1. Chapter 17 STRUCTURAL EQUATION MODELING

2. Structural Equation Modeling (SEM)  Relatively new statistical technique used to test theoretical or causal models. Unlike path analysis, SEM allows estimation of measurement error.  Sometimes referred to as Covariance Structure Models because a covariance matrix is analyzed in SEM, and as LISREL because this is the name of the first commercially available software for SEM.

3. Research questions addressed with SEM: 1. Measurement issues A. As discussed in chapter 15, SEM allows the researcher to test a specific model of an instrument when conducting a CFA. B. Simultaneous testing of an instrument’s factor structure in more than one age, gender, ethnic, illness, etc. subgroup can be done to determine if factor structure is equivalent in the different subgroups

4. 2. Model testing SEM allows: A. estimation of path coefficients in a theoretical model while controlling for the effect of measurement error on the hypothesized relationships; A. testing of the overall fit of a model to the data A. testing different assumptions about measurement error

5. 3. Group Analyses Just as different factor structures can be tested and compared in different groups, so too can different theoretical models. SEM allows you to test whether the magnitude of the paths in a theoretical model are significantly different, and even the model itself in different groups.

6. Data Requirements: 1. Continuous, normally distributed variables. 2. Multiple measures (“indicators”) of theoretical constructs – ideally 3 or more. 3. Big sample – research suggests a sample size of 500 or more (per group if analyzing more than one group) is needed to avoid obtaining misleading results.

7. Assumptions: 1. Theoretical Many theoretical assumptions! Assumptions are made regarding both the presence and absence of relationships among variables in the measurement and theoretical models. All these assumptions are statistically evaluated in SEM.

8. 2. General Statistical A. Regression assumptions B. Assumptions regarding residuals C. Sample size is asymptotic (so large as to approach infinity)

9. 3. Estimation Method Specific Maximum Likelihood (ML) is a common estimation method and appears most robust, particularly as sample size increases.  No single variable or group of variables perfectly explains another in the dataset (cannot have highly correlated variables).  Variables in the analysis have a distribution that is multivariate normal.

10. Terminology measured variables (indicators) responses to answers on a questionnaire or from a data collection protocol directly observable X

11. theoretical variables (latent variables) not directly observable represent abstract constructs that are responsible for creating the pattern of data observed in the indicator variables

12. measurement model model diagramming the relationship between a theoretical construct (latent variable) and its indicator variables contains assumptions about measurement error needs to be tested prior to testing the theoretical model or comparing this model in different groups..... Why?

13. HEALTH Well-being Happiness Karnofsky’s Perf. Status % ADLs Treadmill Perf. Resting Heart Rate PSYCHOLOGICAL PERFORMANCE PHYSIOLGICAL Example of a measurement model (  represents measurement error) ww hh tt rr aa kk

14. theoretical model model of relationships between latent variables if no path is present between two latent variables, this means that the researcher assumes there is no relationship between these two variables

15. Example of a theoretical model HEALTH RESOURCES STRESSFUL LIFE EVENTS LIFESTYLE

16. Parameters the various coefficients estimated during SEM: 1. error variances 2. variances for indicators and latent variables 3. paths in the measurement and theoretical models 4. correlations between measurement error and latent variables 5. “disturbances” -- measurement error in equations predicting endogenous latent variables

17. Fit Indices Statistics used to evaluate validity of model. “Good fit” indicated by: non-significant Chi-Square* CFI, GFI >.95 RMSEA, RMR <.05 * May be significant due to sample size bias and not indicative of a poor fit.

18. Steps in conducting an SEM: 1. Specify the full model to be tested and check identification. Note: Models that appear identified on paper may prove to not be statistically or empirically identifiable due to properties of the data. 2. Test fit of measurement model. 3. Re-specify and refit measurement model if model fit statistics, etc. indicate this is necessary. 4. Once the measurement model is determined to have a good fit, test the fit of the theoretical model. 5. Re-specify and refit theoretical model if model fit statistics, etc. indicate this is necessary.

19. Issues & Concerns 1. A large sample size increases the likelihood that the Chi-Square test of model fit will be statistically significant. Models should be evaluated with a variety of fit indices.

20. 2. Type II error becomes inflated as models are re-specified and tested. Consider how many models were run to arrive at the “best” model when you read an article. If a sample is split into model testing and model confirming sample, and the model replicates with no or only minor re- specification, the results are more trustworthy.

21. 3. Sample sizes of 200 or less are likely to lack sufficient statistical power. Many articles in the literature may be underpowered. Be careful about changing practice on the basis of an SEM with a sample size of less than 200.

22. 4. You may also see models tested that contain categorical variables. We do not really understand how this violation of SEM assumptions influences the path coefficients being estimated in SEM or model fit statistics. Treat results from such models cautiously.