Structural Equation Modeling (SEM) Essentials

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Structural Equation Modeling (SEM) Essentials
Purpose of this module is to provide a very brief
presentation of the things one needs to know
about SEM before learning how apply SEM.
byJim Grace
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Where You can Learn More about SEM
Grace (2006) Structural Equation Modeling and
Natural Systems. Cambridge Univ. Press.
Shipley (2000) Cause and Correlation in Biology.
Cambridge Univ. Press.
Kline (2005) Principles and Practice of
Structural Equation Modeling. (2nd Edition)
Guilford Press.
Bollen (1989) Structural Equations with Latent
Variables. John Wiley and Sons.
Lee (2007) Structural Equation Modeling A
Bayesian Approach. John Wiley and Sons.
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I. Essential Points about SEM
Outline
II. Structural Equation Models Form and Function
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I. SEM Essentials
1. SEM is a form of graphical modeling, and
therefore, a system in which relationships can be
represented in either graphical or equational
form.
2. An equation is said to be structural if there
exists sufficient evidence from all available
sources to support the interpretation that x1 has
a causal effect on y1.
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3. Structural equation modeling can be defined as
the use of two or more structural equations to
represent complex hypotheses.
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4. Some practical criteria for supporting an
assumption of causal relationships in structural
equations
a. manipulations of x can repeatably be
demonstrated to be followed by responses in y,
and/or b. we can assume that the values of x that
we have can serve as indicators for the values of
x that existed when effects on y were being
generated, and/or c. if it can be assumed that a
manipulation of x would result in a subsequent
change in the values of y Relevant
References Pearl (2000) Causality. Cambridge
University Press. Shipley (2000) Cause and
Correlation in Biology. Cambridge
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5. A Grossly Oversimplified History of SEM
Contemporary
Wright (1918)
Joreskog (1973)
path analysis
SEM
factor analysis
Spearman (1904)
Lee (2007)
testing alt. models
likelihood
r, chi-square
Conven- tional Statistics
Pearson (1890s)
Fisher (1922)
Neyman E. Pearson (1934)
Bayesian Analysis
Bayes LaPlace (1773/1774)
Raftery (1993)
MCMC (1948-)
note that SEM is a framework and incorporates new
statistical techniques as they become available
(if appropriate to its purpose)
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6. SEM is a framework for building and evaluating
multivariate hypotheses about multiple processes.
It is not dependent on a particular estimation
method.
7. When it comes to statistical methodology, it
is important to distinguish between the
priorities of the methodology versus those of the
scientific enterprise. Regarding the diagram
below, in SEM we use statistics for the purposes
of the scientific enterprise.
Statistics and other Methodological Tools,
Procedures, and Principles.
The Scientific Enterprise
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The Methodological Side of SEM
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The Relationship of SEM to the Scientific
Enterprise
modified from Starfield and Bleloch (1991)
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8. SEM seeks to progress knowledge through
cumulative learning. Current work is striving to
increase the capacity for model memory and model
generality.
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9. It is not widely understood that the
univariate model, and especially ANOVA, is not
well suited for studying systems, but rather, is
designed for studying individual processes, net
effects, or for identifying predictors.
10. The dominance of the univariate statistical
model in the natural sciences has, in my personal
view, retarded the progress of science.
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11. An interest in systems under multivariate
control motivates us to explicitly consider the
relative importances of multiple processes and
how they interact. We seek to consider
simultaneously the main factors that determine
how system responses behave.
12. SEM is one of the few applications of
statistical inference where the results of
estimation are frequently you have the wrong
model!. This feedback comes from the unique
feature that in SEM we compare patterns in the
data to those implied by the model. This is an
extremely important form of learning about
systems.
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13. Illustrations of fixed-structure protocol
models
Univariate Models
Do these model structures match the causal forces
that influenced the data? If not, what can they
tell you about the processes operating?
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14. Structural equation modeling and its
associated scientific goals represent an
ambitious undertaking. We should be both humbled
by the limits of our successes and inspired by
the learning that takes place during the journey.

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II. Structural Equation Models Form and Function
A. Anatomy of Observed Variable Models
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I have an article on this subject that is brief
and to the point. Grace, J.B. and K.A. Bollen
2005. Interpreting the results from multiple
regression and structural equation models. Bull.
Ecological Soc. Amer. 86283-295.
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II. Structural Equation Models Form and Function
B. Anatomy of Latent Variable Models
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II. Structural Equation Models Form and Function
C. Estimation and Evaluation
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2. Estimation Methods
(a) decomposition of correlations (original path
analysis)
(b) least-squares procedures (historic or in
special cases)
(c) maximum likelihood (standard method)
(d) Markov chain Monte Carlo (MCMC) methods
(including Bayesian applications)
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Model Identification - Summary
1. For the model parameters to be estimated with
unique values, they must be identified. As in
linear algebra, we have a requirement that we
need as many known pieces of information as we do
unknown parameters.
2. Several factors can prevent identification,
including a. too many paths specified in
model b. certain kinds of model specifications
can make parameters unidentified c.
multicollinearity d. combination of a complex
model and a small sample
3. Good news is that most software checks for
identification (in something called the
information matrix) and lets you know which
parameters are not identified.
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The topic of model selection, which focuses on
how you choose among competing models, is very
important. Please refer to additional tutorials
for considerations of this topic.
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While we have glossed over as many details as we
could, these fundamentals will hopefully help you
get started with SEM.
Another gentle introduction to SEM oriented to
the community ecologist is Chapter 30 in McCune,
B. and J.B. Grace 2004. Analysis of Ecological
Communities. MJM. (sold at cost with no profit)