6.8 Canonical Correlation Analysis (CCorA)
6.8.1 Introduction
CCorA is computed from two data tables. The aim of the method is to represent the
observations along canonical axes that maximize the correlations between the two
tables. The solution is found by maximizing the between-set dispersion, expressed
by the covariance matrix between the two sets of variables, with respect to the
within-set dispersion (Legendre and Legendre 2012 Sect. 11.4). The two sets of
variables must be quantitative and are assumed to be multinormally distributed. The
limitation of the method is that the total number of variables in each data table must
be smaller than (n – 1).
In CCorA, one can also test the hypothesis of linear independence of the two
multivariate data tables. Pillai and Hsu (1979) have shown that Pillai’s trace is the
most robust statistic to departures from normality.
The availability of RDA and CCA has made the application of CCorA in ecology
less frequent, since most ecological problems are stated in terms of control-response
hypotheses for which asymmetric ordination should be preferred. CCorA is more
appropriate for exploratory purposes and in cases where the two groups of variables
are likely to influence each other, which may often occur in real ecological systems.
Examples are the study of two groups of competing taxa, a vegetation-herbivore
system, and long-term studies of soil-vegetation relationships during a colonization
process; CCorA is possible as long as the number of species in each data table is
smaller than (n–1).
6.8.2 Canonical Correlation Analysis Using CCorA()
In the variation partitioning example of Sect. 6.3.2.8, we used two subsets of
environmental variables, chemistry and physiography, to explain the structure of
the fish data. Putting aside the variation partitioning of that example, we could study
the structure of correlation of the two complete subsets of explanatory variables.
How does chemistry relate to physiography?
Since the data should be as close as possible to the condition of multinormality,
we will transform some variables in the following example to make them more
symmetrically distributed (we used the Shapiro-Wilk normality test, function
shapiro.test() of package stats; results not shown here). The variables
have different physical dimensions. The CCorA equation includes automatic standardization of the variables. See Legendre and Legendre (2012, Sect. 11.4.1) for
details. However, asking for standardization, or not, changes the RDA results at the
end of the CCorA output file.
The function used for the analysis is CCorA() of package vegan.
6.8 Canonical Correlation Analysis (CCorA)
275
6.8.1 Introduction
CCorA is computed from two data tables. The aim of the method is to represent the
observations along canonical axes that maximize the correlations between the two
tables. The solution is found by maximizing the between-set dispersion, expressed
by the covariance matrix between the two sets of variables, with respect to the
within-set dispersion (Legendre and Legendre 2012 Sect. 11.4). The two sets of
variables must be quantitative and are assumed to be multinormally distributed. The
limitation of the method is that the total number of variables in each data table must
be smaller than (n – 1).
In CCorA, one can also test the hypothesis of linear independence of the two
multivariate data tables. Pillai and Hsu (1979) have shown that Pillai’s trace is the
most robust statistic to departures from normality.
The availability of RDA and CCA has made the application of CCorA in ecology
less frequent, since most ecological problems are stated in terms of control-response
hypotheses for which asymmetric ordination should be preferred. CCorA is more
appropriate for exploratory purposes and in cases where the two groups of variables
are likely to influence each other, which may often occur in real ecological systems.
Examples are the study of two groups of competing taxa, a vegetation-herbivore
system, and long-term studies of soil-vegetation relationships during a colonization
process; CCorA is possible as long as the number of species in each data table is
smaller than (n–1).
6.8.2 Canonical Correlation Analysis Using CCorA()
In the variation partitioning example of Sect. 6.3.2.8, we used two subsets of
environmental variables, chemistry and physiography, to explain the structure of
the fish data. Putting aside the variation partitioning of that example, we could study
the structure of correlation of the two complete subsets of explanatory variables.
How does chemistry relate to physiography?
Since the data should be as close as possible to the condition of multinormality,
we will transform some variables in the following example to make them more
symmetrically distributed (we used the Shapiro-Wilk normality test, function
shapiro.test() of package stats; results not shown here). The variables
have different physical dimensions. The CCorA equation includes automatic standardization of the variables. See Legendre and Legendre (2012, Sect. 11.4.1) for
details. However, asking for standardization, or not, changes the RDA results at the
end of the CCorA output file.
The function used for the analysis is CCorA() of package vegan.
6.8 Canonical Correlation Analysis (CCorA)
275