Combining the fourth‐corner and the RLQ methods for assessing trait responses to environmental variation

Abstract

Assessing trait responses to environmental gradients requires the simultaneous analysis of the information contained in three tables: L (species distribution across samples), R (environmental characteristics of samples), and Q (species traits). Among the available methods, the so‐called fourth‐corner and RLQ methods are two appealing alternatives that provide a direct way to test and estimate trait–environment relationships. Both methods are based on the analysis of the fourth‐corner matrix, which crosses traits and environmental variables weighted by species abundances. However, they differ greatly in their outputs: RLQ is a multivariate technique that provides ordination scores to summarize the joint structure among the three tables, whereas the fourth‐corner method mainly tests for individual trait–environment relationships (i.e., one trait and one environmental variable at a time). Here, we illustrate how the complementarity between these two methods can be exploited to promote new ecological knowledge and to improve the study of trait–environment relationships. After a short description of each method, we apply them to real ecological data to present their different outputs and provide hints about the gain resulting from their combined use.

Fourth-corner correlation is a score test statistic in a log-linear trait–environment model that is useful in permutation testing

“…However, none of their strategies worked satisfactorily under all models they considered (Dray and Legendre 2008). Eventually, ter Braak et al (2012) derived a strategy based on the sequential rejection principle (Goeman and Solari 2010) that controlled the type I error in data generated from any of models considered by Dray et al (2014). This sequential strategy involves both row and column permutation (see Sect.…”

“…In such sampling, there will be intra-specific (and intra-site) variation, which is ignored in the original formulation of Legendre et al (1997), but could certainly be accounted for. Significance testing of the correlation by permutation procedures proceeds similarly to the nominal variables case (Dray et al 2014). The rationale followed by Dolédec et al (1996) to arrive at an equivalent solution is completely different; it is based on ways to constrain row and column scores in statistical triplets (Cailliez and Pagès 1976;Tenenhaus and Young 1985) defining a correspondence analysis.…”

“…Moreover, for multiple traits and environmental variables, the score test statistic is precisely y ++ times the total inertia of a doubly constrained correspondence analysis (Kleyer 2012;Lavorel et al 1999Lavorel et al , 1998, which is the natural generalization of a singly constrained correspondence analysis, known as canonical correspondence analysis (Takane 2013;ter Braak 1986ter Braak , 2014. It is also the natural generalization of RLQ (Dolédec et al 1996;Dray et al 2014) for correlated traits and environmental variables.…”

The front door to the fourth corner: variations on the sample unit × trait matrix in community ecology

“…If we adopt parametric statistics, relationships between the variation in traits and environment can be modeled simultaneously, with the SU × species matrix providing the linkage between the trait matrix and the environmental matrix. Two methods for this are the fourth corner analysis Legendre 2008, ter Braak et al 2012) and RLQ analysis (Dray et al 2003, Dray et al 2014). …”

Caracterización morfofisiológica de semillas de especies leñosas distribuidas en dos zonas secas presentes en el Sur del Ecuador

“…Las asociaciones significativas desaparecieron al ajustar los valores de significación mediante Bonferroni o FDR (método de la tasa de falso descubrimiento; Benjamini y Hochberg, 1995) para así controlar la tasa de error global. El aumento en el número de permutaciones a 50000 (Dray et al, 2014) tampoco condujo a ninguna relación de consideración. Resultados similares se encontraron en otro grupo diferente de especies (hormigas) donde no encontraron relaciones entre rasgos morfológicos con variables de microhábitat (Yates et al 2014).…”