Structural Equation Modeling

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CHAPTER 30 Structural Equation Modeling
Tables, Figures, and Equations
From McCune, B. J. B. Grace. 2002. Analysis
of Ecological Communities. MjM Software Design,
Gleneden Beach, Oregon http//www.pcord.com
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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.
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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. R2 in the simple
correlation column represents the highest R2
obtained for any single variable for other
columns it is the variance explained for the
whole model.
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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.
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Box 30.2 What is a partial correlation?
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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
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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
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Box 30.3. Calculation of R2 in a multiple
regression or path model.
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 (R2) would
simply be the sum of the squared bivariate
correlations. In such a case, the R2 for
richness would be However, when predictors are
correlated, the variance explained differs
markedly from that estimated by a simple
addition.
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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 (R2) would
simply be the sum of the squared bivariate
correlations. In such a case, the R2 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 x1z and x2z are
z-transformed predictor variables. As in
bivariate regression, the values of the betas are
those that satisfy the least squares criterion,
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Box 30.3. cont. Now, the R2 for our example
can be calculated using the formula where ry1
and ry2 are the bivariate correlations between y
and x1 and y and x2. For our example in Figure
30.3,
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Box 30.3. cont. To come full circle in this
illustration, if we had the case where grazing
(x1) and biomass (x2) were uncorrelated as first
mentioned in this Box, then the bivariate and
partial correlations would be equal (i.e., ?1
ry1 and ?2 ry2) and the above equation would
reduce to
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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 PC1 PC2 PC3 PC4 PC5
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
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When moving from a multiple regression to a
structural equation model, the underlying
mathematics changes from y a1 b1x1
b2x2 where y is Rich and x1 and x2 are Graz and
Bio, respectively, to a structured set of
simultaneous regression equations (hence,
structural equation modeling) y1 a1 b11x1
b12y2 y2 a2 b21x1
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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.
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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 x1, x2, and x3 are different
facets of a single underlying causal variable,
A, while x4, x5, and x6 are different facets of
the causal variable, B. Further, y represents an
available estimate of the latent variable C.
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Figure 30.6. Initial (hypothesized) measurement
model relating three latent variables and ten
indicator variables.
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Figure 30.7. Modified measurement model.
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Figure 30.8. Final full model. Here a1 and a2
represent the measured ordination axis scores.
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Table 30.3. Standardized factor loadings
resulting from structural equation model analysis.
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Table 30.4. Correlations among latent variables.
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Figure 30.9. Initial conceptual model (from Gough
et al. 1994).
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Figure 30.10. Initial structural equation model.
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Figure 30.11. Initial results of confirmatory
factor analysis of measurement model. Numbers are
path coefficients, represent partial regression
coefficients and correlation coefficients.
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Figure 30.12. Final results for confirmatory
factor analysis of revised measurement model.
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Figure 30.13. Revised structural equation model.
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Figure 30.14. Final structural equation model.
Path coefficients shown represent partial
regression and correlation coefficients. R2
values specify the amount of variance explained
for the associated endogenous variable.
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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).
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