Appendix A. Further exposition on modeling with a second-order composite.
A primary challenge in the main example of the paper is to relate the overall effects of abiotic stress on richness, light, and biomass. We present in the manuscript three structural equation models, our initial model (Fig. 7), a revised model that shows the significant effects of individual dimensions of stress (Fig. 8), and a final one that shows the collective effects of stress dimensions using a composite for the Species Filter effect on Richness (Fig. 9). The presentation in the paper is actually an abbreviated version of the full process of model consideration. In this appendix, we give a more complete presentation of the models that were considered. Here we capitalize names when they refer specifically to variables in figures for greater clarity.
Figure A1 represents the hypothesis that there exists a single second-order latent composite “Stress” that simultaneously represents the effects of all three stress dimensions (Salinity, Flooding, and Infertility) on Richness, light, and Biomass. In this example, the Stress variable is a latent composite. It is a composite because it has three causal indicators (Salinity, Flooding, and Infertility) and latent because (1) its causal indicators are latent and (2) it has an estimable error ζS. As Grace and Bollen 2008 describe, for this type of model to be appropriate, the effects of the three stress dimensions on Richness, light, and Biomass need to be similar. It is actually possible to estimate the model shown in Figure A1 without resorting to a partially-reduced form model because the latent composite has more than one outflowing arrow (which makes the error term identified). The question remains, however, whether the model shown in Fig. A1 is our most appropriate model.
There are two additional models to consider in deciding whether the model in Fig. 9 in the manuscript is our best choice or whether the model shown in Fig. A1 is more appropriate. First, shown in Fig. A2 is a model in which Stress is represented as a classic second-order latent variable. Implied by this model is the existence of a latent factor that simultaneously causes high values of Salinity, Flooding, and Infertility. Conceptually, for such a model we would expect that places with high salinity would also have high flooding and high levels of infertility (i.e., low soil organic). Such a convergence of stresses is not the case in reality, however, and therefore the model in Fig. A2 is mismatched with our knowledge.
Another alternative to the model shown in Fig. A1 is presented in Fig. A3. Here, we include three composites representing stress effects, Species Filter, Morphological Type, and Growth Inhibition. This model proposes that there are semi-independent effects of Abiotic Stress dimensions on Richness, light, and Biomass. Evaluation of this model requires that we refer to the results from the partially-reduced form model in Fig. 8 in the manuscript. The results expressed in Fig. 8 (and Table 3) show that Richness is influenced by all three dimensions of stress, while light is only influenced by Salinity and Biomass is only influenced by Flooding, We can conclude that the model in Fig. A1 is improper (or at least overcomplicated) as it specifies that all three dimensions of abiotic stress affect Richness, light, and Biomass. Further, since the results show that only one stress dimension affects light and a different one affects Biomass, the model in Fig. A3 simplifies to the one shown in Fig. 9 in the manuscript.
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FIG. A1. Representation of the hypothesis that there exists a single second-order latent composite “Stress” that simultaneously represents the effects of all three stress dimensions (Salinity, Flooding, and Infertility) on Richness, light, and Biomass. |
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FIG. A2. Representation of a model in which “Stress” is represented as a classic second-order latent variable with effect indicators. This model specifies that Stress is a latent factor that explains the covariances between Salinity, Flooding, and Infertility. |
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FIG. A3. Model including three composites representing stress effects, Species Filter, Morphological Type, and Growth Inhibition. This model proposes that there are semi-independent effects of Abiotic Stress dimensions on Richness, light, and Biomass. |
LITERATURE CITED
Grace, J. B., and K. A. Bollen. 2008. Representing general theoretical concepts in structural equation models: the role of composite variables. Environmental and Ecological Statistics 15:191–213.