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 !?!