Ecological Archives E093-014-A3

Etienne Laliberté and Jason M. Tylianakis. 2012. Cascading effects of long-term land-use changes on plant traits and ecosystem functioning. Ecology 93:145–155.

Appendix C. Details on generalized multilevel path models.

Testing the validity of a generalized multilevel causal path model consists of: (1) finding the 'basis set' BU of independence claims implied by a directed acyclic causal graph (i.e. a box-and-arrow causal diagram that involves no feedback loops) that, together, expresses the full set of dependence and independence claims implied by the causal graph, (2) obtaining the probability pi associated with each of the k independence claims in BU, using appropriate statistical tests (in our case, linear mixed models, as described above), (3) combining the pi using , and (4) comparing the C statistic to a chi-square (χ2) distribution with 2k degrees of freedom (Shipley 2009). A causal model can be rejected if the P value associated with its C statistic is smaller than the specified α-level (here, α = 0.05), since a significant P value implies that the data depart significantly from what would be expected under such a causal model (Shipley 2009).

When analyzing experimental data under an SEM framework, categorical and ordinal variables corresponding to experimental treatments can be dealt with in several ways (Grace 2006). We ranked the fertilizer/irrigation treatment (resulting in a continuous variable from 1 to 5 which we refer to as "soil resource availability"). For grazing intensity, we used the proportion of ANPP grazed by sheep over the 18-month sampling period (Bagchi and Ritchie 2010). Grazing intensity values above one (i.e. more ANPP was grazed than produced during the measurement period) were possible because the measurement period for ANPP started near peak standing biomass in October 2007 but ended in April 2009, directly after all plots were grazed before the winter resting period.

While soil resource availability and grazing intensity would be orthogonal to each other had they both been treated as ranked variables (by virtue of the experimental design), the proportion of total ANPP grazed increased in a nonlinear fashion with soil resource availability (R2 = 0.585, P < 0.001). This is consistent with patterns found in other grazing ecosystems, where ecosystems with higher rates of primary production sustain a greater level of grazing per unit of primary production (McNaughton et al. 1989). The dependence of grazing intensity on soil resource availability was therefore explicitly considered in all causal path models (Appendix A). An additional linear mixed model that included both soil resource availability and a ranked grazing variable corresponding to the different levels of the grazing intensity treatment showed that both predictors had significant (P < 0.05) positive effects on our continuous grazing intensity measure (i.e. the fraction of total ANPP grazed by sheep). Importantly, no significant interaction was detected (P = 0.303). Therefore, this shows that despite the fact that greater soil resource availability led to a greater proportion of total ANPP being consumed by sheep, increasing levels of the grazing intensity experimental treatment still led to greater herbage consumption once this effect of soil resource availability was accounted for.

We used linear mixed models with random intercepts per block and whole plot to test the k independence claims implied by each causal path model, allowing us to take into account the hierarchical nature of the experiment (Shipley 2009). Data on ecosystem processes (ANPP, BNPP, litter decomposition, and soil respiration) were log-transformed to linearize relationships. All predictors were centered on their means (i.e. subtracting the mean) to facilitate interpretation and to avoid multicolinearity problems due to the inclusion of interactions and polynomials (Aiken and West 1991). Residuals were inspected to verify model assumptions and appropriate variance structures were used in the presence of heterogeneity (Pinheiro and Bates 2000). We used polynomials of main terms to model nonlinear relationships, but only when differences in AIC and results of likelihood ratio (LR) tests (using ML estimation) provided unequivocal support for their inclusion (i.e. ΔAIC > 8 and P < 0.001 for LR tests), because of the liberal nature of the LR test for comparing models with different fixed effects (Pinheiro and Bates 2000).

Individual path coefficients leading to endogenous variables (i.e. variables with arrows leading to them) were fitted using REML and tested for significance. Residuals were inspected to verify model assumptions, and appropriate variance structures were used in the presence of heterogeneity (Pinheiro and Bates 2000). Significance of individual path coefficients was assessed using conditional t tests (Pinheiro and Bates 2000).

Model fits for endogenous variables were assessed via an R2 statistic developed specifically for linear mixed models (Kramer 2005). While acknowledging that the calculation and interpretation of R2 statistics for mixed models are still under debate, and that such statistics cannot be interpreted in the same way as their least squares counterpart (Edwards et al. 2008), we used this R2 statistic simply as an overall measure of fit for the different models. Because our final models used REML estimation, and the R2 measure we used relies on ML estimation (Kramer 2005, Edwards et al. 2008), our R2 statistics should be interpreted with caution. Nevertheless, we still believe that they should give a reasonable absolute indication (rather than a comparative measure such as AIC) of how well or poorly a particular model fits the data.

Because all predictors were centered, unstandardized path coefficients can be interpreted as the amount of change in the response variable following a unit change in the predictor when all other predictors are held constant at their mean values (Aiken and West 1991). However, in the presence of an interaction this interpretation is no longer valid; in that case, the unstandardized path coefficient for a predictor involved in the interaction is interpreted as its average effect on the response variable when the conditioning variable (i.e. the other predictor involved in the interaction) is held at its mean value (Aiken and West 1991). The unstandardized path coefficient for the interaction, on the other hand, represents the amount of change in the slope of the regression of the response variable on one of the predictors involved in the interaction, following a one-unit change in the other predictor (Aiken and West 1991).

Standardized coefficients for main terms and interactions were computed as described by Aiken and West (1991), whereas those for nonlinear relationships involving polynomials were computed from composite variables of the polynomial terms (Grace et al. 2007). To do so, we first created a single composite variable from these polynomial terms by fitting a mixed model with all polynomials as predictors, extracting its regression coefficients (i.e. fixed effects), and then multiplying each polynomial by its regression coefficient and summing them together into one composite variable (Grace et al. 2007). REML estimation was used and pi (for calculating significance of the path model, see above) was taken from the P value associated with the t statistic for the regression coefficient of the composite variable. Only standardized coefficients are meaningful for such composite variables (Grace et al. 2007). Contrary to unstandardized coefficients, standardized coefficients can be directly compared to each other and represent the relative importance of each path. These standardized coefficients are interpreted in a similar way as unstandardized coefficients, except that changes in both predictors and response variables are expressed in standard deviation units.

After finding that our initial causal model (Appendix A) was well supported by the data (χ2 = 15.3, df = 18, P = 0.644), a more parsimonious model was obtained through a backward selection approach (Zuur et al. 2009). Backward selection was applied on each sub-model within the larger model (Appendix A) that involved an endogenous variable and its direct causal parents. In each case, model selection on the fixed component of the model was based on minimizing the Akaike Information Criterion (AIC) and using LR tests on models fitted via maximum likelihood (ML) estimation. All sub-models were then grouped back together and the resulting parsimonious causal model was then tested using generalized multilevel path models, as described above. This parsimonious model provided a good fit to the data (χ2 = 54.4, df = 54, P = 0.458). All of the aforementioned analyzes were conducted using the 'nlme' (Pinheiro et al. 2010) package in the R environment (R Development Core Team 2010).


Literature Cited

Aiken, L. S., and S. G. West. 1991. Multiple Regression: Testing and Interpreting Interactions. Sage Publications, Newbury Park, USA.

Bagchi, S., and M. E. Ritchie. 2010. Introduced grazers can restrict potential soil carbon sequestration through impacts on plant community composition. Ecology Letters 13:959–968.

Edwards, L. J., K. E. Muller, R. D. Wolfinger, B. F. Qaqish, and O. Schabenberger. 2008. An R2 statistic for fixed effects in the linear mixed model. Statistics in Medicine 27:6137–6157.

Grace, J. B., T. Michael Anderson, M. D. Smith, E. Seabloom, S. J. Andelman, G. Meche, E. Weiher, L. K. Allain, H. Jutila, M. Sankaran, J. Knops, M. Ritchie, and M. R. Willig. 2007. Does species diversity limit productivity in natural grassland communities? Ecology Letters 10:680–689.

Grace, J. B. 2006. Structural Equation Modeling and Natural Systems. Cambridge University Press, Cambridge, UK.

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McNaughton, S. J., M. Oesterheld, D. A. Frank, and K. J. Williams. 1989. Ecosystem-level patterns of primary productivity and herbivory in terrestrial habitats. Nature 341:142–144.

Pinheiro, J., D. Bates, and S. DebRoy. 2010. nlme: Linear and Nonlinear Mixed Effects Models. The Comprehensive R Archive Network (CRAN), Vienna, Austria.

Pinheiro, J. C., and D. M. Bates. 2000. Mixed-Effects Models in S and S-PLUS. Springer, New York, USA.

R Development Core Team. 2010. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.

Shipley, B. 2009. Confirmatory path analysis in a generalized multilevel context. Ecology 90:363–368.

Zuur, A. F., E. N. Ieno, Walker, Neil J., A. A. Saveliev, and G. M. Smith. 2009. Mixed Effects Models and Extensions in Ecology with R. Springer, New York, USA.


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