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1 Composites and Nonlinear Modeling data from Grace and Keeley (2006) Ecological Applications

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2 Bivariate relationship between total cover and richness How do we model the fit to the nonlinear line?

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3 A Simple Nonlinear Model

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4 A Simple Nonlinear Model - Results Note that these are standardized coefficients. What do they mean? Are the two main path coefficients significant contributors to the model?

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5 Amos Output p-values imply both paths contribute significantly. we can also test to see if the model fit is poor with paths omitted.

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6 Model Evaluation Chi-square increased by more than 3.84 when a single path was omitted, thus missing path deemed important to include.

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7 Using Composite to Summarize Nonlinear Effect A composite is created using a latent variable having a zero variance. Identification achieved by fixing one incoming path to 1.0. latent variable zero variance fixed path

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8 Nonlinear Model Results R-square consistent with path coef. standardized coefficient Test for nonlinearity performed prior to introduction of composite.

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9 What about when the nonlinear effect is endogenous?

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10 Endogenous Nonlinear Model Note that the polynomial term is treated as an exogenous variable.

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11 Endogenous Nonlinear Model Results

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12 Endogenous Nonlinear Model with Composite

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13 We can also use composites for other purposes. soil prop1 soil prop2 soil prop3 flooding frequency earthworm density Step 1 - test that all paths should be in model. soil prop1 soil prop2 soil prop3 flooding frequency earthworm density Step 2 - estimate collective effect using composite. soil effects e1 0 Now we can compare the collective effects of soil properties to the effects of flooding frequency on earthworm density.

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