SEM Introduction Latent Factors Mediation analysis

Structural Equation Modeling

Statistical Consulting Services:Yulia Sidi, Renjie Chen, Joochul Lee

Department of StatisticsUniversity of Connecticut

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SEM Introduction Latent Factors Mediation analysis

What is SEM?

SEM is an "umbrella" term that captures different types ofmodels, including t-test, simple linear regression, multipleregression, analysis of variance, path analysis, confirmatoryanalysis, generalized linear models, latent factor analysis,mediation analysis and many others.

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SEM Introduction Latent Factors Mediation analysis

SEM, Example

Let’s say we are interested to study adolescents substance usefor families with alcoholic parents.

Alcoholic Parent Adolescent Substance

Use

This SEM example represents a simple T-Test.

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SEM Introduction Latent Factors Mediation analysis

SEM, Example continued

What if we would like to add other control variables?

Alcoholic Parent Adolescent Substance

Use

Antisocial Personality

Disorder

Parent Education

Socio-economic Status

Adolescent’s Age

This SEM example represents Analysis of Covariance.SCS Department of Statistics, UCONN

SEM Introduction Latent Factors Mediation analysis

SEM

Both T-Test and Analysis of Covariance are very simpleexamples of SEM.

The "classical" SEM is based on the following twocomponents:

I Latent variablesI Path diagrams

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SEM Introduction Latent Factors Mediation analysis

Latent variableLatent variables cannot be directly observed. We can usespecial questionnaires to capture such variables indirectly.

Example:I Happiness→ Oxford Happiness Questionnaire

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SEM Introduction Latent Factors Mediation analysis

Latent variable

More examples:I IntelligenceI Depression levelI Quality of lifeI Business confidenceI Conservatism

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SEM Introduction Latent Factors Mediation analysis

Path diagram

We can think about path diagram as of flow chart. Theyrepresent relationships between variables, such as correlation,or causal relationship.

Alcoholic Parent Adolescent Substance

Use

Antisocial Personality

Disorder

Parent Education

Socio-economic Status

Adolescent’s Age

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SEM Introduction Latent Factors Mediation analysis

Path diagram - example

Alcoholic Parent Adolescent Substance

Use Stress Depression

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SEM Introduction Latent Factors Mediation analysis

Latent variable & Path diagram

The combination of latent variables and path diagramscorresponds to a "classical" SEM.Before we continue, let’s introduce the following SEM notation:

e

Observed variable

Latent variable

Path/ Causal effect

Correlation

Error/Disturbance

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SEM Introduction Latent Factors Mediation analysis

SEM, Notation example

SAT MATH SCORE

High School Student

Intelligence

e1

1

Parents Intelligence

SAT READING SCORE

e2

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SEM Introduction Latent Factors Mediation analysis

SEM, Example

SAT MATH

SAT READING

GRE GPA

U V a

b

c

GRE = aSATMATH + bSATREADING + U

GPA = cGRE + V

⇓GPA = caSATMATH + cbSATREADING + cU + V

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SEM Introduction Latent Factors Mediation analysis

SEM, Example

Y1 X2

Dem60

Y2

Y3

Y4

Y5

Y6

Y8

Y7

Dem65

Ind60

X1 X3

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SEM Introduction Latent Factors Mediation analysis

SEM, Example variablesVariable DescriptionInd60 Industrialization in 1960Dem60 Political democracy in 1960Dem65 Political democracy in 1965X1 The gross national product (GNP) per capita in 1960X2 The inanimate energy consumption per capita in 1960X3 The percentage of the labor force in industry in 1960Y1 Expert ratings of the freedom of the press in 1960Y2 The freedom of political opposition in 1960Y3 The fairness of elections in 1960Y4 The effectiveness of the elected legislature in 1960Y5 Expert ratings of the freedom of the press in 1965Y6 The freedom of political opposition in 1965Y7 The fairness of elections in 1965Y8 The effectiveness of the elected legislature in 1965

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SEM Introduction Latent Factors Mediation analysis

Data

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SEM Introduction Latent Factors Mediation analysis

Path diagram

I Domain knowledge + statistical verification

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SEM Introduction Latent Factors Mediation analysis

Confirmative factor analysis (CFA) vs Multipleregression

I Latent factor explain variables with errorI MLR variables explain a variable with error

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SEM Introduction Latent Factors Mediation analysis

CFA

library(lavaan); data(PoliticalDemocracy)model <- ’# measurement modelind60 =~ x1 + x2 + x3’summary(cfa(model, data=PoliticalDemocracy))Latent Variables:Estimate Std.Err z-value P(>|z|)ind60 =~x1 1.000x2 2.193 0.142 15.403 0.000x3 1.824 0.153 11.883 0.000

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SEM Introduction Latent Factors Mediation analysis

Models with latent variables

I Coefficient correlationI Multiple linear regressionI Moderation AnalysisI Mediation Analysis

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SEM Introduction Latent Factors Mediation analysis

Mediation Analysis : Example

I Two variables : Test Score (Independent), Happiness(Dependent)

I Mediator : Self esteem

I Mediator that explains the underlying mechanism of therelationship between test Score (Independent) andhappiness (Dependent)

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SEM Introduction Latent Factors Mediation analysis

Direct effect and Indirect effect

I Mediation Analysis- M = aX+ ε- Y = dX+ ε- Y = cX+ bM+ ε

I Indirect effect : a× b

I Direct effect : cI Total effect (d) = Indirect effect + Direct effect

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SEM Introduction Latent Factors Mediation analysis

Analysis with example in ’lavaan’ package

I Mediation Analysis- Dem60 = a · Ind60+ ε- Dem65 = d · Ind60+ ε- Dem65 = c · Ind60+ b ·Dem60+ ε

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SEM Introduction Latent Factors Mediation analysis

Code

model <- # Factor analysis‘ind60 =~ x1 + x2 + x3dem60 =~ y1 + y2 + y3 + y4dem65 =~ y5 + y6 + y7 + y8# Mediation analysisdem60 ~ ind60dem65 ~ ind60 + dem60’fit <- sem(model, data=PoliticalDemocracy)summary(fit)

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SEM Introduction Latent Factors Mediation analysis

Result

I Mediation Analysis Result

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SEM Introduction Latent Factors Mediation analysis

Statistical significance?

I Mediation Analysis Result

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SEM Introduction Latent Factors Mediation analysis

Bootstrapping

I Bootstrap replications of the original data can be createdby resampling with replacement

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SEM Introduction Latent Factors Mediation analysis

Code for bootstrapping

set.seed(1234)model <- # Factor analysis’ind60 =~ x1 + x2 + x3dem60 =~ y1 + y2 + y3 + y4dem65 =~ y5 + y6 + y7 + y8# Mediation analysisdem60 ~ a * ind60dem65 ~ c * ind60 + b * dem60indirect := a * btotaleff := c + indirect’fit <- sem(model,data=PoliticalDemocracy,se="bootstrap",bootstrap=100)parameterEstimates(fit)

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SEM Introduction Latent Factors Mediation analysis

Result using bootstrapping

I Result in R

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SEM Introduction Latent Factors Mediation analysis

Diagram

I Result

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SEM Introduction Latent Factors Mediation analysis

Mediation analysis summary

I Mediator that explains the underlying mechanism of therelationship between Independent variable and Dependentvariable

I Keep in mind that mediation analysis does not implynecessarily causal relationships

I Indirect effect, direct effect, total effect in mediationanalysis

I For the statistical significance, bootstrapping can beconsidered

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SEM Introduction Latent Factors Mediation analysis

Summary

I "Classical" SEM is based on two components: latentvariables and path diagrams.

I Latent factors: analysis and interpretation using lavaanpackage.

I Mediation analysis: analysis and interpretation usinglavaan package.

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