Key ideas, terms and concepts in SEM
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This document provides an overview of key concepts in structural equation modeling (SEM) including: 1) Path diagrams are used to represent structural equations and the relationships between latent and observed variables. 2) SEM analyzes the variance/covariance matrix of observed variables rather than raw data. Maximum likelihood estimation is used to estimate model parameters by maximizing the likelihood of the sample data. 3) Model identification, fit, and constraints are important concepts. A model must be over-identified to yield a likelihood value for assessing fit. Parameter constraints can be used to make a just-identified model over-identified.
Key ideas, terms and concepts in SEM
- 1. Key ideas, terms & concepts in SEM Professor Patrick Sturgis
- 2. Plan • Path diagrams • Exogenous, endogenous variables • Variance/covariance matrices • Maximum likelihood estimation • Parameter constraints • Nested Models and Model fit • Model identification
- 3. Path diagrams • An appealing feature of SEM is representation of equations diagrammatically e.g. bivariate regression Y= bX + e
- 4. Path Diagram conventions Error variance / disturbance term Measured latent variable Observed / manifest variable Covariance / non-directional path Regression / directional path
- 5. Reading path diagrams A latent variable Causes/measured by 3 observed variables With 3 error variances
- 6. Reading path diagrams 2 latent variables, each measured by 3 observed variables Correlated
- 7. Reading path diagrams 2 latent variables, each measured by 3 observed variables Regression of LV1 on LV2 Error/disturbance
- 8. Exogenous/Endogenous variables • Endogenous (dependent) – caused by variables in the system • Exogenous (independent) – caused by variables outside the system • In SEM a variable can be a predictor and an outcome (a mediating variable)
- 9. 2 (correlated) exogenous variables
- 10. η1 endogenous, η2 exogenous 1 1
- 11. Data for SEM • In SEM we analyse the variance/covariance matrix (S) of the observed variables, not raw data • Some SEMs also analyse means • The goal is to summarise S by specifying a simpler underlying structure: the SEM • The SEM yields an implied var/covar matrix which can be compared to S
- 12. Variance/Covariance Matrix (S) x1 x2 x3 x4 x5 X6 x1 0.91 -0.37 0.05 0.04 0.34 0.31 x2 -0.37 1.01 0.11 0.03 -0.22 -0.23 x3 0.05 0.11 0.84 0.29 0.14 0.11 x4 0.04 0.03 0.29 1.13 0.11 0.06 x5 0.34 -0.22 0.14 0.11 1.12 0.34 x6 0.31 -0.23 0.11 0.06 0.34 0.96
- 13. Maximum Likelihood (ML) • ML estimates model parameters by maximising the Likelihood, L, of sample data • L is a mathematical function based on joint probability of continuous sample observations • ML is asymptotically unbiased and efficient, assuming multivariate normal data • The (log)likelihood of a model can be used to test fit against more/less restrictive baseline
- 14. Parameter constraints • An important part of SEM is fixing or constraining model parameters • We fix some model parameters to particular values, commonly 0, or 1 • We constrain other model parameters to be equal to other model parameters • Parameter constraints are important for identification
- 15. Nested Models • Two models, A & B, are said to be ‘nested’ when one is a subset of the other (A = B + parameter restrictions) e.g. Model B: yi= a + b1X1 + b2X2 +ei • Model A: yi= a + b1X1 + b2X2 +ei (constraint: b1=b2) • Model C (not nested in B): yi= a + b1X1 + b2Z2 +ei
- 16. Model Fit • Based on (log)likelihood of model(s) • Where model A is nested in model B: LLA-LLB = , with df = dfA-dfB • Where p of > 0.05, we prefer the more parsimonious model, A • Where B = observed matrix, there is no difference between observed and implied • Model ‘fits’! 2 χ 2 χ
- 17. Model Identification • An equation needs enough ‘known’ pieces of information to produce unique estimates of ‘unknown’ parameters X + 2Y=7 (unidentified) 3 + 2Y=7 (identified) (y=2) • In SEM ‘knowns’ are the variances/ covariances/ means of observed variables • Unknowns are the model parameters to be estimated
- 18. Identification Status • Models can be: – Unidentified, knowns < unknowns – Just identified, knowns = unknowns – Over-identified, knowns > unknowns • In general, for CFA/SEM we require over- identified models • Over-identified SEMs yield a likelihood value which can be used to assess model fit
- 19. Assessing identification status • Checking identification status using the counting rule • Let s = number of observed variables in the model • number of non-redundant parameters = • t=number of parameters to be estimated t> model is unidentified t< model is over-identified )1( 2 1 +ss )1( 2 1 +ss )1( 2 1 +ss
- 20. Example 1 - identification )1( 2 1 +ss = 6 Non-redundant parameters parameters to be estimated 3 * error variance + 2 * factor loading + 1 * latent variance = 6 6 - 6 = 0 degrees of freedom, model is just-identified
- 21. Controlling Identification • We can make an under/just identified model over-identified by: – Adding more knowns – Removing unknowns • Including more observed variables can add more knowns • Parameter constraints remove unknowns • Constraint b1=b2 removes one unknown from the model (gain 1 df)
- 22. Example 2 – add knowns )1( 2 1 +ss = 10 Non-redundant parameters parameters to be estimated 4 * error variance + 3 * factor loading + 1 * latent variance = 8 10 - 8 = 2 degrees of freedom, model is over-identified
- 23. Example 3 – remove unknowns )1( 2 1 +ss = 6 Non-redundant parameters parameters to be estimated 3 * error variance + 0 * factor loading + 1 * latent variance = 4 6 - 4 = 2 degrees of freedom, model is over-identified Constrain factor loadings = 1
- 24. Summary • SEM requires understanding of some ideas which are unfamiliar for many substantive researchers: – Path diagrams – Analysing variance/covariance matrix – ML estimation – global ‘test’ of model fit – Nested models – Identification – Parameter constraints/restrictions
- 25. for more information contact www.ncrm.ac.uk