1. Structural Equation
Modeling
Douglas Gunzler, PhD
Adam Carle, PhD
Adam T. Perzynski, PhD
Adam.Perzynski@case.edu
@atperzynski

2. SEM QUIZ TIME!!

3. Why are SEM Researchers
so Fashionable?

4. We have lots of Models!

5. Why were the latent
variables so sad?

6. Why are the latent variables
so sad?
Because they are unobserved!!

7. Why did the structural
equation model need new
shoes?

8. Why did the structural
equation model need new
shoes?
It had poor fit.
CFI = 0.5
TLI= 0.71
RMSEA = 0.73 (0.62-0.94)

9. Why are SEM Researchers
so confident?

10. Why are SEM Researchers
so confident?
They estimate with the maximum
likelihood.

11. SEM Resource: Statmodel.com

12. SEM Resources

13. SEM Resources

14. SEM Resources

15. SEM Resources

16. “SEM allows researchers to take advantage
of reasonable causal assumptions”

17. Model Specification Proposition
No amount of model specification testing will identify if the model is a
good fit, only which is better between the choices you are offering.
17

18. Role of Statistical Testing in Model Specification
minimize specification errors between an initial model and the unknown,
“true” model
1. logic, theory and prior empirical evidence to choose the initial model
2. statistical testing to compare the initial model to competing models
3. combination of theory, prior evidence, and the results of the statistical
testing to decide upon which model or models are appropriate for a
given study
18

19. (Good) Longitudinal
Models (should) Start with
a “Theory of Change”

20. Teaser Holmes MR, Yoon S, Berg KA, Cage JL, Perzynski AT. Promoting the development of resilient academic functioning in maltreated children. Child
abuse & neglect. 2018 Jan 1;75:92-103.

21. (Bio)ecological Developmental Systems Theory
W. Thomas Boyce et al. PNAS 2020;117:38:23235-23241 Image by Paquette and Ryan, adapted from Bronfrenbrenner

22. Longitudinal Structural
Equation Models

23. Basic Models

24. Longitudinal Mediation

25. Autoregressive Models
• In common SEM applications, the autoregressive model can be
further extended:
• Systolic Blood Pressure: 7AM (X1), 10AM (X3), Noon (X5), 7PM (X7)
• Salt Intake: 9AM (Z2), 11AM (Z4), 6PM (Z6)
• This data collection structure results in a data file with 7 separate
time points.

26. Autoregressive models

27. Latent Growth (and growth mixture) Models
• Involve a growth assumption
• Are considerably flexible
• Are adept at modeling non-
linearity
• Can accommodate categorical
latent change

28. Latent Growth vs Latent Growth Mixture

29. Quick Segway: Latent Categorical vs. Latent
Continuous
Figure 1. Example data of Accuracy and Response Time,
subjected to (A) correlation analysis, and (B) latent profile
analysis.

30. Growth Models are also often called
trajectories
• Regrettably, the term “trajectory” has taken on multiple meanings
across disciplines and research studies.
• A broad, inclusive definition of trajectory modeling is the analysis of
patterns of change or stability.
• Confusion is possible between aggregate trajectories which
summarize an overall average pattern of change for a population and
disaggregated trajectories which examine multiple potential
trajectories of different shapes (George 2006).

31. Continuous Latent Growth Curve Analysis
• LGA / LGCA
• Studies in older adults (ie George and Lynch 2003) typically find that
the slope of the latent growth curve for depressive symptoms is small
and positive, and that the slope of the curve is steepest in the oldest
cohorts.

32. Example from George and Lynch (2003)

33. Example of a single growth trajectory

34. What is Latent Class Growth Analysis?
• Latent Class Growth Analysis (LCGA), one form of growth mixture
modeling, belongs to a family of statistical techniques referred to as
general latent variable modeling or GLVM.

35. Why would we ever think we should use
LCGA?
• Studying the mean change or using a single trajectory for everyone
assumes uniform heterogeneity in the population.
• Researchers use familiar methods and typically assume that the underlying
(latent or real) distribution of variables is continuous.
• We have theoretical reasons to suspect that underlying distributions could
be categorical.
• Life course theorists (Dannefer) specifically caution that intracohort
differentiation is unlikely to be homogeneous.

36. Growth Mixture models seek determine underlying empirical
subgroups characterized by different patterns of growth.

37. Health and Retirement Study Example
• 5,195 age-eligible respondents from the 1992 Health and Retirement
Study cohort, who completed interviews in all seven waves through
2004.
• Depressive symptoms in HRS are measured an 8-item version of the
CES-D
• Using MPlus, we fitted LCGA models.

38. Results
• How many classes/trajectories are there?
• What do the classes look like?
• Are variables associated with being in a particular class?

39. Rule for Determining the number of Latent
Classes
•“How many trajectories are there?”
• Measures of model fit including:
• Lo-Mendell-Rubin Test (LMR test)
• log-likelihood (LL)
• Bayesian Information Criteria (BIC) (Vuong, 1989; Muthen, 2004; Muthen, & Muthen,
2005; Nylund et al, 2007).
• Here we will use the LMR Test
• Where k is the number of latent classes, this test gives a p-value for the k-1
versus the k-class model when running the k-class model (Vuong, 1989;
Muthen, B. 2005).
• The first time p > .05, k-1 is the preferred number of classes.

40. How many Classes are there?
K LL BIC Adjusted BIC LMR Test LMR p Entropy
2 -56367.49 112983.09 112890.94 10525.69 0.000 0.955
3 -55146.65 110618.41 110497.66 2410.38 0.000 0.922
4 -54652.99 109708.09 109558.74 974.66 0.015 0.925
5 -54357.08 109193.27 109015.32 519.88 0.149 0.901
6 -54090.08 108736.27 108529.72 397.39 0.354 0.912
7 -54079.98 108793.06 108557.91 97.55 0.392 0.920
8 -53895.87 108501.85 108238.10 307.84 0.314 0.732
Table 1. Depressive Symptoms LCGA Model Fit Comparison, N = 5,195

41. What do the classes look like?
0
1
2
3
4
5
6
1994 1996 1998 2000 2002 2004
Mean
#
of
Depressive
Symptoms
HRS Study Wave
Figure 1: Four Latent Classes of Depressive Symptoms over 12
Years of the HRS
Many Persistent Symptoms = 5.4% Decreasing Symptoms = 9.6%
Increasing Symptoms = 11.5% Almost No Symptoms = 73.5%
N = 5195

42. Does anything influence the chances of being in a
particular class?
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Latent
Class
Probability
Years of Education
Figure 2. Relationship between Years of Education and
Depressive Symptoms Trajectory/Latent Class Membership
Many Symptoms Decreasing Symptoms
Increasing Symptoms Almost No Symptoms
N = 5195

43. Does anything influence the chances of being in a
particular class?
• Females, African Americans and those with fewer
years of education have a higher probability of
being in the Many Symptoms trajectory.
N = 5195
OR b p OR b p OR b p
Age 0.94 -0.057 0.010 0.96 -0.043 0.014 1.02 0.016 0.422
Female 2.19 0.785 0.000 1.53 0.428 0.001 1.41 0.346 0.002
Black 1.89 0.635 0.000 1.90 0.641 0.000 1.54 0.429 0.001
Hispanic 1.12 0.113 0.655 1.59 0.461 0.018 1.19 0.178 0.461
Low Education 1.32 0.274 0.000 0.84 -0.173 0.000 0.90 -0.105 0.000
Table. Effects of Demographics on the Likelikhood of a Depressive Symptoms Trajectory
vs. Almost No Symptoms (reference category)
Many Symptoms Decreasing Increasing

44. (Good) Longitudinal
Models (should) Start with
a “Theory of Change”

47. Structural Equation Models are Powerful for
Understanding Diverse & Distinct Outcomes over Time
Academic Competence
Neglect
Physical Abuse
Internalizing Symptoms
Sexual Abuse
Neglect
Early Substance Use
Physical Abuse
Diverse groupings of developmental
pathways can be explicitly modeled as
opposed to ignored or assumed away.
Need to carefully consider the
consequences of modeling decisions.
Even highly sophisticated techniques
(univariate latent growth models and
growth mixture models) can conceal
important variation of interest to
researchers and policy makers.

48. Common questions about longitudinal SEM
How big of a
sample size do I
need?
How many waves
of data do I need?
Is my model too
complicated?
Bigger!
(~200 minimum but
it depends)
More!
(4 waves is a general
minimum, but it
depends)
Probably
(I usually start by
building simpler
subsets of models
and gradually
growing complexity)

49. Common questions about longitudinal SEM
How big of a
sample size do I
need?
How many waves
of data do I need?
Is my model too
complicated?

50. Selected Strengths & Limitations of SEM
• Strengths
• Very flexible
• Estimate and correct for measurement error
• Limitations
• Large sample sizes
• Challenging to learn
• Need lots of hands-on experience to learn
• Need a strong theoretical basis
• It’s easy to mis-specify a model if you have no idea what you are doing.
• Recommendations
• Try it!
• Ask the experts
• Don’t give up right away

51. Thank you. More questions? Let’s connect!
Adam.Perzynski@case.edu
@atperzynski