1. CHAPTER 30 Structural Equation Modeling From: McCune, B. & J. B. Grace. 2002. Analysis of Ecological Communities. MjM Software Design, Gleneden Beach, Oregon http://www.pcord.comhttp://www.pcord.com Tables, Figures, and Equations

2. Figure 30.1. Example of an ordination biplot showing results of nonmetric multidimensional scaling, group identities for individual plots, and vectors indicating environmental correlations (modified from Grace et al. 2000). Elev is elevation; C, Ca, K, Mg, Mn, N, P, and Zn are elements in soil samples. Ellipses represent ordination space envelopes for vegetation groups A, B, C, D, and E while an envelope is not given for the heterogeneous group F.

4. Figure 30.2. Illustration of the regression relationships between environmental parameters and Axes 1 and 2 of the ordination. Simple bivariate regression, multiple regression, and stepwise regression results are shown for comparison. “----“ denotes nonsignificant coefficients. Double-headed arrows represent correlations between independent variables, which are dealt with differently in the three methods of correlation analysis. R 2 in the simple correlation column represents the highest R 2 obtained for any single variable; for other columns it is the variance explained for the whole model.

5. Figure 30.3. Offsetting pathways are represented differently by bivariate correlation and regression, multiple regression, and path models. Graz = grazing (yes or no), Bio = standing community biomass, and Rich = plant species richness. In this example, the path model shows how offsetting negative and positive effects of grazing on richness can result in a zero bivariate correlation.

6. Box 30.2 What is a partial correlation?

7. The strength of the path mediated through biomass is calculated based on the formula: strength of a compound path= product of path components which in the case of Graz  Bio  Rich is -0.5  -0.8 = +0.40

8. Further, the total effect of grazing on richness is the sum of the various paths that connect a predictor variable with a response variable. In this case, total effect = sum of individual paths = -0.40 + 0.40 = 0.0

9. In multiple regression and path models, results compensate for the correlations among predictors. For the example in Figure 30.3, variation in richness is explained by both variation in grazing and biomass. If grazing and biomass were uncorrelated, the variance explained (R 2 ) would simply be the sum of the squared bivariate correlations. In such a case, the R 2 for richness would be However, when predictors are correlated, the variance explained differs markedly from that estimated by a simple addition. Box 30.3. Calculation of R 2 in a multiple regression or path model.

10. In multiple regression and path models, results compensate for the correlations among predictors. For the example in Figure 30.3, variation in rich- ness is explained by both variation in grazing and biomass. If grazing and biomass were uncorrelated, the variance explained (R 2 ) would simply be the sum of the squared bivariate correlations. In such a case, the R 2 for richness would be However, when predictors are correlated, the vari-ance explained differs markedly from that estimated by a simple addition. Box 30.3. cont. Estimates of the predicted values can be calculated as follows: where  1 and  2 are standardized partial regression coefficients (see Box 30.2) and x 1z and x 2z are “z-transformed” predictor variables. As in bivariate regression, the values of the betas are those that satisfy the least squares criterion,

11. Box 30.3. cont. Now, the R 2 for our example can be calculated using the formula where r y1 and r y2 are the bivariate correlations between y and x 1 and y and x 2. For our example in Figure 30.3,

12. Box 30.3. cont. To come full circle in this illustration, if we had the case where grazing (x 1 ) and biomass (x 2 ) were uncorrelated as first mentioned in this Box, then the bivariate and partial correlations would be equal (i.e.,  1 = r y1 and  2 = r y2 ) and the above equation would reduce to

13. Table 30.2. Principal component loadings for the first five principal components (PC1 to PC5). Loadings greater than 0.3 are shown in bold to highlight patterns. Variable PC1PC2PC3PC4PC5 elev-.18.44-.04.55-.26 Ca.19.47-.29-.14.04 Mg.22.36-.60-.29.25 Mn.04.52.23.32-.11 Zn.38.20.38-.14.22 K.41.10.29-.11-.26 P.42.01.32-.13-.01 pH-.33.20.35.03.77 C.40-.19-.09.41.12 N.35-.24-.22.52.37

14. When moving from a multiple regression to a structural equation model, the underlying mathematics changes from y = a 1 + b 1 x 1 + b 2 x 2 where y is Rich and x 1 and x 2 are Graz and Bio, respectively, to a structured set of simultaneous regression equations (hence, “structural equation” modeling) y 1 = a 1 + b 11 x 1 + b 12 y 2 y 2 = a 2 + b 21 x 1

15. Figure 30.4. Use of principal components in combination with multiple regression. PC1 through PC5 represent principal components of the environmental variables at left. Axis1 and Axis2 represent scores on NMS ordination axes of community data. Numbers along arrows are partial correlation coefficients.

16. Figure 30.5. Development of a structural equation model in contrast to a regression model. Boxes represent measured or indicator variables while ellipses represent conceptual or latent variables. In the structural equation model, the indicator variables are organized around the hypothesis that x 1, x 2, and x 3 are different facets of a single underlying causal variable, A, while x 4, x 5, and x 6 are different facets of the causal variable, B. Further, y represents an available estimate of the latent variable C.

17. Figure 30.6. Initial (hypothesized) measurement model relating three latent variables and ten indicator variables.

18. Figure 30.7. Modified measurement model.

19. Figure 30.8. Final full model. Here a1 and a2 represent the measured ordination axis scores.

20. Table 30.3. Standardized factor loadings resulting from structural equation model analysis.

21. Table 30.4. Correlations among latent variables.

22. Figure 30.9. Initial conceptual model (from Gough et al. 1994).

23. Figure 30.10. Initial structural equation model.

24. Figure 30.11. Initial results of confirmatory factor analysis of measurement model. Numbers are path coefficients, represent partial regression coefficients and correlation coefficients.

25. Figure 30.12. Final results for confirmatory factor analysis of revised measurement model.

26. Figure 30.13. Revised structural equation model.

27. Figure 30.14. Final structural equation model. Path coefficients shown represent partial regression and correlation coefficients. R 2 values specify the amount of variance explained for the associated endogenous variable.

28. Table 30.5. Standardized total effects of predictors on predicted variables for the model in Figure 30.14. Total effects include both indirect and direct effects, and represent the sum of the strengths of all pathways between two variables. Numbers in parentheses are standard errors. Numbers in brackets are t values. Reprinted with permission from Grace and Pugesek (1997).