Structural Equation Modeling A Network Modeling Framework a

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Structural Equation Modeling: A Network Modeling Framework (a) the observed variable model y =

Structural Equation Modeling: A Network Modeling Framework (a) the observed variable model y = α + Γx+ Βy + ζ α = p x 1 vector of intercepts Φ = cov (x) = q x q matrix of covariances among xs Β = p x p coefficient matrix of ys on ys Ψ = cov (ζ) = q x q matrix of y = p x 1 vector of responses Γ = p x q coefficient matrix of ys on xs covariances among errors x = q x 1 vector of exogenous predictors ζ = p x 1 vector of errors for the elements of y 4

(b) the latent variable model η = α + Γξ+ Β η + ζ

(b) the latent variable model η = α + Γξ+ Β η + ζ x = Λ xξ + δ y = Λ yη + ε where: η is a vector of latent responses, ξ is a vector of latent predictors, Β and Γ are matrices of coefficients, ζ is a vector of errors for η, and α is a vector of intercepts for η and: Λx is a vector of loadings that link observed x variables to latent predictors, Λy is a vector of loadings that link observed y variables to latent responses, and δ and ε are vectors are errors Equations can be hierarchical. Estimates can be obtained either using likelihood or Bayesian methods. 5

A Graphical Modeling Perspective Hypothesized Probabilistic Network x 1 x 2 ? y 1

A Graphical Modeling Perspective Hypothesized Probabilistic Network x 1 x 2 ? y 1 ? tests for conditional independence y 2 In network models, each directed pathway between two variables, simple or compound, represents a different process.

SEM Perspective: The concept of mediation - topographic wetness plant species richness what do

SEM Perspective: The concept of mediation - topographic wetness plant species richness what do we think the causal link might be? Hypothesize that wet areas produce taller grasses, which reduces richness So, we might postulate: topographic wetness + herbaceous biomass - plant species richness If the second model holds, herbaceous biomass explains (mediates) effect of topographic wetness on plant richness. Note topographic wetness drops out of univariate model when the mediating variable(s) are present. If the second model does not hold, this implies another process is operating.

(A) (B) Environmental factors Phylogenetic distance Grass species richness Environmental factors Grass species richness

(A) (B) Environmental factors Phylogenetic distance Grass species richness Environmental factors Grass species richness Trait distance (C) Phylogenetic distance Trait distance (D) Environmental factors Phylogenetic distance Grass species richness Trait distance

Structural equation meta-model Water availability Rainfall - PET Drought index Moisture Rainfall concentration Disturbance

Structural equation meta-model Water availability Rainfall - PET Drought index Moisture Rainfall concentration Disturbance Fire frequency Soil fertility Percent sand Organic carbon Soil Biomass production Woody Biomass Minimum temperature Temperature Maximum temperature Climatic growing conditions Tree biomass

Landscape Curvature Large-scale Influences Results Local-scale Interactions -0. 30 -0. 45 Distance to Rivers

Landscape Curvature Large-scale Influences Results Local-scale Interactions -0. 30 -0. 45 Distance to Rivers 0. 34 R 2 = 0. 67 Topographic Wetness Index 0. 29 -0. 40 Herbaceous Biomass -0. 17 0. 30 -0. 50 Leaf Nitrogen -0 Rainfall Hotspot . 2 9 6. 4 -0 0. 14 -0. 48 0. 32 Soil Fertility Leaf Sodium 0. 15 0. 27 Leaf Magnesium

Landscape Curvature Large-scale Influences We can consolidate paths if we wish. Local-scale Interactions -0.

Landscape Curvature Large-scale Influences We can consolidate paths if we wish. Local-scale Interactions -0. 30 -0. 45 Distance to Rivers 0. 34 R 2 = 0. 67 Topographic Wetness Index 0. 29 -0. 40 Herbaceous Biomass -0. 17 -0. 50 . 2 9 N 6. 4 -0 -0. 48 0. 32 Soil Fertility 0. 45 Leaf Nitrogen -0 Rainfall Hotspot Leaf Sodium 0. 27 Leaf Magnesium

env. growing conditions biomass soils par ratio disturbance plant species richness species pool

env. growing conditions biomass soils par ratio disturbance plant species richness species pool

env. growing conditions biomass soils par ratio disturbance plant species richness species pool

env. growing conditions biomass soils par ratio disturbance plant species richness species pool

- Disturbance (grazing/fire) - Soils - Non-linear effects - Hierarchical design (plots within sites)

- Disturbance (grazing/fire) - Soils - Non-linear effects - Hierarchical design (plots within sites) p = 0. 85