and eigenvectors of that matrix. The eigenvalues represent a partitioning of the
total co-inertia.
• Project the objects and variables of the two original data tables on the co-inertia
axes. By means of graphs, compare the projections of the two data tables in the
common co-inertia space.
Basically, CoIA requires site-by-variables input matrices. The equations are
provided by Legendre and Legendre (2012 Sect. 11.5). An attractive feature of
CoIA is the possibility to adapt it to the mathematical type of the variables of the
two matrices, using the various transformations presented in Chaps. 2 and 3:
standardization of the variables expressed in different physical units; Hellinger,
chord or other appropriate transformations for species presence-absence or abundance data. If the Euclidean distance among object computed from the raw data is to
be preserved, no transformation is necessary.
A function called coinertia() is available in package ade4 to compute
CoIA. However, this function implements the data transformations in a particular
way. For internal technical reasons, the two data tables must first be submitted to an
ade4 ordination function: dudi.pca() for PCA, dudi.pco() for PCoA, and
so on
5 . This intermediate step acts as a transformation route. For untransformed data
(or pretransformed species data), use dudi.pca() with argument
scale ¼ FALSE; to standardize the variables, use the same function with
scale ¼ TRUE. If the required transformation involves computation of a dissimilarity index, it can be computed by the appropriate function (see Chap. 3); dudi.
pco() is then called to extract its principal coordinates. Function coinertia()
retrieves the centred or transformed data matrices from the dudi.xxx() output
objects and computes coinertia analysis from them.
Note also that the row weights must be equal in the two separate ordinations, a
condition that makes the use of CoIA after correspondence analysis (dudi.coa)
difficult. CA is a weighted regression approach, and weights depend on the data, so
that the two data tables are likely to have different row weights. To produce a
symmetric analysis based on CA, we suggest to use the function coca() of
package cocorresp with method ¼ "symmetric" (see Sect. 6.6.2).
6.9.2 Co-inertia Analysis Using Function coinertia()
of ade4
In the code below, we apply CoIA to the chemical and physiographic subsets of
environmental variables of the Doubs data set. Following the route explained above
5 Note that ade4 has been developed around a general mathematical framework involving entities
that will not be described here, called duality diagrams (Escoufier 1987); hence the dudi part of the
function names. Readers are invited to consult the original publication to learn more about this
framework.
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6 Canonical Ordination
total co-inertia.
• Project the objects and variables of the two original data tables on the co-inertia
axes. By means of graphs, compare the projections of the two data tables in the
common co-inertia space.
Basically, CoIA requires site-by-variables input matrices. The equations are
provided by Legendre and Legendre (2012 Sect. 11.5). An attractive feature of
CoIA is the possibility to adapt it to the mathematical type of the variables of the
two matrices, using the various transformations presented in Chaps. 2 and 3:
standardization of the variables expressed in different physical units; Hellinger,
chord or other appropriate transformations for species presence-absence or abundance data. If the Euclidean distance among object computed from the raw data is to
be preserved, no transformation is necessary.
A function called coinertia() is available in package ade4 to compute
CoIA. However, this function implements the data transformations in a particular
way. For internal technical reasons, the two data tables must first be submitted to an
ade4 ordination function: dudi.pca() for PCA, dudi.pco() for PCoA, and
so on
5 . This intermediate step acts as a transformation route. For untransformed data
(or pretransformed species data), use dudi.pca() with argument
scale ¼ FALSE; to standardize the variables, use the same function with
scale ¼ TRUE. If the required transformation involves computation of a dissimilarity index, it can be computed by the appropriate function (see Chap. 3); dudi.
pco() is then called to extract its principal coordinates. Function coinertia()
retrieves the centred or transformed data matrices from the dudi.xxx() output
objects and computes coinertia analysis from them.
Note also that the row weights must be equal in the two separate ordinations, a
condition that makes the use of CoIA after correspondence analysis (dudi.coa)
difficult. CA is a weighted regression approach, and weights depend on the data, so
that the two data tables are likely to have different row weights. To produce a
symmetric analysis based on CA, we suggest to use the function coca() of
package cocorresp with method ¼ "symmetric" (see Sect. 6.6.2).
6.9.2 Co-inertia Analysis Using Function coinertia()
of ade4
In the code below, we apply CoIA to the chemical and physiographic subsets of
environmental variables of the Doubs data set. Following the route explained above
5 Note that ade4 has been developed around a general mathematical framework involving entities
that will not be described here, called duality diagrams (Escoufier 1987); hence the dudi part of the
function names. Readers are invited to consult the original publication to learn more about this
framework.
278
6 Canonical Ordination