Structural Equation Modeling
H o s s e i n S a l e h iJ e n n y L e h m a nJ a c o b Te n n e yO c t o b e r , 2 0 1 5
LEARNING OBJECTIVES
Understand Latent Variables (Ghost Chasing)
Definition of Structural Equation Modeling (SEM)
SEM Model
Goals in PFP
SEM Assumptions
Basic Components of SEM
Calculate Implied Covariance Matrix
SEM Approach
SEM in R
SEM’s Advantages
GHOST CHASING
We are in the business of Chasing “Ghosts”
• “Ghost” diagnoses
• Measuring “Ghosts”
• Exchanging one “Ghost” for another “Ghost”
(Ainsworth 2006)
LATENT VARIABLES▪ Variables of Interest
▪ Not directly measured or manifest
▪ Common
▪ Intelligence
▪ Trust
▪ Democracy
▪ Disturbance variables
(Paxton)
FAMILY TREE OF SEM Factor Analysis
Exploratory Factor Analysis
Confirmatory Factor Analysis
Now it is …
Structural Equation Modeling
(SEM)’s
turn !!!
(Hubona)
▪ The structural model is :
e.g.
▪ The measurement models are:
SEM MODEL
…Question
Question
Question
Risk Requirement…
Question
Question
Question
Risk Tolerance
SEM IN PFP
Let’s run some data in R.
?
(FinaMetrica)
Observed (or manifest, measures, indicators)
Latent (or factor, constructs)
PATH DIAGRAM SYMBOLS Direction of influence, relationship from one
variable to another Reciprocal effects Correlation or covariance
(Sudano & Perzvnski, 2013)
…
Question
Question
Question
Risk Requirement
𝜹𝟓
…
𝜹𝟐
𝜹𝟏
…
Question
Question
Question
Risk Tolerance
𝜺𝟐𝟒
…
𝜺𝟐
𝜺𝟏𝟏
…
𝟐
𝟐𝟒
𝜸𝟏
𝜸𝟐
𝜸…
𝜸𝟓
𝜷
ζ 𝟏
Structural Model
Two Measurement ModelsESTABLISHING PATH DIAGRAM
𝐹=𝐵𝐹+ζ
𝑋=Γ 𝑓 2+𝛿𝑌=Λ 𝑓 1+ε
GOALS OF SEM▪ To determine whether the theoretical model is supported by sample
data or the model fits the data well.
▪ To understand the complex relationships among constructs.
▪ To compare the covariance matrix from all manifest variables (from the data collected) to the model-implied covariance matrix of the manifest variables.
(Oct. 1 Class Presentation)
SEM ASSUMPTIONS Univariate and multivariate normality (In theory but never in
practice)
Independence of observations
Linearity in the relationships between your variables
Adequate sample size
The factors and measurement errors are uncorrelated.• Cov(F, ) = 0ε
(Oct. 1 Class Presentation)
▪ The structural model is :
e.g.
▪ The measurement models are:
Let’s unpack the two measurement models:
SEM MODEL
(Steiger)……
▪ The structural model is :
▪ The measurement models are:
SEM GENERAL MODEL
(Steiger)
Let’s unpack the structural model:
SEM GENERAL MODEL
Let’s unpack the two measurement models:
(Steiger)
▪ Error terms covariance matrix
SEM GENERAL MODEL
𝚯𝜹=(𝛿1 0 … 00 𝛿2 … 0… … … …0 0 … 𝛿𝑚
)𝒎×𝒎
𝚯𝜺=(𝜺1 0 … 00 𝜺2 … 0… … … …0 0 … 𝜺𝑛
)𝒏×𝒏
(Steiger)
▪ Implied covariance matrix
SEM GENERAL MODEL
(Steiger)
…
Question
Question
Question
Risk Requirement
𝜹𝟓
…
𝜹𝟐
𝜹𝟏
…
Question
Question
Question
Risk Tolerance
𝜺𝟐𝟒
…
𝜺𝟐
𝜺𝟏𝟏
…
𝟐
𝟐𝟒
𝜸𝟏
𝜸𝟐
𝜸…
𝜸𝟓
𝜷
ζ 𝟏
ESTABLISHING PATH DIAGRAM
𝐹=𝐵𝐹+ζ
𝑋=Γ 𝑓 2+𝛿𝑌=Λ 𝑓 1+ε
POLITICAL DEMOCRACY▪ The two latent variables :
• DEM60 = Democracy measure in 1960
• IND60 = Industrialization measure in 1960
▪ The two observation series are:• X variables are macroeconomic measures:
o = GNP per capita, 1960o = Energy consumption per capita, 1960o = Percentage of labor force in industry, 1960
• Y variables are macroeconomic measures: o = Freedom of the press, 1960o = Freedom of political opposition, 1960o = Fairness of elections, 1960o = Effectiveness of elected legislature, 1960 (Bollen, 1989)
EXAMPLE: POLITICAL DEMOCRACY MODEL
𝜺𝟒
𝜺𝟑
𝜺𝟐
𝜺𝟏𝒙𝟏
𝒙𝟐
𝒙𝟑
IND 60
𝜹𝟑
𝜹𝟐
𝜹𝟏
𝒚𝟑
𝒚𝟏
𝒚𝟐
𝒚𝟒
DEM 60
𝒚 𝟏
𝒚 𝟑
𝒚 𝟐
𝒚 𝟒
𝒙𝟏
𝒙𝟐
𝒙𝟑
𝜷
ζ 𝟏
(Bollen, 1989)
▪ The structural model is :
The measurement models are:
Let’s unpack the two measurement models:
SEM MODEL FOR DEMOCRACY EXAMPLE
LATENT VARIABLE MODELS 212/20/2006
IMPORTANT MATRICES▪ We can rewrite the two measurement models in a matrix form :
▪ And the implied covariance matrix would be:
LATENT VARIABLE MODELS 222/20/2006
IMPORTANT MATRICES▪ Next, we need to compare the observed covariance matrix (S) and implied
covariance matrix () and calculate the residual matrix.
▪ Let’s simulate some data in R.
APPROACH TO SEM Model Specification
Creating a hypothesized model that you think explains the relationships among multiple variablesConverting the model to multiple equations
Model EstimationTechnique used to calculate parametersE.G. - Maximum Likelihood (ML), Ordinary Least Squares (OLS), etc.
(Stevens, 2009)
SEM can address the directional effects between latent
variables, whereas factor analysis does not model relations
because it assumes factors are independent.
Unlike factor analysis, SEM allows you to restrict some of
loadings to zero to see how this changes the outcome.
(Dr. Westfall)
SEM ADVANTAGES
Missing data Can be dealt with in the typical ways (e.g. regression, EM
algorithm, etc.) Most SEM programs will estimate missing data and run the
model simultaneously
CONSIDERATION IN APPLYING SEM
CONCLUSION
Now we know how
to use SEM to find
the ghosts !!!!!!
REFERENCES▪ Ainsworth, A. (2006). "Ghost Chasing": Demystifying Latent Variables and SEM. Retrieved from UCLA.
▪ Bollen, K.A. (1989). Structural Equations with Latent Variables. John Wiley & Sons.
▪ Hubona, G. (2015). Structural Equation Modeling (SEM) with Lavaan. Udem.
▪ Iacobucci, D. (2009). Everything you always wanted to know about SEM (structural equations modeling) but were afraid to ask. Journal of Consumer Psychology, 19(Oct), 673-680.
▪ Paxton, P. (n.d.). Structural Equation Modeling: An Overview.
▪ PIRE. (2007). Structural Equation Modeling Workshop.
▪ Rosseel, Y. (2012). Lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 47(May), 2-36.
▪ Stevens, J. (2009). Structural Equation Modeling (SEM). University of Oregon.
▪ Steiger, J.H. (n.d.). LISREL Models and Methods.
▪ Sudano, & Perzynski. (2013). Applied Structural Equation Modeling for Dummies, by Dummies. Retrieved from Indiana University, Bloomington.
▪ FinaMetrica
▪ Wikipedia
▪ Oct. 1 Group
▪ Dr. Westfall
THANK YOU
QUESTIONS !?!