1. Structural Equation Modeling using MPlus
04.11
Yaeeun Kim

2. Characteristics of SEM
The term structural equation modeling (SEM) does not designate a single statistical technique but instead refers to a family of related procedures. Other terms such as covariance structure analysis, covariance structure modeling, or analysis of covariance structures are also used in the literature to classify these techniques together under a single label.
There are two broad classes of variables in SEM, observed and latent. The observed class represents your data—that is, variables for which you have collected scores and entered in a data file.
The whole of the general linear model (GLM) can be seen as just a restricted case of SEM.

3. Research Questions & SEM
Confirmatory Factor Analysis.
Path Models with Latent Variables.
Latent Class Analysis.
Models of Growth Trajectories.
Model Differences between Groups.

4. Model Estimation
Use a model-fitting program to derive estimates of model p arameters (Amos, EQS, LISREL, Mplus).
Maximum Likelihood (ML) is by far the most widely used estimation procedure.
However, ML assumes multivariate normality. If the data a re not MVN, then other procedures can be used (e.g. Satorr a-Bentler adjustment)

5. ML
The term maximum likelihood describes the statistical principle that underlies the derivation of parameter estimates; the estimates are the ones that maximize the likelihood (the continuous generalization) that the data (the observed covariances) were drawn from this population.
The method of ML estimation method is the default in most SEM computer programs, and most structural equation models described in the literature are analyzed with this method.

6. Assessing Model Fit
Determine how well the model accounts for the observed variances and covariances of the measured variables.
Fit Indices: chi square, GFI, CFI, TLI, RMSEA, SRMR, a nd many others.
These indicate only the overall or average fit of the model , and do not indicate whether the results are theoretically meaningful.

7. Assumptions within SEM
Large samples: try to obtain a 10:1 ratio of number of subjects to number of model parameters.
Variables are typically at interval or ratio level of meas urement, although not necessarily.
Mplus program designed to analyze categorical variables
Approximately multivariate normal distribution.

8. Flow chart of basic steps of SEM

9. Model diagram symbols
1. Observed variables with squares or rectangles (e.g., ).
2. Latent variables with circles or ellipses (e.g., ).
3. Hypothesized directional effects of one variable on another, or direct effects, with a line with a single arrowhead (e.g., →).
4. Covariances (in the unstandardized solution) or correlations (in the standardized one) between independent variables—referred to in SEM as exogenous variables—with a curved line with two arrowheads ( ).

10. CFA Anxiety Depression latent variable r1 r2 error r3 r4 r5 r6 r7 r8
Nervous
Not Calm r2
error r3
Tired Out
Anxiety
Effort r4
Can’t Sit Still r5
Restless r6
Worthless r7
Latent variable
;cannot be measured; can’t have measurement error
Depression
Can’t Cheer Up r8
Depressed r9
Hopeless
r10

11. Path Model with Latent Variables
Family Conflict
Child
Observer
Child Depression
Parent
Child Age
Child
Observer
Child Gender
Parent

12. Unstandardized estimates

13. Standardized estimates

15. Maximum Likelihood Estimates for a Recursive Path Model of Causes and Effects
-.384;
This means that a 1-point increase on the school support variable predicts a .384-point decrease on the burnout variable, controlling for coercive control.
z = –.384/.079 = 4.86, which exceeds the critical value for two-tailed statistical significance at the .01 level, or 2.58.
A level of school support one full standard deviation above the mean predicts a burnout level just over
.40 standard deviations below the mean, holding coercive control constant

16. Model fit result
Root Mean Square Error of Approximation (RMSEA): Badness of fit indicator
Close-fit hypothesis
Is RMSEA <= .05
Poor-fit hypothesis
Is RMSEA >= .10
Goodness of Fit Index (GFI): Estimates the proportion of data covariances explained by the model. Similar to R2 in OLS regression
Comparative Fit Index (CFI): Measures the relative improvement of your model over that of a baseline model.
Standardized Root Mean Square Residual (SRMR): Assesses the overall difference between the observed and predicted correlations. Values > .08 indicate poor fit

17. Mplus User’s Guide Ex 3.11

18. Mplus Syntax for Ex 3.11
TITLE: this is an example of a path analysis with continuous dependent variables DATA: FILE IS ex3.11.dat; VARIABLE: NAMES ARE y1-y3 x1-x3; MODEL: y1 y2 ON x1 x2 x3; y3 ON y1 y2 x2;

19. Testing Indirect Effects
TITLE: this is an example of a path analysis with continuous dependent variables DATA: FILE IS ex3.11.dat; VARIABLE: NAMES ARE y1-y3 x1-x3; MODEL: y1 y2 ON x1 x2 x3; y3 ON y1 y2 x2; MODEL INDIRECT: y3 IND y1 x1; y3 IND y2 x1;

20. With categorical dependent variables (binary or ordered responses)
TITLE: this is an example of a path analysis with categorical dependent variables DATA: FILE IS ex3.12.dat; VARIABLE: NAMES ARE u1-u3 x1-x3; CATEGORICAL ARE u1-u3 MODEL: u1 u2 ON x1 x2 x3; u3 ON u1 u2 x2;

21. Reference
Kline, R. B. (2011). Principles and practice of structural equation modeling (3rd Ed). New York: Guilford Press).
Psychology 8032 Structural Equation Modeling course lecture note.