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Structural equation modeling is a multivariate statistical analysis technique that is used to analyze structural relationships. This technique is the combination of factor analysis and multiple regression analysis, and it is used to analyze the structural relationship between measured variables and latent constructs.

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