Ecology, 95(1), 2014, pp.

14–21
Ó 2014 by the Ecological Society of America

Combining the fourth-corner and the RLQ methods for assessing

trait responses to environmental variation
STÉPHANE DRAY,1,9 PHILIPPE CHOLER,2,3 SYLVAIN DOLÉDEC,4 PEDRO R. PERES-NETO,5 WILFRIED THUILLER,2
SANDRINE PAVOINE,6,7 AND CAJO J. F. TER BRAAK8
1
Universite´ de Lyon, F-69000, Lyon, Université Lyon 1, CNRS, UMR5558, Laboratoire de Biome´trie et Biologie Evolutive,
F-69622 Villeurbanne, France
2
Laboratoire d’Ecologie Alpine, UMR CNRS-UJF 5553, Universite´ J. Fourier, BP 53, F-38041 Grenoble, France
3
Station Alpine J. Fourier, UMS CNRS-UJF 3370, Universite´ J. Fourier, BP 53, F-38041 Grenoble, France
4
Universite´ de Lyon, F-69000 Lyon, Universite´ Lyon 1, CNRS, UMR5023, Laboratoire d’Ecologie
des Hydrosystèmes Naturels et Anthropisés, F-69622 Villeurbanne, France
5
Canada Research Chair in Spatial Modelling and Biodiversity, De´partement des Sciences Biologiques,
Universite´ du Québec à Montréal, Case Postale 8888, Succursale Centre-Ville, Montre´al, Quebec H3C 3P8 Canada
6
UMR CNRS-MNHN 7204, Muse´um National d’Histoire Naturelle, De´partement d’Ecologie et Gestion de la Biodiversité,
Paris, France
7
Mathematical Ecology Research Group, Department of Zoology, University of Oxford, South Parks Road,
Oxford OX1 3PS United Kingdom
8
Biometris, Wageningen University, Wageningen, The Netherlands

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
Reports

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.
Key words: Alps; fourth-corner matrix; functional ecology; permutation procedures; RLQ tables; trait–
environment relationship.

INTRODUCTION gration, emigration), which in turn might affect com-

Recent increasing interest in trait-based approaches munity structure and dynamics and ecosystem
has renewed community ecology both on the theoretical functioning. Among a set of traits, the identification of
(McGill et al. 2006) and the applied side (Vandewalle et response traits, i.e., ‘‘which vary in response to changes
al. 2010). By using species traits instead of their in environmental conditions’’ (Violle et al. 2007) is a key
identities, these approaches improve the ability to issue for functional ecology (Bernhardt-Römermann et
understand the structure and dynamics of ecological al. 2008). The methodological challenge associated to
communities and potentially predict their response to this goal relies on the analysis of the information
natural or human disturbances (Keddy 1992, Diaz and contained in three tables: a table Q ( p 3 s) describing s
Cabido 1997). Functional traits are usually defined as traits for p species, a table R (n 3 m) with the
any measurable features at the individual level that measurements of m environmental variables in n samples
directly or indirectly affect overall fitness or perfor- (e.g., plot or site), and a third n 3 p table L with the
mance (e.g., growth, fecundity, survival; Violle et al. abundances (or occurrences) of the p species within n
2007). Change in performance might affect demographic samples. Several approaches have been developed to
characteristics of populations (e.g., birth, death, immi- examine the link among these tables. Some authors (e.g.,
Pakeman and Marriott 2010) combined Q and L to
build a sample-by-trait table that contains for each
Manuscript received 31 January 2013; revised 25 July 2013;
sample the (weighted by the species abundances)
accepted 4 September 2013. Corresponding Editor: N. J.
Gotelli. averages of numerical traits over all species present or
9 E-mail: stephane.dray@univ-lyon1.fr the (weighted) frequencies of categorical traits. The link
14
January 2014 ANALYSIS OF TRAIT–ENVIRONMENT LINK 15

between the sample-by-trait and the R matrices can then (Baptist and Choler 2008), the annual variation of soil
be investigated using a two-table ordination method. temperature and soil water content (Campbell et al.
Legendre et al. (1997) and Dolédec et al. (1996) 2005), or the disturbance regime by rodents (Choler
independently developed two methods that consider 2005). The study site was located in the South Western
simultaneously the information contained in tables R, L, Alps (Lieu-dit Aravo, Commune de Valloire, France;
and Q: the fourth-corner approach and the RLQ 45.0678 N, 6.3948 E; see Plate 1). It covers 2 ha between
analysis, respectively. Legendre et al. (1997) combined 2700 m and 2750 m elevation. Community composition
the three original tables into a matrix describing trait– of vascular plants was determined in 75 5 3 5 m plots.
environment associations (the so-called fourth-corner Each site was described by six environmental variables:
matrix) and proposed statistics and permutation proce- mean snowmelt date over the period 1997–1999, slope
dures to evaluate the significance of these associations. inclination, aspect, index of microscale landform, index
RLQ analysis extends coinertia analysis (a two-table of physical disturbance due to cryoturbation and
method, Dolédec and Chessel 1994) to produce a solifluction, and an index of zoogenic disturbance due
simultaneous ordination of three tables (Dray et al. to trampling and burrowing activities of the Alpine
2003). Mathematically, it corresponds to the generalized marmot. All variables are quantitative except the
singular value decomposition (e.g., Greenacre 1984) of landform and zoogenic disturbance indices that are
the fourth-corner matrix. categorical variables with five and three categories,
Today, RLQ analysis and the fourth-corner approach respectively. Eight quantitative functional traits (i.e.,
represent the most integrated methods to analyze trait– vegetative height, lateral spread, leaf elevation angle, leaf
environment relationships (Kleyer et al. 2012). Even if area, leaf thickness, specific leaf area, mass-based leaf
their mathematical principles are quite similar (both nitrogen content, and seed mass) were measured on the
consider the fourth-corner matrix), their objectives 82 most abundant plant species (out of a total of 132
(ordination vs. hypothesis testing) and their outputs recorded species). See Appendix A for species and
are quite different. On the one hand, the ordination variables codes and Choler (2005) for further details on

Reports
provided by RLQ analysis assigns scores to species, data collection.
samples, traits, and environmental variables along
orthogonal axes and yields graphical summary of the IDENTIFYING MAIN PATTERNS OF VARIATION
main structures. On the other hand, the fourth-corner Separate ordinations on each table allow characteriz-
approach measures and tests the multiple associations ing the main environmental gradients (R), understand-
between one trait and one environmental variable at a ing how species communities are organized (L), or
time. These differences imply several drawbacks associ- identifying trait syndromes (Q). Correspondence anal-
ated either to the RLQ (e.g., only a global test that does ysis (CA), which provides a joint ordination of species
not allow identifying which environmental variables is and samples, is routinely applied to the table L.
acting on which combination of trait, complexity of the According to the type of variables, R and Q can be
graphical outputs) or to the fourth-corner analyses (e.g., treated by different methods: principal component
high number of tests, no consideration of the covaria- analysis for quantitative variables, multiple correspon-
tion among traits or among environmental variables, no dence analysis (Tenenhaus and Young 1985) for
information about samples and species). Here, we qualitative variables or Hill-Smith analysis (Hill and
propose some methodological adjustments to overcome Smith 1976) for a mix of qualitative and quantitative
these drawbacks and to integrate the two approaches variables. Missing values and other types of variables
into a single framework. We adopt a data-driven (e.g., ordinal, circular) can also be considered if the
presentation and use an ecological example to illustrate original variables are first transformed into a distance
each method and show how their combined use matrix (Pavoine et al. 2009) and then analyzed by a
improves the analysis of ecological data. We further principal coordinate analysis (Vallet et al. 2010).
provide a detailed tutorial (Supplement) guiding users RLQ combines the three separate analyses of R, L,
through the new integrated framework conducted using and Q and aims at identifying the main relationships
the ade4 package (Dray and Dufour 2007) for the R between environmental gradients and trait syndromes
software (R Core Team 2013). mediated by species abundances. The analysis computes
an s 3 m matrix X (see Appendix B) containing
ECOLOGICAL EXAMPLE: RESPONSE OF PLANT TRAITS measures of the intensity of the link between species
TO A SNOW-MELTING GRADIENT
traits and environmental variables (Dray and Legendre
Choler (2005) examined the functional diversity 2008). The further eigendecomposition of X> X allows
patterns of alpine plants and tested for a significant identifying the main associations between traits and
relationship between plant functional traits and habitat environmental variables (see Appendix B, Dray et al.
heterogeneity along a snow melting gradient. Snow cover [2002], and Dolédec et al. [1996] for mathematical
duration may impact the structure and the dynamics of details). For the first dimension, this analysis finds a
alpine grasslands through a variety of factors including vector u1 containing coefficients for the environmental
the length of the favorable period for carbon uptake variables and a vector v1 of coefficients for the species
 16 STÉPHANE DRAY ET AL. Ecology, Vol. 95, No. 1

traits. These loadings measure the contributions of adequate testing procedure. Dray and Legendre (2008)
individual variables and are used to compute sample showed that none of the procedures proposed by Dolédec
(a1 ¼ RDmu1) and species scores (b1 ¼ QDsv1) where Dm et al. (1996) and Legendre et al. (1997) truly controlled
and Ds are diagonal matrices of variable weights (see the type I error and they proposed an alternative
Appendix B). RLQ chooses the coefficient vectors u1 combining two permutation models (see Appendix B
and v1 in such a way that the derived sample and species for a description of the different models):
scores have maximum squared cross-covariance covP(a1,
b1)2 ¼(a>1 Pb1)2 ¼ k1 where k1 is the first RLQ eigenvalue. 1) Model 2: Permute the n samples (i.e., rows of R or L)
In other words, RLQ finds linear combinations of to test the null hypothesis that the distribution of
environmental variables (i.e., environmental gradient) species with fixed traits is not influenced by the
and of traits (i.e., trait syndrome) such that their environmental conditions. In other words, the null
squared cross-covariance is maximum. The same quan- hypothesis assumes no relationship between R and L
tity is maximized for the k dimensions with the (given that the L-Q link is preserved). The alterna-
additional constraints of orthogonality (u>i Dmuj ¼ v>i tive hypothesis considers that the environment
Dsvj ¼ 0 for i 6¼ j ). Results are stored in matrices U ¼ influences the distribution of species with fixed traits.
[u1 j  j uk], V ¼ [v1 j  j vk], A ¼ RDmU ¼ [a1 j  j ak] and 2) Model 4: Permute the p species (i.e., rows of Q or
B ¼ QDsV ¼ [b1 j  j bk]. columns of L) to test the null hypothesis that the
species composition of samples with fixed environ-
Ecological application mental conditions is not influenced by the species
In our example (response of plant traits to a snow characteristics. In other words, the null hypothesis
melting gradient), the relationships between traits and assumes no relationship between L and Q (given that
environmental variables can be summarized by the first the R-L link is preserved). The alternative hypothesis
two RLQ axes (86.7% and 9.8% of the cross-covariance considers that the traits influence the composition of
between traits and environment for axis 1 and 2, species assemblages found in samples with given
Reports

respectively). The left (negative) part of the first RLQ environmental conditions.
axis identifies species (Poa supina, Alchemilla pentaphyl-
Combining outputs produced by these two models
lea, or Taraxacum alpinum; Fig. 1a) with higher specific
allows testing the null hypothesis that at least one table
leaf area (SLA) and mass-based leaf nitrogen content
(R or Q) is not linked to L against the alternative
(NMass), lower height, and a reduced seed mass (Fig.
hypothesis that both traits and environment influence
1c). These species were mostly found in late-melting
habitats (Fig. 1b). The right part of the axis highlights species distributions (i.e., the links L-Q and R-L are
trait attributes (upright and thick leaves) associated with significant). Dray and Legendre (2008) proposed to
convex landforms, physically disturbed and mostly perform separate tests using Models 2 and 4 with a
pffiffiffi
early-melting sites. Corresponding species are Semper- significance level equal to a to obtain a global
vivum, montanum, Androsace adfinis, or Lloydia serotina. combined test with a significance level a (product of
The second RLQ axis outlined zoogenic disturbed sites separate significance levels). This combined approach
located in concave slopes. These habitats were charac- clearly improves the type I error compared to simple
terized by large-leaved species (Cirsium acaule, Geum permutation models. However, the simulation study
montanum, Alchemilla vulgaris). carried out by Dray and Legendre (2008) showed that
this procedure is slightly liberal when R, L, and Q are
TESTING BIVARIATE ASSOCIATIONS not linked and that the type I error varies between 0.198
Similarly to RLQ analysis, the fourth-corner method and 0.258 (with a ¼ 0.05) when L is only linked to one
computes an s 3 m matrix X containing measures of other table (R or Q). As an alternative, ter Braak et al.
trait–environment associations (see details in Legendre et (2012) suggested a sequential test that controls the type I
al. [1997] and Dray and Legendre [2008]). While RLQ error in all cases. This new test also consists of two steps,
analysis provides a summary of the multivariate associ- but differs conceptually from Dray and Legendre (2008)
ations, the fourth-corner method allows evaluating the proposal in that the second test is only performed if the
significance of bivariate associations (i.e., one single trait first test rejects the null hypothesis. In practice, both
and one single environmental variable at a time) approaches are very similar: the only difference is that
corresponding to cells of X. In other terms, if we consider separate tests are performed using a significance level a
pffiffiffi
a table of quantitative variables for which a correlation instead of a. Hence, an association between a trait and
matrix can be computed, RLQ analysis would be similar an environmental variable is considered significant with
to the PCA performed on this table whereas the fourth- the sequential approach if the largest of the two P values
corner method could be related to the correlation tests (obtained from Models 2 and 4) is lower than a. As the
computed for each pair of variables. Since the fourth- sequential test (or equivalently Model 6) fixes the level
corner method considers variables measured on different of type I error, we strongly advocate its use in future
statistical units (species and samples), appropriate ran- applications of the fourth-corner method and use it as
domization procedures should be used to obtain an the default in the ade4 package.
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Reports
FIG. 1. Results of the first two axes of RLQ analysis: (a) eigenvalues and scores of species (the insert shows eigenvalues, with
the first two axes in black), (b) coefficients for environmental variables, and (c) traits. The values of d give the grid size. Codes for
species and variables are available in Appendix A.

The fourth-corner method only deals with bivariate and Legendre 2008) approach. Using the sequential
associations (one trait and one environmental variable at approach (significance level a ¼ 0.05), 26 significant
a time) implying that s 3 m statistical tests are performed associations remained significant. When a ¼ 0.05 (sequen-
simultaneously. Hence, when more traits (s) and envi- tial approach) and P values are adjusted for multiple
ronmental variables (m) are considered, the number of testing, 18 significant associations remained significant
tests increases and it becomes more likely to find (Fig. 2). SLA and NMass showed the same trend (positive
‘‘significant’’ associations. This multiple testing issue
was not discussed by Legendre et al. (1997) nor by Dray
and Legendre (2008) but testing procedures clearly
require an adjustment of P values to control for the
overall error rate. In practice, adjusting P values
necessarily implies that randomization tests should be
performed with a very high number of permutations to
detect significant associations. For instance, if we use a
Bonferroni correction and test the associations between
10 traits and 10 environmental variables, each individual
hypothesis should be tested at a significance level a/100
because 100 tests are performed simultaneously. If a ¼
0.05, then we should use a significance level of 0.0005 and
2000 permutations at least are required to obtain a P
value of this level.

Ecological application
The fourth-corner method has been used to test the FIG. 2. Results of the fourth-corner tests. Significant (P ,
significance of bivariate associations (see Supplement). In 0.05) positive associations are represented by red cells, and
this paper, we used 49 999 permutations in all randomiza- significant negative associations correspond to blue cells.
tion procedures and the false discovery rate method (FDR; Nonsignificant associations are in green. Black lines separate
different variables; white lines separate different modalities for
Benjamini and Hochberg 1995) to adjust P values for categorical variables. P values were adjusted for multiple
multiple testing. Among the 96 possible associations, 51 comparisons using the FDR (false discovery rate) procedure.
were found significant with the original combined (Dray Codes for traits and variables are explained in Appendix A.
 18 STÉPHANE DRAY ET AL. Ecology, Vol. 95, No. 1

correlation with snow (Snow) and landform concavity solve all the problems described above because the
(Form.5), negative correlation with right slope (Form.3) computation of each analysis is performed separately
and physical disturbance (PhysD). This high number of and their outputs are combined a posteriori.
significant tests is linked to the strong snow-melting Last, we propose a new approach that applies the
gradient (also depicted by RLQ axis 1). Other significant fourth-corner tests directly on the outputs of RLQ
bivariate tests could be identified, e.g., the associations analysis. The complete procedure associated to this
between plant height (Height) and right slopes (Form.3), approach consists of the following:
and between leaf area (Area) and zoogenic disturbance
(ZoogD.high). This last relationship was indeed described 1) Perform RLQ analysis to summarize the main
by the RLQ axis 2. structures. Select k, the number of dimensions that
should be kept for the interpretation, by a visual
COMBINING BOTH RLQ AND FOURTH-CORNER METHODS inspection of the bar plot of RLQ eigenvalues.
RLQ and fourth-corner methods have been already Compute the sample scores A ¼ RDmU (environ-
used jointly in some trait–environment studies (e.g., mental gradients) and species scores B ¼ QDsV (trait
Lacourse 2009, Brind’amour et al. 2011). This joint use syndromes).
demonstrates the complementarity of the two approach- 2) Apply the fourth-corner tests to evaluate the
es to describe multivariate patterns and to test the statistical significance of the associations between
significance of bivariate associations. However, it also traits and environmental gradients (Q and A) and/or
highlights the drawbacks of each method and suggests trait syndromes and environmental variables (B and
that using only one approach is not sufficient to R). Here, RLQ scores (A, B) are treated as the
interpret ecological results. On one hand, RLQ summa- variables in the fourth-corner instead of the original
rizes multivariate structures but it does not provide raw data tables and thus the testing procedure
significance tests. Moreover, the produced factorial should be slightly modified. We describe the algo-
maps could be unreadable when a large number of rithm only for the case of the associations between
Reports

variables is considered. On the other hand, the fourth- traits and environmental gradients (Q and A) but the
corner only tests the significance of bivariate associa- same logic is applied for the study of the link
tions and it does not consider the covariation among between R and B. The steps are:
traits or among environmental variables. The resulting
high number of statistical tests is also difficult to 2.1) Compute observed values for the fourth-corner
summarize. To take advantage of both methods that statistics (i.e., bivariate associations between the
share the analysis of a matrix of trait–environment k RLQ environmental scores and the s traits).
associations, it is important to consider a single 2.2) Repeat a large number of times (e.g., 999 times).
framework that allows summarizing and simultaneously 2.2.a) Permute the n samples using Model 2, leading
testing the main ecological structures. Three approaches to the new table R* and recompute scores by
can be envisaged to achieve this goal. multiplying the permuted table and the
First, one can use a multivariate statistic that coefficients matrices U (A* ¼ R*DmU). Com-
measures the global association among the three tables pute the fourth-corner statistics using the
R, L, and Q. This statistic is equal to the sum of the permuted scores A* and the original table Q.
slightly modified bivariate fourth-corner statistics over 2.2.b) Permute the p species using Model 4 leading
all possible pairs of traits and environmental variables to the new table Q*. Compute the fourth-
(Dray and Legendre 2008). This statistic also equals the corner statistics using the permuted table Q*
sum of eigenvalues of RLQ analysis as originally and the original score A.
proposed by Dolédec et al. (1996). As for bivariate 2.3) Estimate P values by comparing observed
statistics, this multivariate measure should be tested with values of the statistics to the distributions of
the sequential testing procedure to avoid inflation of the 999 values obtained under the null models in
type I error. 2.2.a and 2.2.b. For the association between an
Second, an alternative approach consists in represent- environmental gradient and a trait, two P
ing the results of the fourth-corner tests onto the values, P2 and P4, are computed corresponding
factorial map produced by the RLQ analysis. In that to Models 2 and 4.
case, RLQ scores are used to position traits and 2.4) For a given bivariate association, combine P
environmental variables on a biplot and significant values of the two models by taking the
associations detected by the fourth-corner tests are maximum value between P2 and P4.
depicted by lines. This procedure results in a global 2.5) Consider all the k 3 s bivariate associations and
representation of the significant links as edges of a correct the combined P values using an
correlation network. It has the main advantage of adjustment method for multiple testing.
summarizing the results of the two analyses using a 3) Represent significant associations between RLQ axes
single biplot that facilitates the interpretation of and traits and/or environmental variables on the
ecological structures. However, the approach does not RLQ factorial map or as a table.
January 2014 ANALYSIS OF TRAIT–ENVIRONMENT LINK 19

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FIG. 3. Combination of fourth-corner and RLQ results. (a) Representation of significant (P , 0.05) associations identified by
the fourth-corner method on the factorial map of RLQ analysis. The values of d give the grid size. (b) Fourth-corner tests between
the first two RLQ axes for environmental gradients (AxR1/AxR2) and traits. (c) Fourth-corner tests between the first two RLQ
axes for trait syndromes (AxQ1 and AxQ2) and environmental variables. Positive significant associations are represented by red
lines and cells, and negative significant associations by blue lines and cells. In panel (a), traits are in boldface type and are
represented by circles; environmental variables are in lightface type and are represented by triangles. In panels (b) and (c), black
lines separate different variables; white lines separate different modalities for categorical variables. Variables with no significant
associations are shown in green. P values were adjusted for multiple comparisons using the FDR procedure. Codes for traits and
variables are explained in Appendix A.

The results of a simulation study (see Appendix C) sentation of the significant associations identified by the
demonstrate that this new approach has correct type I fourth-corner method onto the RLQ factorial map helps
error rates. interpreting the main patterns of variation and correla-
tion (Fig. 3a). Compared to the classical RLQ outputs
Ecological application (Fig. 1b and 1c), the interpretation focuses only on traits
In our example, the global testing procedure (i.e., and environmental variables that are significantly
multivariate statistic equal to the sum of eigenvalues of related. Groups of significant positive associations can
RLQ analysis) was highly significant (P ¼ 0.00002 for be identified (e.g., SLA, NMass with snow and
both permutation Models 2 and 4 and thus their concavity, leaf area with high zoogenic disturbance).
maximum), indicating a global relationship between However, it is much harder to summarize the high
species traits and environmental variables. The repre- number of significant negative associations (blue lines in
 20 STÉPHANE DRAY ET AL. Ecology, Vol. 95, No. 1
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PLATE 1. Autumn view of the studied Alpine meadows showing the contrasting mesotopographical situations that control snow
cover duration. Flat areas in the foreground with reddish patches of Salix herbacea correspond to late snowmelting sites. Early
snowmelting sites in the upper slopes and ridges are covered by turf meadows dominated by Kobresia myosuroides. The background
shows the high summits of the Massif du Grand Galibier, France. Photo credit: P. Choler.

Fig. 3a). Testing directly the associations between RLQ appropriately analyze the response of organism traits
axes and traits/environmental variables clearly improves to environmental changes. These methods are quite
the interpretation of RLQ and fourth-corner results flexible, and several developments can be foreseen. For
(Fig. 3b and c). The first axis is significantly negatively instance, RLQ has been recently extended to introduce
correlated with snow cover and concavity (late melting) spatial (Brind’amour et al. 2011) and/or phylogenetic
and positively with physical disturbance and slope (early (Pavoine et al. 2011) information or to partial out the
melting). Associated traits are higher specific leaf area effects of covariables (Wesuls et al. 2012). Considering
and nitrogen content for late-melting sites and higher these aspects in the fourth-corner testing procedure
angle and plant height for early melting sites. Choler would allow, among other things, to evaluate how a
(2005) hypothesized that high leaf angle in the physically common evolutionary history (i.e., phylogenetic signal)
disturbed, early-melting habitats limits nocturnal radi- influences trait–habitat relationships (Ernst et al. 2012).
ative loss of leaf surfaces and ensures a better structural LITERATURE CITED
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tion. The second axis opposes convex sites with no importance of length of growing season and canopy
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mots are present. Communities found in these disturbed tion of temperate alpine meadows. Annals of Botany 101:
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SUPPLEMENTAL MATERIAL
Appendix A
Description and codes for the variables and species of the plant data set (Ecological Archives E095-002-A1).

Appendix B
Detailed description of RLQ and fourth-corner methods (Ecological Archives E095-002-A2).

Appendix C
Results of the simulation study (estimation of Type I error) for the new approach that combines the fourth-corner and RLQ
methods (Ecological Archives E095-002-A3).

Supplement
A tutorial to perform fourth-corner and RLQ analyses in R (Ecological Archives E095-002-S1).