This type of study poses methodological challenges to researchers because to test
hypotheses in this context, one must evaluate the relationships between species traits
and environmental characteristics, as mediated by the species presence-absence or
abundance data. Two related methods have been proposed to answer this type of
questions: the RLQ (Dolédec et al. 1996) and the fourth-corner methods (Legendre
et al. 1997, Dray and Legendre 2008, Legendre and Legendre 2012). As explained
by Dray et al. (2014), the two methods are complementary: the former is an
ordination method allowing the visualization of the joint structure resulting from
the three data tables, but with a single global test, whereas the latter consists in a
series of statistical tests of individual trait-environment relationships, without consideration of the covariation among traits or environmental variables and no output
about the sites and the species. These authors proposed some adjustments to improve
the complementarity of the methods.
The data consist of three matrices (Fig. 6.21):
• a matrix of n sites by p species (presence-absence or abundance data), called
A (Legendre en al. 1997) or L (Dray and Legendre 2008; Dray et al. 2014);
• a matrix of p species by s biological or behavioural traits, called B or Q;
• a matrix of n sites by m habitat characteristics, called C or R.
Therefore, the data contains information in three matrices, about the relationships
between species and environment and about the species traits. The purpose of both
methods is to gain knowledge about the relationships between the species traits and
the environment. RLQ analyses the joint structure of R and Q mediated by matrix L,
which serves as a link between R and Q. The s  m matrix crossing traits and
environmental variables is the fourth one (hence the name “fourth corner”; see the
matrix arrangement in Fig. 6.21), called D.
6.11.1 The Fourth-Corner Method
The original fourth-corner method (Legendre et al. 1997) was limited to presenceabsence data and the analysis of a single trait and a single environmental variable at a
time. An extension to abundance data, several traits and environmental variables,
with improvements on the testing procedures, was proposed by Dray and
Legendre (2008).
The principle of the fourth-corner method consists in (1) estimating the parameters found in matrix D and (2) testing the significance of these parameters, choosing
the most appropriate permutational model among 6 possibilities. The parameters of
D can be estimated by matrix product as follows (Legendre and Legendre 2012), the
two approaches producing the same D matrix:
288
6 Canonical Ordination
hypotheses in this context, one must evaluate the relationships between species traits
and environmental characteristics, as mediated by the species presence-absence or
abundance data. Two related methods have been proposed to answer this type of
questions: the RLQ (Dolédec et al. 1996) and the fourth-corner methods (Legendre
et al. 1997, Dray and Legendre 2008, Legendre and Legendre 2012). As explained
by Dray et al. (2014), the two methods are complementary: the former is an
ordination method allowing the visualization of the joint structure resulting from
the three data tables, but with a single global test, whereas the latter consists in a
series of statistical tests of individual trait-environment relationships, without consideration of the covariation among traits or environmental variables and no output
about the sites and the species. These authors proposed some adjustments to improve
the complementarity of the methods.
The data consist of three matrices (Fig. 6.21):
• a matrix of n sites by p species (presence-absence or abundance data), called
A (Legendre en al. 1997) or L (Dray and Legendre 2008; Dray et al. 2014);
• a matrix of p species by s biological or behavioural traits, called B or Q;
• a matrix of n sites by m habitat characteristics, called C or R.
Therefore, the data contains information in three matrices, about the relationships
between species and environment and about the species traits. The purpose of both
methods is to gain knowledge about the relationships between the species traits and
the environment. RLQ analyses the joint structure of R and Q mediated by matrix L,
which serves as a link between R and Q. The s  m matrix crossing traits and
environmental variables is the fourth one (hence the name “fourth corner”; see the
matrix arrangement in Fig. 6.21), called D.
6.11.1 The Fourth-Corner Method
The original fourth-corner method (Legendre et al. 1997) was limited to presenceabsence data and the analysis of a single trait and a single environmental variable at a
time. An extension to abundance data, several traits and environmental variables,
with improvements on the testing procedures, was proposed by Dray and
Legendre (2008).
The principle of the fourth-corner method consists in (1) estimating the parameters found in matrix D and (2) testing the significance of these parameters, choosing
the most appropriate permutational model among 6 possibilities. The parameters of
D can be estimated by matrix product as follows (Legendre and Legendre 2012), the
two approaches producing the same D matrix:
288
6 Canonical Ordination