1. Advanced Statistical Methods: Continuous Variables http://statisticalmethods.wordpress.com
Structural Equation Modeling_Part I
Materials for this session are based on Dr. Bart Meuleman’s introductory lectures on SEM, delivered at the
2011 QMSS2 Summer School in Leuven

2. Ex: how are education, income & threat related?
SEM = Multivariate analytical technique: to gain insight in the relations between multiple variables
Ex: how are education, income & threat related?
OBS Education Income Ethnic threat

3. Rather than single equations, systems of equations are modeled
Ex. regression:
Y = a + b1X1 + b2X2 + e
Ex. SEM: Y1 = λ1F + ε1
Y2 = λ2F + ε2
Y3 = λ3F + ε3
F = b1X1 + b2X2 + ε4
X2 = b1X1 + ε5

4. Advantages of modeling systems of equations:
– Latent variable estimation
E.g.: values, attitudes, socioeconomic status, IQ
Multiple indicators that contain measurement error (random & non‐random)
SEM allows to estimate relations btw. latent variables instead of btw. unreliable indicators

5. Ex: perceived economic threat scale in the European Social Survey:
Would you say it is generally bad or good for [country’s]
economy that people come to live here from other countries?
(0 = bad for the economy ‐ 10 = good for the economy)
Would you say that [country’s] cultural life is generally
undermined or enriched by people coming to live here from
other countries?
(0 = cultural life undermined – 10 = cultural life enriched)
Is [country] made a worse or a better place to live by people
coming to live here from other countries?
(0 = worse place to live – 10 = better place to live)

6. X1 = education
X2 = income
F = perceived economic threat
Y1, Y2, Y3 = ESS indicators

8. Graphical notation of SEM – Types of variables
Path Diagrams
Graphical notation of SEM – Types of variables
Latent unobserved variable (measured indirectly)
Observed (manifest) variable (measured directly)
Measurement/stochastic error F X1 e1

9. Graphical notation of SEM – Types of relations
Uni‐directional relation / Effect
Correlation
Feedback relation
No relation

10. Exogenous vs. Endogenous variables exogenous: not explained by model;
endogeous: explained by model
X1 = education
X2 = income
F = perceived economic threat
Y1, Y2, Y3 = ESS indicators

14. Types of Models
1. Path models: direct & indirect effects btw. manifest variables

15. 2. Confirmatory Factor Analysis: Construct Validation

16. 3. Full SEM: combination of 1 & 2

17. 4. Advanced Models

18. Confirmatory Factor Analysis (CFA)
- way of representing latent constructs that are measured through multiple indicators containing measurement error
X1 = T + E1
X2 = T + E2
CFA vs. sumscores and indices: CFA tests assumptions which
are assumed a priori in sumscores and indices:
– equal weighting
– equal measurement errors
– unidimensionality

19. Notation of CFA model:

20. Parameters in the model:
– Factor loadings: relations btw. latent construct & observed indicators ‐ λ
– Residual variances: measurement error in the observed indicators ‐ Var(δ)
– Variance of the latent construct – Var(ξ)

21. Model Estimation – 1
SEM (CFA) models covariance (and mean) structures (rathe rthan working with raw data)
OBS Edu Inc. Ethnic threat
Educ Inc EthnT
Educ 2.00
Inc
EthnT

22. Model Estimation – 2
Every set of parameters implies a certain covariance (& mean) structure
= implied or reproduced covariance (mean)
structure
Estimate the parameters so that they approach the observed covariance (mean) structure as good as possible
Mostly: Maximum Likelihood Estimation

23. Model Estimation – 3

24. Model Estimation – 4

25. Evalutation of Model Fit– 1
Look at discrepancy btw. observed & implied covariance matrices
Chi‐square test
– tests whether the assumed linear structure holds in the
population
– if chi2 value is statistically significant, reject the model
– degrees of freedom: if there are k observed variables in a
linear model with t unconstrained parameters to be
estimated, then:
df={k(k+1)/2} – t
However: sensitive for large sample sizes (& deviations
from multivariate normality)
– Chi2 difference test for nested models

26. Evalutation of Model Fit– 2
Alternative fit indices:

27. Evalutation of Model Fit– 3
Previous measures are indices of global fit (the fit of the model as a whole)
One should also look at measures of local (mis)fit:
Modification indices:
- available for constrained & fixed parameters (except from the one for scaling the latent variable)
- give the expected chi² reduction if the correspoding parameter is set free
- Chi² tests with 1 df
- also check the corresponing expected parameter change (EPC)!

28. Restrictions & identification – 1
3 possibilities for parameters:
– free parameters;
– fixed parameters (parameters = fixed to a certain constant);
– constrained parameters (parameters that have been set equal)
RESTRICTIONS reflect a priori knowledge
Functions of restrictions:
– identification of the model
– statistical assumptions
of the model
– transformations of theoretical
assumptions

29. Restrictions & identification - 2
Is there enough information available in the data to estimate all free parameters?
Model needs to be identified!
A model is identified if the sample variance and covariances include enough information to obtain unique estimates of all free model parameters.

30. Conditions for model identification:
1: Latent variables should be scaled by…
– …constraining the factor loading of one item (i.e.the marker item) to 1 OR
– …constraining the variance of the latent variable to 1
2: Statistical identification
– the no. of freely estimated parameters (t) should not exceed the number of pieces of information [k(k+1)/2], where k = no. of observed variables
– the no. of dfs should not be negative
– in terms of equations: the number of unknowns cannot be larger than the number of knowns

31. Under‐identified models:
– less pieces of information than free parameters; df < 0
– infinite number of solutions
– model is more complex than the data structure
Just‐identified models:
– as many pieces of information as free parameters; df = 0
– a unique perfect solution
– the model is as complex as the data structure
Over‐identified models:
– more pieces of information than free parameters; df > 0
– no perfect solution possible
– the model is less complex than the data structure

32. -what kind of model would this be?
Excercise:
how many degrees of freedom does a CFA model with 1 latent variable & 4 indicators have?
-what kind of model would this be?
Remember:
df={k(k+1)/2} – t, where k = no. of observed variables & t = no of free parameters