What is? Structural Equation Modeling ( A Very Brief Introduction )

1. What is?Structural Equation Modeling (A Very Brief Introduction) Patrick Sturgis University of Surrey

2. What is SEM? • SEM is not one statistical ‘technique’. • It integrates a number of different multivariate techniques into one model fitting process. • It is essentially an integration of: • Measurement theory • Factor analysis • Regression • Simultaneous equation modeling • Path analysis

3. SEM is essentially Path Analysis using Latent Variables

4. What are Latent Variables? • Most/all variables in the social world are not directly observable. • This makes them ‘latent’ or hypothetical constructs. • We measure latent variables with observable indicators, e.g. questionnaire items. • We can think of the variance of an observable indicator as being partially caused by: • The latent construct in question • Other factors (error)

5. True score and measurement error x = t + e Error Measured True Score True point on continuum Systematic Error Mean of Errors ≠0 Random Error Mean of Errors =0

6. error True score The True Score Equation X = t + e Can be expressed diagrammatically Observed item Problem – with one indicator, the equation is unidentified. We can’t separate true score and error.

7. Identifying True Score & Error • This means we need multiple indicators of each latent variable • With multiple indicators we can use Factor Analysis to estimate these parameters • Factor analysis transforms correlated observed variables into uncorrelated components • We can then use a subset of components to summarise the observed relationships

8. Latent Construct A Common Factor Model Indicator 1 Indicator 2 Indicator 3 Indicator 4 .6 .5 .5 .7 Indicators become conditionally independent Factor loadings = regression of factor on indicators

9. Factor Analysis • So the factor loading is the standardised regression of the latent variable on the indicator. • Squaring the factor loading gives us the % of variance ‘explained’ by the latent variable (factor). • This can be considered as the true score component of the item. • 1-the % variance explained by the factor gives us the residual or ‘error’ variance. • Thus, the variance of the factor contains only the true score component of each item.

10. Benefits of Latent Variables • Most social concepts are complex and multi-faceted • Using single measures will not adequately cover the full conceptual map • Systematic error biases descriptive and causal inferences • Stochastic error in dependents leaves estimates unbiased but less efficient • Stochastic error in independents attenuates associational effect sizes estimates

11. Remember SEM is essentially Path Analysis using Latent Variables We now know about latent variables, what about path analysis?

12. Path Analysis • Sewell Wright, a biologist, developed the fundamental ideas of path analysis in the 1920s. • The diagrammatic representation of a theoretical model. • Standardised Notation. • Estimation of a series of regression models to ‘decompose’ effects: • Direct • Indirect • Total

13. - + + + + Direct, Indirect and Total Effects Example: effect of exam nerves on exam performance Physical/mental anxiety Exam performance Exam stress Exam preparation

14. Effect Decomposition • We can break down effects of X on Y into direct, indirect and total. • Direct = a • Indirect = b x c • Total = a + b xc Y1 b X1 c a Y2

15. Latent variable Observed variable Residual or Error Term Causal Effect Covariance Path Standard Symbols for Path Analysis

16. e1 e2 e3 1 1 1 O1 O2 O3 e7 e8 e9 1 1 1 1 gender O4 O5 O6 1 1 O10 e10 1 1 attitude behaviour O11 e11 1 O12 e12 1 1 class 1 e13 e14 O7 O8 O9 1 1 1 e6 e5 e4 So when a path diagram includes latent variables… …it becomes a SEM

17. Simultaneous Equations We might estimate this as 3 separate models factor model, run out factor score variable Regression Y on X In SEM, we estimate the equations simultaneously Regression Y on X, Z

18. Estimation & Model Fit • Variety of estimators available but predominantly maximum likelihood (ML) • Model fit can be tested by comparison of Likelihoods for specified v baseline model • Tests significance of difference in likelihood between specified and observed variance covariance matrices • With large n, no model fits! • ‘Adjusted’ indices (RMSEA, CFI, etc. etc..) • Perhaps more useful for testing nested models, where the comparator is more substantively meaningful.

19. Other things we can do with SEM… • Panel data models (tomorrow!) • Categorical endogenous variables • Multiple group models • Latent variable interactions • Model missing data • Complex sample data • Multi-level SEM