1. 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.
by Jim Grace

2. 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.

3. I. Essential Points about SEM
Outline
I. Essential Points about SEM
II. Structural Equation Models: Form and Function

4. y1 = γ11x1 + ζ1 1 11 x1 y1 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.
y1 = γ11x1 + ζ1
equational
form x1 y1 1
11
graphical
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.

5. y2 y1 x1 y3 ζ1 ζ3 y1 = γ11x1 + ζ1 y2 = β 21y1 + γ 21x1 + ζ 2
3. Structural equation modeling can be defined as the use of two or more structural equations to represent complex hypotheses. y2 y1 x1 y3 ζ1 ζ2 ζ3
Complex
Hypothesis
e.g.
y1 = γ11x1 + ζ1
y2 = β 21y1 + γ 21x1 + ζ 2
y3 = β 32y2 + γ31x1 + ζ 3
Corresponding
Equations

6. 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

7. 5. A Grossly Oversimplified History of SEM
Contemporary
Wright
(1918)
path analysis
Joreskog
(1973)
SEM
factor analysis
Spearman
(1904)
Lee
(2007)
testing
alt. models
r, chi-square
likelihood
Pearson
(1890s)
Conven-
tional
Statistics
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)

8. Statistics and other Methodological Tools, Procedures, and Principles.
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

9. The Methodological Side of SEM

10. The Relationship of SEM to the Scientific Enterprise
Understanding of Processes
univariate
descriptive statistics
exploration,
methodology and
theory development
realistic
predictive models
simplistic
models
multivariate
detailed process
univariate data
modeling
Data
structural
equation
modified from Starfield and Bleloch (1991)

11. structural equation modeling
8. SEM seeks to progress knowledge through cumulative learning. Current work is striving to increase the capacity for model memory and model generality.
exploratory/
model-building
applications
structural equation modeling
confirmatory/
hypothesis-testing
one aim of
SEM

12. 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.

13. 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.

14. 13. Illustrations of fixed-structure protocol models:
Univariate Models
Multivariate Models x1 x2 x3 x4 x5 F y1 y2 y3 y4 y5 x1 x2 x3 x4 x5 y1
Do these model structures match the causal forces that influenced
the data? If not, what can they tell you about the processes operating?

15. 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.

16. II. Structural Equation Models: Form and Function
A. Anatomy of Observed Variable Models

17. x1 y1 y2 1 2 Some Terminology 21 11 21 exogenous variable
endogenous
variables
21
11
21
path
coefficients
direct effect of x1 on y2
indirect effect of x1 on y2
is 11 times 21

18. nonrecursive y1 x2 x1 y2 ζ1 ζ2 C D A B
model B, which has paths between all variables is “saturated” (vs A, which is “unsaturated”)

19. x1 y1 x2 First Rule of Path Coefficients: the path coefficients for
unanalyzed relationships (curved arrows) between exogenous variables are simply the correlations (standardized form) or covariances (unstandardized form). x1 x2 y1
.40
x1 x2 y1
x1 1.0 x y

20. x1 y1 y2
11 = .50
21 = .60
 (gamma) used to represent effect of exogenous on endogenous.
 (beta) used to represent effect of endogenous on endogenous.
Second Rule of Path Coefficients: when variables are
connected by a single causal path, the path
coefficient is simply the standardized or unstandardized
regression coefficient (note that a standardized regression
coefficient = a simple correlation.)
x1 y1 y2
x1 1.0 y y

21. Third Rule of Path Coefficients: strength of a compound path is the product of the coefficients along the path. x1 y1 y2
.50
.60
Thus, in this example the effect of x1 on y2 = 0.5 x 0.6 = 0.30
Since the strength of the indirect path from x1 to y2 equals the
correlation between x1 and y2, we say x1 and y2 are
conditionally independent.

22. What does it mean when two separated variables
are not conditionally independent?
x1 y1 y2
x1 1.0 y y x1 y1 y2
r = .55
r = .60
0.55 x 0.60 = 0.33, which is not equal to 0.50

23. x1 y1 y2 The inequality implies that the true model is
Fourth Rule of Path Coefficients: when variables are
connected by more than one causal pathway, the path
coefficients are "partial" regression coefficients.
additional process
Which pairs of variables are connected by two causal paths?
answer: x1 and y2 (obvious one), but also y1 and y2, which are connected by the joint influence of x1 on both of them.

24. x1 y1 x2 And for another case:
A case of shared causal influence: the unanalyzed relation
between x1 and x2 represents the effects of an unspecified
joint causal process. Therefore, x1 and y1 connected by two
causal paths. x2 and y1 likewise.

25. x1 y1 y2 .31 .40 .48 How to Interpret Partial Path Coefficients:
- The Concept of Statistical Control
The effect of y1 on y2 is controlled for the joint effects of x1.
I have an article on this subject that is brief and to the point.
Grace, J.B. and K.A. Bollen Interpreting the results from multiple regression and structural equation models. Bull. Ecological Soc. Amer. 86:

26. Interpretation of Partial Coefficients
Analogy to an electronic equalizer
from Sourceforge.net
With all other variables in model held to their means, how much does a response variable change when a predictor is varied?

27. x1 y1 y2
Fifth Rule of Path Coefficients: paths from error variables are correlations or covariances.
R2 = 0.16
.92
R2 = 0.44
.73 2 1
.31
.40
.48
equation for path
from error variable
.56
alternative is to
show values for zetas,
which = 1-R2
.84

28. y2 2 x1 y1 1 .50 .40 Now, imagine y1 and y2 are joint responses
x1 y1 y2
x1 1.0 y y
Now, imagine y1 and y2
are joint responses
Sixth Rule of Path Coefficients: unanalyzed residual correlations between endogenous variables are partial correlations or covariances.

29. x1 y1 y2
R2 = 0.16
R2 = 0.25 2 1
.50
.40
This implies that some other factor is influencing y1 and y2
the partial correlation between y1 and y2 is typically
represented as a correlated error term

30. Seventh Rule of Path Coefficients: total effect one variable has on another equals the sum of its direct and indirect effects. y1 x2 x1 y2 ζ1 ζ2
.80
.15
.64
-.11
.27
x1 x2 y1 y y
Total Effects:
Eighth Rule of Path Coefficients:
sum of all pathways between two variables (causal and noncausal) equals the correlation/covariance.
note: correlation between
x1 and y1 = 0.55, which
equals *0.11

31. path coefficient for x2 to y1 very different from correlation,
Suppression Effect - when presence of another variable causes path coefficient to strongly differ from bivariate correlation.
x1 x2 y1 y2
x1 1.0 x y y y1 x2 x1 y2 ζ1 ζ2
.80
.15
.64
-.11
.27
path coefficient for x2 to y1 very different from correlation,
(results from overwhelming influence from x1.)

32. II. Structural Equation Models: Form and Function
B. Anatomy of Latent Variable Models

33. Latent Variables
Latent variables are those whose presence we suspect or theorize, but for which we have no direct measures.
Intelligence
IQ score
*note that we must specify some parameter, either error,
loading, or variance of latent variable. ζ
latent variable
observed indicator
error
variable
1.0
fixed loading*

34. Latent Variables (cont.)
Purposes Served by Latent Variables:
(2) Allow us to estimate and correct for measurement error.
(3) Represent certain kinds of hypotheses.
(1) Specification of difference between observed data
and processes of interest.

35. Range of Examples single-indicator multi-method repeated measures
Elevation
estimate
from map
multi-method
Soil
Organic
soil C
loss on
ignition
Territory
Size
singing range, t1
singing range, t2
singing range, t3
repeated measures
Caribou
Counts
observer 1
observer 2
repeatability

36. The Concept of Measurement Error
the argument for universal use of latent variables
1. Observed variable models, path or other, assume all
independent variables are measured without error.
2. Reliability - the degree to which a measurement is
repeatable (i.e., a measure of precision).
error in measuring x is ascribed to error in
predicting/explaining y x y
0.60
R2 = 0.30
illustration

37. Example Imagine that some of the observed variance in x is
due to error of measurement.
calibration data set based
on repeated measurement trials
plot x-trial1 x-trial2 x-trial3 n
average correlation between trials = 0.90
therefore, average R-square = 0.81
reliability = square root of R2
measurement error variance =
(1 - R2) times VARx
imagine in this case VARx = 3.14, so error variance = 0.19 x 3.14 = 0.60
LV1 x
LV2 y
.90
.65
1.0
.60
R2 = .42

38. II. Structural Equation Models: Form and Function
C. Estimation and Evaluation

39. y = α + Βy + Γx + ζ 1. The Multiequational Framework
(a) the observed variable model
We can model the interdependences among a set of predictors and responses using an extension of the general linear model that accommodates the dependences of response variables on other response variables.
y = p x 1 vector of responses
α = p x 1 vector of intercepts
Β = p x p coefficient matrix of ys on ys
Γ = p x q coefficient matrix of ys on xs
x = q x 1 vector of exogenous predictors
ζ = p x 1 vector of errors for the elements of y
Φ = cov (x) = q x q matrix of covariances among xs
Ψ = cov (ζ) = q x q matrix of covariances among errors
y = α + Βy + Γx + ζ

40. η = α + Β η + Γξ + ζ x = Λxξ + δ y = Λyη + ε
The LISREL Equations
Jöreskög 1973
(b) the latent variable model
η = α + Β η + Γξ + ζ
x = Λxξ + δ
y = Λyη + 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
(c) the measurement model
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

41. 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)

42. Bayesian References:
Bayesian Networks:
Neopolitan, R.E. (2004). Learning Bayesian Networks. Upper Saddle River, NJ, Prentice Hall Publs.
Bayesian SEM:
Lee, SY (2007) Structural Equation Modeling: A Bayesian Approach. Wiley & Sons.

43. SEM is Based on the Analysis of Covariances!
Why?
Analysis of correlations represents loss of information. A B
r = 0.86
r = 0.50
illustration with regressions having same slope and intercept
Analysis of covariances allows for estimation of both standardized and unstandardized parameters.

44. { } Σ S = σ11 σ12 σ22 σ13 σ23 σ33 Model-Implied Correlations
Observed Correlations*
1.0 S
* typically the unstandardized correlations, or covariances
2. Estimation (cont.) – analysis of covariance structure
The most commonly used method of estimation over the past 3 decades has been through the analysis of covariance structure (think – analysis of patterns of correlations among variables).
compare

45. (e.g., maximum likelihood)y1 y2
Hypothesized Model Σ = {
σ11
σ12 σ22
σ13 σ23 σ33 }
Implied Covariance Matrix
Observed Covariance Matrix
1.3 S
compare
Model Fit
Evaluations +
Parameter
Estimates
(e.g., maximum likelihood)
estimation
3. Evaluation

46. 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.

47. The most commonly used fitting function in maximum likelihood estimation of structural equation models is based on the log likelihood ratio, which compares the likelihood for a given model to the likelihood of a model with perfect fit.
Fitting Functions
Note that when sample matrix and implied matrix are equal, terms 1 and 3 = 0 and terms 2 and 4 = 0. Thus, perfect model fit yields a value of FML of 0.

48. Maximum likelihood estimators, such as FML, possess several important properties: (1) asymptotically unbiased, (2) scale invariant, and (3) best estimators.
Assumptions:
(1)  and S matrices are positive definite (i.e., that they do not have a singular determinant such as might arise from a negative variance estimate, an implied correlation greater than 1.0, or from one row of a matrix being a linear function of another), and
(2) data follow a multinormal distribution.
Fitting Functions (cont.)

49. One of the most commonly used approaches to performing such tests (the model Χ2 test) utilizes the fact that the maximum likelihood fitting function FML follows a X2 (chi-square) distribution.
The Χ2 Test
X2 = n-1(FML)
Here, n refers to the sample size, thus X2 is a direct function of sample size.
Assessment of Fit between Sample Covariance and Model- Implied Covariance Matrix

50. x y2 y1 1.0 0.4 1.0 0.35 0.5 1.0 Illustration of the use of Χ2
X2 = 3.64 with 1 df and 100 samples
P = 0.056
X2 = 7.27 with 1 df and 200 samples
P = 0.007 x y1 y2
1.0
rxy2 expected to be 0.2 (0.40 x 0.50)
X2 = 1.82 with 1 df and 50 samples
P = 0.18
correlation matrix
issue: should there be
a path from x to y2?
0.40
0.50
Essentially, our ability to detect significant differences from our base model, depends as usual on sample size.

51. Additional Points about Model Fit Indices:
1.The chi-square test appears to be reasonably effective at sample sizes less than 200.
2. There is no perfect answer to the model selection problem.
4. A lot of attention is being paid to Bayesian model selection methods at the present time.
3. No topic in SEM has had more attention than the development of indices that can be used as guides for model selection.
5. In SEM practice, much of the weight of evidence falls on the investigator to show that the results are repeatable (predictive of the next sample).

52. Alternatives when data extremely nonnormal
Robust Methods:
Satorra, A., & Bentler, P. M. (1988). Scaling corrections for chi-square statistics in covariance structure analysis Proceedings of the Business and Economics Statistics Section of the American Statistical Association,
Bootstrap Methods:
Bollen, K. A., & Stine, R. A. (1993). Bootstrapping goodness-of-fit measures in structural equation models. In K. A. Bollen and J. S. Long (Eds.) Testing structural equation models. Newbury Park, CA: Sage Publications.
Alternative Distribution Specification: - Bayesian and other:

53. Diagnosing Causes of Lack of Fit (misspecification)
Residuals: Most fit indices represent average of residuals between observed and predicted covariances. Therefore, individual residuals should be inspected.
Correlation Matrix to be Analyzed
y y x y y x
Fitted Correlation Matrix x
residual = 0.15
Diagnosing Causes of Lack of Fit (misspecification)
Modification Indices: Predicted effects of model modification on model chi-square.

54. 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.

55. 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 Analysis of Ecological Communities. MJM. (sold at cost with no profit)