• Model 3: lottery. Permute the species data within each row (site). The null
hypothesis of this model is that the distribution of presences of the species at a
site is the result of a random allocation process.
• Model 4: random species attributes. Permute entire columns of matrix L (A). H 0
states that species are distributed according to their preferences for site conditions
(this is preserved through the permutations), but independently from their traits.
• Model 5: permute rows and columns. In turn, permute entire rows then entire
columns (or the reverse). H 0 states that the species distributions are not related to
their traits or to the site conditions. This is equivalent to permuting the rows of
R (C) and the columns of Q’ (B), as done by Dolédec et al. (1996) in their RLQ
method.
• Model 6: this is actually a combination of models 2 and 4. A first form of this
combination was proposed by Dray and Legendre (2008), who noted, however,
that it suffered from a strong inflation of type I error rate when L (A) is only
linked to one other table (R or Q). ter Braak et al. (2012) proposed to overcome
this problem by considering the two tests sequentially and rejecting the overall
null hypothesis (i.e., traits and environment unrelated) only if both tests (models
2 and 4) reject H 0 at the α level. The maximum p-value becomes the overall pvalue. These authors showed that this procedure ensures a correct level of type I
error and a good power if the number of species is sufficient (at least 30).
Dray et al. (2014) raised the following issue: the fourth-corner method involves
multiple tests, a situation where the overall rate of type I error (i.e., the risk of finding
at least one false rejection of H 0 ) is increased. This calls for a correction of the pvalues, which Legendre et al. (1997) were already advocating. To improve the
testing procedure, Dray et al. (2014) proposed a sequential computation of the
tests, followed by a correction for multiple testing. The Holm (1979) correction or
the false discovery rate method (FDR; Benjamini and Hochberg 1995) can be used.
ter Braak (2017) revisited the fourth-corner analysis by focussing on the fourthcorner correlation statistic used with species abundances, and showing that “the
squared fourth-corner correlation times the total count is precisely the score test
statistic for testing the linear-by-linear interaction in a Poisson log-linear model
that also contains species and sites as main effects”. Thus, he bridged the gap
between the fourth-corner analysis and an alternative approach based on generalized
mixed models.
6.11.2 RLQ Analysis
RLQ analysis (Dolédec et al. 1996) is an extension of co-inertia analysis (CoIA,
Sect. 6.9) producing a simultaneous ordination of three tables. The method works
upon three separate ordinations, one for each data matrix and adapted to its mathematical type, and combines the three to identify the main relationships between the
environment and the traits, as mediated by the species. It computes a generalized
290
6 Canonical Ordination
hypothesis of this model is that the distribution of presences of the species at a
site is the result of a random allocation process.
• Model 4: random species attributes. Permute entire columns of matrix L (A). H 0
states that species are distributed according to their preferences for site conditions
(this is preserved through the permutations), but independently from their traits.
• Model 5: permute rows and columns. In turn, permute entire rows then entire
columns (or the reverse). H 0 states that the species distributions are not related to
their traits or to the site conditions. This is equivalent to permuting the rows of
R (C) and the columns of Q’ (B), as done by Dolédec et al. (1996) in their RLQ
method.
• Model 6: this is actually a combination of models 2 and 4. A first form of this
combination was proposed by Dray and Legendre (2008), who noted, however,
that it suffered from a strong inflation of type I error rate when L (A) is only
linked to one other table (R or Q). ter Braak et al. (2012) proposed to overcome
this problem by considering the two tests sequentially and rejecting the overall
null hypothesis (i.e., traits and environment unrelated) only if both tests (models
2 and 4) reject H 0 at the α level. The maximum p-value becomes the overall pvalue. These authors showed that this procedure ensures a correct level of type I
error and a good power if the number of species is sufficient (at least 30).
Dray et al. (2014) raised the following issue: the fourth-corner method involves
multiple tests, a situation where the overall rate of type I error (i.e., the risk of finding
at least one false rejection of H 0 ) is increased. This calls for a correction of the pvalues, which Legendre et al. (1997) were already advocating. To improve the
testing procedure, Dray et al. (2014) proposed a sequential computation of the
tests, followed by a correction for multiple testing. The Holm (1979) correction or
the false discovery rate method (FDR; Benjamini and Hochberg 1995) can be used.
ter Braak (2017) revisited the fourth-corner analysis by focussing on the fourthcorner correlation statistic used with species abundances, and showing that “the
squared fourth-corner correlation times the total count is precisely the score test
statistic for testing the linear-by-linear interaction in a Poisson log-linear model
that also contains species and sites as main effects”. Thus, he bridged the gap
between the fourth-corner analysis and an alternative approach based on generalized
mixed models.
6.11.2 RLQ Analysis
RLQ analysis (Dolédec et al. 1996) is an extension of co-inertia analysis (CoIA,
Sect. 6.9) producing a simultaneous ordination of three tables. The method works
upon three separate ordinations, one for each data matrix and adapted to its mathematical type, and combines the three to identify the main relationships between the
environment and the traits, as mediated by the species. It computes a generalized
290
6 Canonical Ordination