6.4 Canonical Correspondence Analysis (CCA)
6.4.1 Introduction
The canonical counterpart of CA, canonical correspondence analysis, has been
acclaimed by ecologists ever since its introduction (ter Braak, 1986, 1987, 1988).
It shares many characteristics with RDA, so that a detailed description is not
necessary here. Basically, it is a weighted form of RDA applied to the same matrix
Q of contributions to the χ
2 statistic as used in CA (Legendre and Legendre 2012
Sect. 11.2). CCA shares the basic properties of CA, combined with those of a
constrained ordination. It preserves the χ
2 distance among sites, and species are
represented as points in the triplots. ter Braak (1986) has shown that, provided that
some conditions are fulfilled
4 , CCA is a good approximation of a multivariate
Gaussian regression. One particularly attractive feature of a CCA triplot is that
species are ordered along the canonical axes following their ecological optima.
This allows a relatively easy ecological interpretation of species assemblages.
Also, species scores can be used as synthetic descriptors in a clustering procedure
(for instance k-means partitioning) to produce a typology of the species in
assemblages.
CCA does have some drawbacks, however, related to the mathematical properties
of the χ
2 distance. Legendre and Gallagher (2001) state that “a difference between
abundance values for a common species contributes less to the distance than the
same difference for a rare species, so that rare species may have an unduly large
influence on the analysis”. Despite its widespread use, “the χ
2 distance is not
unanimously accepted among ecologists; using simulations, Faith et al. (1987)
concluded that it was one of the worst distances for community composition data”
(Legendre and Gallagher 2001). Its use should be limited to situations where rare
species are well sampled and are seen as potential indicators of particular characteristics of an ecosystem; the alternative is to eliminate rare species from the data table
before CCA. These problems, among other points, have led to the development of
the species pre-transformations to open these data to the realm of RDA, ANOVA
and other linear methods. The proportion of total inertia represented by explained
inertia (inertia is the measure of variation of the data in CCA), which can be
interpreted as an R
2 , is also biased, but Ezekiel’s adjustment cannot be used.
However, a bootstrap procedure has been developed for its estimation (Peres-Neto
et al. 2006).
Despite these shortcomings, CCA is still widely used and deserves an illustration.
4 Two important conditions are that the species must have been sampled along their whole
ecological range and that they display unimodal responses towards their main ecological constraints. These conditions are difficult to test formally, but graphs of species abundances in sites
arranged along their scores on the first few CA ordination axes may help visualize their distributions
along the main ecological gradients.
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6 Canonical Ordination
6.4.1 Introduction
The canonical counterpart of CA, canonical correspondence analysis, has been
acclaimed by ecologists ever since its introduction (ter Braak, 1986, 1987, 1988).
It shares many characteristics with RDA, so that a detailed description is not
necessary here. Basically, it is a weighted form of RDA applied to the same matrix
Q of contributions to the χ
2 statistic as used in CA (Legendre and Legendre 2012
Sect. 11.2). CCA shares the basic properties of CA, combined with those of a
constrained ordination. It preserves the χ
2 distance among sites, and species are
represented as points in the triplots. ter Braak (1986) has shown that, provided that
some conditions are fulfilled
4 , CCA is a good approximation of a multivariate
Gaussian regression. One particularly attractive feature of a CCA triplot is that
species are ordered along the canonical axes following their ecological optima.
This allows a relatively easy ecological interpretation of species assemblages.
Also, species scores can be used as synthetic descriptors in a clustering procedure
(for instance k-means partitioning) to produce a typology of the species in
assemblages.
CCA does have some drawbacks, however, related to the mathematical properties
of the χ
2 distance. Legendre and Gallagher (2001) state that “a difference between
abundance values for a common species contributes less to the distance than the
same difference for a rare species, so that rare species may have an unduly large
influence on the analysis”. Despite its widespread use, “the χ
2 distance is not
unanimously accepted among ecologists; using simulations, Faith et al. (1987)
concluded that it was one of the worst distances for community composition data”
(Legendre and Gallagher 2001). Its use should be limited to situations where rare
species are well sampled and are seen as potential indicators of particular characteristics of an ecosystem; the alternative is to eliminate rare species from the data table
before CCA. These problems, among other points, have led to the development of
the species pre-transformations to open these data to the realm of RDA, ANOVA
and other linear methods. The proportion of total inertia represented by explained
inertia (inertia is the measure of variation of the data in CCA), which can be
interpreted as an R
2 , is also biased, but Ezekiel’s adjustment cannot be used.
However, a bootstrap procedure has been developed for its estimation (Peres-Neto
et al. 2006).
Despite these shortcomings, CCA is still widely used and deserves an illustration.
4 Two important conditions are that the species must have been sampled along their whole
ecological range and that they display unimodal responses towards their main ecological constraints. These conditions are difficult to test formally, but graphs of species abundances in sites
arranged along their scores on the first few CA ordination axes may help visualize their distributions
along the main ecological gradients.
256
6 Canonical Ordination