segments has a strong influence on the result. DCA results presented in the
literature suggest that the scores along the second axis are essentially meaningless. The authors of this book strongly warn against the use of this form of DCA
as an ordination technique; however, it may be used to estimate the “gradient
length” of the first ordination axis, expressed in standard deviation units of
species turnover. A gradient length larger than 4 indicates that some species
have a unimodal response along the axis (ter Braak and Šmilauer 2002).
• Detrending by polynomials: another line of reasoning about the origin of the arch
effect leads to the observation that when an arch occurs, the second axis can be
seen as quadratically related to the first (i.e. it is the first axis to the power 2). This
explains the parabolic shape of the scatter of points. Hence, a solution is to make
the second axis not only linearly, but also quadratically independent from the
first. Although intuitively attractive, this method of detrending should be applied
with caution because it actually imposes a constraining model on the data.
DCA by segments is available in package vegan (function decorana()). In
the output of this function, the gradient length of the axes is called “Axis lengths”.
Given all its problems (see discussion in Legendre and Legendre 2012,
p. 482–487), we will not describe this method further here. We now know that,
even with long ecological gradients, a meaningful ordination can be obtained by
using PCA of chord, Hellinger, or log-chord-transformed species data; see Sects.
3.5, 5.3.3, and 5.3.4.
An even more extreme effect of the same kind exists in PCA. It is called the
horseshoe effect because, in the case of strong gradients, the sites at both ends bend
inwards and appear closer than other pairs. This is due to the fact that PCA considers
double zeros as resemblances. Consequently, sites located at opposite ends of an
ecological gradient, having many double zeros, “resemble” each other in this
respect. The Hellinger or chord transformations of the species data partly alleviate
this problem.
5.4.5 Multiple Correspondence Analysis (MCA)
Multiple correspondence analysis (MCA) is the counterpart of PCA for the ordination of a table of categorical variables, i.e. a data frame in which all variables are
factors. It is a special form of correspondence analysis where the variables are
categorical. It has been designed primarily to analyse a series of individuals
(e.g. persons in a survey, specimens in a taxonomic study) characterized by qualitative variables (e.g. questions with a choice of answers, or morphological characteristics). MCA can also be useful in environmental studies if the sites are described
by qualitative variables.
In MCA, the variation of the data is expressed as inertia, as in CA. In most real
cases, the inertia of the first few axes is relatively low when compared to the inertia
of a CA axis or the variance of a PCA axis, because MCA computation involves the
5.4 Correspondence Analysis (CA)
183
literature suggest that the scores along the second axis are essentially meaningless. The authors of this book strongly warn against the use of this form of DCA
as an ordination technique; however, it may be used to estimate the “gradient
length” of the first ordination axis, expressed in standard deviation units of
species turnover. A gradient length larger than 4 indicates that some species
have a unimodal response along the axis (ter Braak and Šmilauer 2002).
• Detrending by polynomials: another line of reasoning about the origin of the arch
effect leads to the observation that when an arch occurs, the second axis can be
seen as quadratically related to the first (i.e. it is the first axis to the power 2). This
explains the parabolic shape of the scatter of points. Hence, a solution is to make
the second axis not only linearly, but also quadratically independent from the
first. Although intuitively attractive, this method of detrending should be applied
with caution because it actually imposes a constraining model on the data.
DCA by segments is available in package vegan (function decorana()). In
the output of this function, the gradient length of the axes is called “Axis lengths”.
Given all its problems (see discussion in Legendre and Legendre 2012,
p. 482–487), we will not describe this method further here. We now know that,
even with long ecological gradients, a meaningful ordination can be obtained by
using PCA of chord, Hellinger, or log-chord-transformed species data; see Sects.
3.5, 5.3.3, and 5.3.4.
An even more extreme effect of the same kind exists in PCA. It is called the
horseshoe effect because, in the case of strong gradients, the sites at both ends bend
inwards and appear closer than other pairs. This is due to the fact that PCA considers
double zeros as resemblances. Consequently, sites located at opposite ends of an
ecological gradient, having many double zeros, “resemble” each other in this
respect. The Hellinger or chord transformations of the species data partly alleviate
this problem.
5.4.5 Multiple Correspondence Analysis (MCA)
Multiple correspondence analysis (MCA) is the counterpart of PCA for the ordination of a table of categorical variables, i.e. a data frame in which all variables are
factors. It is a special form of correspondence analysis where the variables are
categorical. It has been designed primarily to analyse a series of individuals
(e.g. persons in a survey, specimens in a taxonomic study) characterized by qualitative variables (e.g. questions with a choice of answers, or morphological characteristics). MCA can also be useful in environmental studies if the sites are described
by qualitative variables.
In MCA, the variation of the data is expressed as inertia, as in CA. In most real
cases, the inertia of the first few axes is relatively low when compared to the inertia
of a CA axis or the variance of a PCA axis, because MCA computation involves the
5.4 Correspondence Analysis (CA)
183