Appendix S5. Hypothetical causal relationship between grasshopper traits and their
impact on plant biomass
The confirmatory multilevel path analysis (Shipley, 2009) is based on directed acyclic graphs
(i.e. a box-and-arrow diagrams without feedback loops). The graphs are used to specify the
direct, indirect and independent relationships between the variables of the system implied by
each competing hypothesis (Fig. S5). The validity of each path model is tested by deriving
from each graph the set of independence claims (Table S5). Using multilevel/mixed effect
models the probabilities pi of each of the k independence claims are obtained, which are then
combined into a C statistic:
k
C  2 ln( p i )
(1)
i 1
The resulting value is compared to a χ2 distribution with 2k degrees of freedom (Shipley
2013). If the value of the C-statistic is lower than the specified significance level (here, α =
0.05) the path model is rejected, as the data depart significantly from what expected under the
tested causal model.
To test the independence claims, we used linear mixed models with block as random
effect, using the lmer function of the lme4 package version 1.1-6. (Bates et al. 2014) within
the R language and software environment for statistical computing version 3.0.2, following
Shipley (2013). Model assumptions were tested by inspecting the residuals Pinheiro & Bates
(2000). All variables were standardised (mean-centred and divided by the standard deviation)
and standardised path coefficients were obtained. Direct, indirect, and total effects were
computed using standardised path coefficients following Grace & Bollen (2005).
We tested here two alternative hypotheses using confirmatory path analysis (Shipley 2013)
see main text: (i) direct effect of grasshopper traits on plant community biomass; (ii) indirect
effect mediated by grasshopper feeding niche (see main text for detailed hypotheses on each
arrow). Fig. S5 describes the two a priori hypotheses for all links (arrows) leading to
endogenous variables. Table S5 details the conditional independence between variables tested
in the analyses. For each hypothesis, we also tested alternative models which included one
grasshopper trait only or the two traits. Hypothesis 1 was rejected while Hypothesis 2 was
accepted (Table S5; Fig. 5 in the main text).
Table S5 Conditional independence tests in the basis sets implied by the hypothesized path models (Fig. S5).
D-sep claim of independence
Hypothesis 1 : Direct effect
(BS,Predicted LNRR)|{Ø}
(IS, Predicted LNRR)|{Ø}
(Survival, Predicted LNRR)|{Ø}
(Predicted LNRR, Observed LNRR)|{IS,Survival}
(BS, IS)|{Ø}
(BS, Survival)|{Ø}
(IS, Survival)|{Ø}
(BS, LNRRobs)|{IS,Survival}
Model formula
Ho
P value
Predicted LNRR ~ BS
Predicted LNRR ~ IS
Predicted LNRR ~ Survival
Observed LNRR ~ Predicted LNRR + BS + IS + Survival
IS ~ BS
Survival ~ BS
Survival ~ IS
Observed LNRR ~ BS + IS + Survival
BS=0
IS=0
Survival=0
Predicted LNRR=0
BS=0
BS=0
IS=0
BS=0
0.25
<0.0001
0.02
0.07
0.45
0.60
0.11
0.045
IS ~ BS
Survival ~ BS + Predicted LNRR
Survival ~ IS + Predicted LNRR
Observed LNRR ~ BS + Survival + Predicted LNRR
Observed LNRR ~ IS + Survival + Predicted LNRR
Predicted LNRR ~ BS + IS
BS=0
BS=0
IS=0
BS=0
IS=0
BS=0
0.45
0. 94
0. 78
0.07
0.36
0.40
C statistic : 60.28
P value of C (DF) : < 0.0001 (16)
Hypothesis 2 : Mediated effect
(BS,IS)|{Ø}
(BS, Survival)|{ Predicted LNRR }
(IS, Survival)|{ Predicted LNRR }
(BS, Observed LNRR)|{Survival, Predicted LNRR}
(IS, Observed LNRR)|{Survival, Predicted LNRR}
(BS, Predicted LNRR)|{IS}
C statistic : 11.30
P value of C (DF) : 0.5 (12)
Notes: Key to variables: BS = grasshopper body size, IS = grasshopper incisive strength, Survival = grasshopper survival within each cage at the
end of the experiment, Observed LNRR and predicted LNRR referred to equations 1 and 6 respectively in the main text. (Xi,Xj)|{Xk} means
that variables Xi and Xj are independent conditional on variable Xk (thus variation in Xi does not imply variation in Xj if Xk is held constant,).†
The associated mixed model regression for each d-sep claim using the lmer function in the lme4 package in R to test the independence claims. *
The p-value is obtained by comparing the value of the C statistic for each hypothesis to a chi-square distribution with the same degrees of
freedom – note that a model is rejected if the C statistic is significantly different from the χ2 value.