
grouping of the objects, principal response curves (PRC), developed for the analysis
of multivariate results of designed experiments that involve repeated measurements,
and co-correspondence analysis (CoCA), which is devoted to the simultaneous
ordination of two communities sampled at the same sites. Then, we turn to three
symmetric methods that compute eigenvectors describing the common structure of
two or several data sets: canonical correlation analysis (CCorA), co-inertia analysis
(CoIA) and multiple factor analysis (MFA). We conclude the chapter by visiting two
methods devoted to the study of the relationship between species traits and the
environment: the fourth-corner method and the RLQ analysis.
6.3 Redundancy Analysis (RDA)
6.3.1 Introduction
RDA is a method combining regression and principal component analysis (PCA). It
is a direct extension of multiple regression analysis to model multivariate response
data. RDA is an extremely powerful tool in the hands of ecologists, especially since
the introduction of the Legendre and Gallagher (2001) transformations that opened
RDA to the analysis of community composition data (transformation-based RDA, or
tb-RDA).
204
6 Canonical Ordination