singular value decomposition of the fourth-corner matrix D (Dray et al. 2014). For
the first ordination axis, RLQ finds coefficients for the environmental variables and
species traits. These coefficients measure the contributions of individual variables
and are used to compute site and species scores; they are chosen to maximize the first
eigenvalue. The analysis proceeds in the same way for the next, orthogonal ordination axes. For mathematical details see Dray et al. (2014).
6.11.3 Application in R
Due to a lack of significant results, we will refrain from using the Doubs data set in
this application. The R code for the Doubs data is provided in the accompanying
material, however. The example proposed here is extracted from a tutorial written by
Stéphane Dray and provided as a Supplement to the Dray et al. (2014) paper (ESA
Ecological Archives E095–002-S1). The example concerns the ecological data
analysed in the Dray et al. (2014) paper, which describes the response of plant traits
to a snow-melting gradient in the French Alps. The main question is: how does the
snow cover duration, with all its consequences, impact the alpine grasslands, as
assessed by functional traits of the plant species?
The data come from 75, 5 Â 5 m plots located in the South-Western Alps at about
2700 m elevation. They consist in the three following matrices: community composition (82 species, abundance scale from 0 to 5), traits (8 quantitative variables) and
environment (4 quantitative and 2 categorical variables). To simplify this presentation, only some of the variables that are identified as significant are presented below.
Readers are referred to Dray et al. (2014) and Choler (2005) for a more complete
interpretation of the results.
The data are available in ade4. The script below shows the computation of an
RLQ analysis followed by a fourth-corner analysis. We then conclude with a
combined representation of the results of the two analyses.
After having loaded the data, the first step of the RLQ analysis is to compute
separate ordinations of the three data sets, which are computed by function rlq()
of ade4 using the same general framework as in co-inertia analysis (Sect. 6.9). The
ordination methods are chosen in accordance with the mathematical types of the
variables. Here, following Dray et al. (2014), we compute a CA on the species data, a
PCA on the (quantitative) trait data; for the environmental data, which are quantitative and categorical, we will apply a special type of PCA that can handle such types
of data, called a Hill-Smith analysis (Hill and Smith, 1976).
The RLQ analysis is then computed on the basis of the three ordinations. A single
plot() command allows one to plot all results in a single graphical window, but
the individual plots are rather crowded, so we also provide the code to plot the results
separately. These plots are assembled in Fig. 6.22. The script below concludes with a
global “model 6” test (after ter Braak et al. 2012). The two tests included in “model
6” yielded a combined p-value ¼ 0.001, hence the null hypothesis is rejected, which
means that both links, L À Q and R À L, are significant.
6.11 Relating Species Traits and Environment
291
the first ordination axis, RLQ finds coefficients for the environmental variables and
species traits. These coefficients measure the contributions of individual variables
and are used to compute site and species scores; they are chosen to maximize the first
eigenvalue. The analysis proceeds in the same way for the next, orthogonal ordination axes. For mathematical details see Dray et al. (2014).
6.11.3 Application in R
Due to a lack of significant results, we will refrain from using the Doubs data set in
this application. The R code for the Doubs data is provided in the accompanying
material, however. The example proposed here is extracted from a tutorial written by
Stéphane Dray and provided as a Supplement to the Dray et al. (2014) paper (ESA
Ecological Archives E095–002-S1). The example concerns the ecological data
analysed in the Dray et al. (2014) paper, which describes the response of plant traits
to a snow-melting gradient in the French Alps. The main question is: how does the
snow cover duration, with all its consequences, impact the alpine grasslands, as
assessed by functional traits of the plant species?
The data come from 75, 5 Â 5 m plots located in the South-Western Alps at about
2700 m elevation. They consist in the three following matrices: community composition (82 species, abundance scale from 0 to 5), traits (8 quantitative variables) and
environment (4 quantitative and 2 categorical variables). To simplify this presentation, only some of the variables that are identified as significant are presented below.
Readers are referred to Dray et al. (2014) and Choler (2005) for a more complete
interpretation of the results.
The data are available in ade4. The script below shows the computation of an
RLQ analysis followed by a fourth-corner analysis. We then conclude with a
combined representation of the results of the two analyses.
After having loaded the data, the first step of the RLQ analysis is to compute
separate ordinations of the three data sets, which are computed by function rlq()
of ade4 using the same general framework as in co-inertia analysis (Sect. 6.9). The
ordination methods are chosen in accordance with the mathematical types of the
variables. Here, following Dray et al. (2014), we compute a CA on the species data, a
PCA on the (quantitative) trait data; for the environmental data, which are quantitative and categorical, we will apply a special type of PCA that can handle such types
of data, called a Hill-Smith analysis (Hill and Smith, 1976).
The RLQ analysis is then computed on the basis of the three ordinations. A single
plot() command allows one to plot all results in a single graphical window, but
the individual plots are rather crowded, so we also provide the code to plot the results
separately. These plots are assembled in Fig. 6.22. The script below concludes with a
global “model 6” test (after ter Braak et al. 2012). The two tests included in “model
6” yielded a combined p-value ¼ 0.001, hence the null hypothesis is rejected, which
means that both links, L À Q and R À L, are significant.
6.11 Relating Species Traits and Environment
291