Ecological Archives E095-216-A1
Eric G. Lamb, Kerrie L. Mengersen, Katherine J. Stewart, Udayanga Attanayake, Steven D. Siciliano. 2014. Spatially explicit structural equation modeling. Ecology 95:2434–2442. http://dx.doi.org/10.1890/13-1997.1
Appendix A. Additional results for the Truelove Lowland spatially explicit structural equation models (SE-SEM), and example applications of SE-SEM to two additional data sets.
This supplemental file contains additional results from the Truelove Lowland example and descriptions of two additional datasets: Alexandra Fiord Lowland (irregular lag distance design) and Plant Competition (ad hoc lag distance design). For those datasets we provide data descriptions, descriptions of the non-spatial models, results, and interpretation. Raw data and annotated R-code to perform all of these analyses are available in the compressed file "Lamb et al. 2014. Example Scripts.zip". Within that file, the file "Lamb et al. 2014. Example Scripts.r" contains annotated step by step instructions to replicate all analyses. The file "sesem1.0.0.r" contains all of the functions required to perform SE-SEM. The latest version of this software is available as package "sesem" from the R project for statistical computing http://www.r-project.org/.
Alexandra Fiord Lowland (irregular lag distance design). This study examined the same relationships between soil moisture, vascular plants, common nitrogen fixing associations and N2-fixation rates as the Truelove study and employed the same non-spatial model (Stewart et al. 2011b). However, the sampling design used for Alexandra Fiord was a variable lag distance design intended to capture small scale changes in microbial patterns (Banerjee and Siciliano 2012). The study site was on Alexandra Fiord Lowland, an 8 km² lowland oasis on the eastern side of central Ellesmere Island (78°53'N, 75°55'W). The oasis is a deglaciated lowland delimited by a glacier to the south, cliffs and talus slopes (ca. 500 m) to the west and east and by the fiord waters to the north (Muc et al. 1989). The lowland has an extensive vegetation cover dominated by dwarf shrubs, heaths, cushion plants and hydric sedges. Transects were placed perpendicular to the slope of the lowland and over relatively flat terrain with the presence of some hummocky areas. It was initially fitted as a multi-group model between four sites, but here we analyze this SEM as a single-group model. We chose to use a-priori lag distances (0.01, 1, 2, 2.2, 4, 5, 8, 16, 32, 64, 96, 128, and 160 m) similar to those used in past spatial analyses of data collected using this sampling design (Banerjee and Siciliano 2012).
Competition in Fescue Grassland (ad-hoc distance design). This study examined the importance of plant competition in structuring plant community diversity in a fescue grassland. Total competition intensity, plant community biomass, diversity, species composition, and environmental variables were measured at 192 locations distributed on a variety of topographic positions in a ∼200 m – 350 m area. Including a spatial structure in this data set was not intended, but clumping in the distribution of samples was unavoidable due to the presence of several small aspen stands in the field. Sample locations were measured using a differentially corrected GPS. Here we evaluate a simplified version of the published structural equation model (Lamb and Cahill 2008). This model was intended to assess whether competition is an important factor controlling plant community richness by including competition intensity in a structural equation model relating diversity and composition to environmental conditions and plant community biomass. The ad hoc sampling design precluded the identification of sensible a priori lag distance bins as in the Alexandra Fiord study. We therefore used a minimum sample size (200) criterion to automatically identify lag distance bins. This resulted in 62 bins with a mean size of 2.86 m (range 1.81 m to 5.97 m) up to a maximum lag distance of 177.49 m.
Rational for non-spatial model structures
Truelove (regular lag distances) and Alexandra Fiord Lowland (irregular lag distances). The rationale for the non-spatial Truelove structural equation model presented in the main text is largely identical to that for the Alexandra Fiord model. The non-spatial structural equation model for was based on the final path model used by Stewart et al. (2011b). The SEM consisted of directly observed measures of soil moisture, shrubs, gramminoids, forbs, bryophytes, lichens, biological soil crusts and their link to N2 fixation. Briefly, Stewart et al. (2011b) developed a path model to describe how soil moisture and functional vascular plant community composition (i.e., proportion graminoids, forbs and shrubs) directly influenced cover of bryophytes, soil crust, and lichen abundance and hence indirectly influenced N2-fixation. Direct paths from potential N2-fixing cyanobacteria associations (Bryophyte, Lichen and Bare ground) to N2-fixation were included. Soil crusts were included as a potential N2-fixing association because "bare ground" as recorded in the field supported communities composed of bacteria, cyanobacteria, algae, mosses, liverworts, fungi and small lichens (Stewart et al. 2011b). Paths from soil moisture to all plant components were included because (1) soil moisture is a key environmental factor determining the distribution of vegetation types in arctic environments (Oberbauer and Dawson 1992) and (2) interactions between plant communities and soil moisture can be important in determining the operating environment of N2-fixing associations (Zielke et al. 2002, Zielke et al. 2005, Stewart et al. 2011a).
In the Truelove and Alexandra fiord non-spatial SEMs each path represents a spatially dependent ecological mechanism linked back to soil moisture. Variation in soil moisture at Truelove lowland, the key underlying environmental factor in the SEM, is driven by changes in topography and soil drainage patterns along the beach ridge cantinas occurring at scales of tens to hundreds of meters (Lev and King 1999; Fig. A1). Spatially dependent moisture patterns at Alexandra Fiord are driven by smaller-scale hummock and hollow topography (Ma et al. 2007). Numerous studies have demonstrated moisture driven spatial dependence in arctic bryophyte, vascular plant, and soil communities (e.g., Peterson and Billings 1980, Muc and Bliss 1987, Ostendorf and Reynolds 1998, Banerjee et al. 2011). There are both direct and indirect spatially dependent ecological mechanisms linking vascular plants (Shrubs, Gramminoids, and Forbs) to cryptogams (Bryophyte, Lichen, Soil Crust) in this community. The direct mechanisms such as shading and competitive displacement are likely most important at very small scales. The indirect mechanisms linking moisture effects on the vascular plants to bryophytes likely occur at scales of tens to hundreds of meters in Truelove lowland and at meters to tens of meters at Alexandra fiord (Muc and Bliss 1987, Lev and King 1999, Ma et al. 2007, Banerjee et al. 2011) and thus are appropriate for testing in an SE-SEM context across scales.
Fescue Grassland Plant Competition (ad hoc lag distances).The non-spatial structural equation model for the plant competition data set is a simplified version of the published structural equation model (Lamb and Cahill 2008). This model was intended to evaluate whether competition is an important factor controlling plant community richness by including competition intensity in a structural equation model relating diversity and composition to environmental conditions and plant community biomass. The published model included several complications including a composite variable (Grace and Bollen 2008), a hierarchical data structure (species – level competitive ability) fit using a two-level hierarchical model using M-Plus (Muthén and Muthén 2010) and a nitrogen addition treatment randomly applied to half of the plots. We have simplified the present model by removing these features as evaluating the performance of spatially explicit (SE-SEM) for models with latent variables, composite variables, and hierarchical data structure is beyond the scope of the present paper. We did not include the nitrogen treatment because it was randomly applied to plots and hence had no direct spatial component. As plant productivity at the study site is nitrogen limited (Lamb et al. 2007, Lamb 2008), the effects of the nitrogen treatment are incorporated into the present data through plant community biomass responses. In the Plant Competition dataset spatial dependence from topographic position, soil moisture, and total soil nitrogen resulted in indirect spatially causal relationships throughout the path model.
Results and Interpretation for the Alexandra Fiord and Plant Competition data sets.
Alexandra Fiord Lowland (irregular lag distance design). The initial non-spatial Alexandra Fiord model was identical to the Truelove model described above. The non-spatial nitrogen fixation model had an adequate fit for the Alexandra Fiord data (χ²5 = 2.37, p = 0.80) (Fig. A3). The model explained a relatively low percentage of the variation in N2-fixation (R² = 0.12); however N2-fixation in this community appears to be driven by the abundance of bare ground, which also includes N2-fixing biological soil crusts. While lichen abundance was poorly explained in our non-spatial model (R² = 0.02), bryophyte abundance was much better explained (R² = 0.14) and clearly linked to moisture conditions. The importance of increasing moisture on gramminoid, shrub and bryophyte abundance is evident in the non-spatial model.
The SE-SEM models at lag distances of less than 50 m had good fit, but models fit rapidly declined at larger distances (Fig. A4). The small-scale hummock-hollow topography typical of the Alexandra Fiord lowland is likely to be a major driver of the strong spatial patterns at shorter lag distances (Ma et al. 2007). The poor model fit at larger lag distances resulted in a large number of modification indices (Table A1). We chose not to modify this path model; the issue of whether or not to modify a model in these cases is explored in the main text discussion.
The Alexandra Fiord SE-SEM demonstrates similar patterns of soil moisture influence on gramminoids, shrubs and bryophytes (Fig. A5). Path coefficients indicating the influence of moisture on shrubs were significant at all lag distances examined; for gramminoids path coefficients were significant at all but three lag distances and for bryophytes all but one lag distance. Intermediate lag distances were important in predicting N2-fixation. Unlike Truelove lowland, bryophytes at Alexandra Fiord only had a significant influence on N2-fixation in the 96 m and 128 m bins. In the Alexandra Fiord lowland N2-fixing associations other than bryophytes may play a more important role driving N2-fixation. For example, while bare ground had little influence on N2-fixation at the Truelove lowland, bare ground contributed to N2-fixation rates at most spatial scales at Alexandra Fiord. This likely indicates an abundance of cyanobacteria associated with biological soil crusts throughout the lowland acting as an important driver of N2-fixation (Alexander et al. 1978, Dickson 2000). Lichens show a similar pattern of influence on N2-fixation across a range of spatial scales, however positive influences on N2-fixation were only detected at distances less than 8 m. Both the use of a priori lag distances and more variability in microtopographical patterns may have resulted in a great number of insignificant or weak path coefficients between variables in the Alexandra Fiord data set. A second set of lag distances was generated with a minimum sample size criteria of 100 per bin and the SE-SEM models refit. Model χ² were substantially lower for the larger bin sizes (Fig. A5), suggesting that the larger a priori bin sizes (32 m spans) were overly large and masked substantial patterns.
Competition in Fescue Grassland (ad-hoc distance design). The non-spatial plant competition model had an adequate fit to these data (χ²8 = 9.7, p = 0.287) (Fig. A3). As in the published SEM (Lamb and Cahill 2008) there was no substantive link between competition intensity and species richness. Plant productivity was tightly linked to environmental factors (topographic position, soil moisture, and soil total N), but those factors were weak predictors of both competition intensity and species richness.
The SE-SEM models generally had good fit across all lag distances (Fig. A7). Only weak modification indices were identified (Table A2), indicating that the initial path model was appropriate across all lag distances. Several important relationships not evident in the non-spatial model were revealed by SE-SEM (Fig. A8). The strong relationships between environmental factors (topographic position, soil moisture, and soil total N) and plant biomass generally remained similar in magnitude and direction in the spatial models. Spatial controls on competition intensity, however, were clearly evident. There was, for example a switch from non-significant effects of total Nitrogen on competition at most lag distances to a clear reduction in intensity with higher N at lag distances between ~50 and 80m and again at distances greater than 140 m. Similarly, increasing root biomass increased competition intensity at lag distances greater than ~100 m. Reduced competition intensity led to slightly increased species richness at distances between 50 m and 100 m, but switched to a negative effect on species richness at distances greater than ~150 m. Note that conventionally in plant completion studies, intensity increases as the values of the log response ratio (lnRR) indices become more negative. Lamb and Cahill (2008) discounted the role of competition based on several non-significant paths involving competition intensity. The spatial SEM, however, revealed both positive and negative landscape level controls on community structure and competition at a range of scales. Competition intensity increased with declining soil moisture across all lag distances, with the strongest effects at lag distances between ~50 and 125 m. Light interception, however, increased competition intensity only at lag distances greater than ~100 m. These patterns suggest a shift from the direct control of competition intensity by moisture at smaller scales to indirect control (through light) at larger scales. The implications of such a shift are not clear, but could suggest a change in the relative importance of root and shoot competition in this community.
Fig. A1. The regular sampling design on the Truelove Lowland transect, Devon Island, Nunavut, Canada. A beach ridge is in the foreground; the transect start point is in a moist area near the lakeshore in the background. Photo Credit: Eric Lamb.
Fig. A2. Change in model fit (χ², CFI, RMSEA, and SRMR) with lag distance for the Truelove Lowland data set. Solid lines are smoothed curves fit using function lowess. Dotted lines indicate the critical χ² value corresponding to p = 0.05 and commonly accepted cut-offs (Kline 2011) for CFI (good fit above 0.9), RMSEA (good fit below 0.05), and SRMR (good fit below 0.08).
Fig. A3. Fitted non-spatial SEM models for the Truelove Lowland, Alexandra Fiord, and Plant Competition data sets. Standardized path coefficients are shown; dotted lines indicate non-significant paths (p > 0.100).
Fig. A4. Change in model fit (χ², CFI, RMSEA, and SRMR) with lag distance for the Alexandra Fiord data set. Dotted lines indicate the critical χ² value corresponding to p = 0.05, and commonly accepted cut-offs (Kline 2011) for CFI (good fit above 0.9), RMSEA (good fit below 0.05), and SRMR (good fit below 0.08).