Jiyoon An Kiran Pedada Structural Equation Modeling

Jiyoon AnKiran Pedada

Structural Equation Modeling

AgendaPart 1 (Presented by Jiyoon An)- SEM and latent variable- Find a model from datasetPart 2 (Presented by Kiran Pedada)- SEM Structural model and measurement

models- How to use Lavaan- Addressing missing values- Path Diagrams

Part 1 (Presented by Jiyoon An)

Structural equation modeling (SEM)Test and estimate the (causal)

relationships among observable measures and non-observable theoretical (or latent) variables, and further to describe relationships between the latent variables themselves with directed arrows

Source: http://davidakenny.net/

Why latent variable?A latent variable, a random variable, differs

from a fixed process parameterMeasuring a person’s characteristics (e.g.

dominance)Everyone has a different level of

dominance. Some are less dominant and some are more dominant

We cannot measure dominance directly and need a latent variable

Source: Borsboom, D., Mellenbergh, G. J., & Van Heerden, J. (2003), The theoretical status of latent variables, Psychological review, 110(2), 203.

Measuring ‘dominance’ by using latent variable

Latent variable

Manifest variables X1: “I would like a job where I

have power over others” X2: “I would make a good

military leader” X3: “I try to control others”

Dominance

Xi

Source: Borsboom, D., Mellenbergh, G. J., & Van Heerden, J. (2003), The theoretical status of latent variables, Psychological review, 110(2), 203.

When do you have a latent variable?

A latent variable is defined as a random variable whose realizations cannot be observed directly

Remind an example of “ROA”Assess of true measure against

measurement error (e.g. age)

Source: Borsboom, D. (2008), Latent variable theory, Measurement 6, 25-53, Howell, R. D. (2014), course materials from MKT 6355 Theory Testing

SEM case in point: Student evaluationInfer from data structure to variable

structureHow to conceptualize latent variables?What are their causal relationships?

Source: Borsboom, D. (2008), Latent variable theory, Measurement 6, 25-53, Howell, R. D. (2014), course martials from MKT 6355 Theory Testing

How to conceptualize latent variables?

Perceived instructor competence (R1, R3, R7, R8, R9, R10)

Perceived instructor interaction (R6, R4, R5)

Perceived course quality (R11, R12, R13, R14, R15, R16)

R2 is removed

Factor analysis and SEMEFA - Find a latent variable which affects observed

variables - Without prior assumption, all loadings are free to vary

CFA - Some loadings are forced to be zero by the

researcher - Factors are allowed to correlated - No direct arrows between factors (Measured model)

SEM - Test and estimate the (causal) relationships

Where is latent variable?

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 F1

(Competence)F2

(Interaction)F3

(Course)Student 1 Student 2

… Student n

Comp.

Inter.Cours

eStudent n

R1 R3 R7 R8 R9 R10 R6 R4 R5 R11 R12 R13 R14 R15 R16

e1 e6 e7 e8 e9 e10 e3 e4 e5 e11 e12 e13 e14 e15 e16

What are their causal relationships?

Criteria for classifying an explanation as causal - Temporal sequentiality, nonspurious correlation, and common sense logic

# of people of drowning and ice cream consumption

Source: Hunt, S. D. (2010), Foundations of marketing theory: Toward a general theory of marketing, ME Sharpe

Applying criteria for choosing a model• Latent variables: Perceived course quality,

perceived instructor competence, and perceived instructor interaction

• Discussion: What are our DV(s) and IV(s)?

A model that does not make sense

A student forms an opinion about interaction, which influences his/her opinion about competence, which in turn influences his/her opinion about course quality.

Remember criteria of causality Course

Inter.Comp

.

A model that makes more sense

A student forms his/her opinion on interaction and competence simultaneously, which influences perceived course quality

Opinions on interaction and competence are correlated because they come from the same student

How the instructor offers and what the instructor offers influence perceived quality of course

Course

Inter.Comp

.

Source: Grönroos, C. (1984), A service quality model and its marketing implications, European Journal of marketing, 18(4), 36-44.

Part 2 (Presented by Kiran Pedada)

SEM Structural ModelSEM model for the

case:

Z = BzU + ez

Here: Z is the endogenous

latent variable, U is a (2x1) matrix of

exogenous latent variables

Bz is a (1x2) matrix of coefficients of exogenous variables,

ez is the error associated with the endogenous variable.

Source: “Factor Analysis, Path Analysis, and Structural Equations Modeling”, Book extract, Jones and Bartlett publishers. http://www.jblearning.com/samples/0763755486/55485_CH14_Walker.pdf Note: The equation is taken from the above mentioned source. However, the symbols are changed for ease and convenience.

Perceived

Course Quality

Perceived

Interaction

Perceived Compete

nce

Model

Exogenous Measurement Model Exogenous measurement model:

X = BxU + ex Here: X is a (9 x 1) matrix of exogenous indicators, Bx is a (9 x 2) matrix of coefficients from the exogenous

variables to exogenous indicators, U is a (2 x 1) matrix of exogenous latent variables, ex is a (9 x 1) matrix for error associated with the

exogenous indicators.


Exogenous Measurement Model

X = BxU + ex

Endogenous Measurement Model Endogenous measurement model:

Y = ByZ + ey Here: Y is a (6x1) matrix of endogenous indicators, By is a (6x1) matrix of coefficients from the endogenous

variable to endogenous indicators, Z is a (1x1) matrix of endogenous latent variable,ey is a (6x1) matrix for error associated with the

endogenous indicators.


Y = ByZ + ey

Endogenous Measurement Model

SEM and Analysis of Covariance

SEM is based on the analysis of covariances

Analysis of covariances allows for estimation of both standardized and unstandardized parameters

Source: www.structuralequations.com/resources/SEM+Essentials.pps

Example of Analysis of Covariance Structure

Source: www.structuralequations.com/resources/SEM+Essentials.pps

Compare

S denotes the observed covariances (typically the unstandardized covariances)

∑ denotes the model-implied covariances

R Packages for SEM – Non-commerical

Source: Rosseel, Yves. "lavaan: An R package for structural equation modeling."Journal of Statistical Software 48.2 (2012): 1-36.Source 2: https://personality-project.org/revelle/syllabi/454/wk6.lavaan.pdf

Why lavaan?A free, open-source for latent variable

modelingEasy and intuitive to useResults are typically very close, to the

results of MplusPowerful, easy-to-use text-based syntax

describing the modelFairly complete

Source: Rosseel, Yves. "lavaan: An R package for structural equation modeling."Journal of Statistical Software 48.2 (2012): 1-36.

Data

#Data

Data = read.csv(file.choose(), header=T)

attach(Data)

#Responses 1 to 16

evals=as.matrix(cbind(RESP_1,RESP_2,RESP_3,RESP_4,RESP_5,RESP_6,RESP_7,RESP_8,RESP_9,RESP_10,RESP_11,RESP_12,RESP_13,RESP_14,RESP_15,RESP_16))

Formulae and Operators

Formula typeFormula type OperatorOperator MnemonicMnemonic

Latent variable =~ is manifested by

Regression ~ is regressed on

Covariance ~~ is correlated with

Defined parameter

: = is defined as

Equality constraint

== is equal to

Inequality constraint

< is smaller than

Inequality constraint

> is larger than

Source: Rosseel, Yves. "lavaan: An R package for structural equation modeling."Journal of Statistical Software 48.2 (2012): 1-36.

Specifying the Modelmodel <- '# Defining the Latent VariablesCompetence =~ RESP_1 + RESP_3 + RESP_7 +

RESP_8 + RESP_9 + RESP_10Course =~ RESP_11 + RESP_12 + RESP_13 +

RESP_14 + RESP_15 + RESP_16Interaction =~ RESP_6 + RESP_4 + RESP_5

#RegressionCourse ~ Interaction + Competence

#covariance of latent variablesInteraction ~~ Competence '

Install Packages

Install.packages(“lavaan”)

Install.packages(“semplot”)

Running the Model

require("lavaan")

#Fitting the data

fit <- sem(model, data = evals, missing = "FIML")

Dealing with Missing Values in Lavaan“listwise” - cases with missing data

removed listwise (before analysis)

“fiml” - the package offers estimation using all available data.This is also called “case-wise” maximum likelihood estimation.

Source: http://cran.r-project.org/web/packages/lavaan/lavaan.pdf

Examining the Results#Examining the resultssummary(fit, fit.measure=TRUE,

standardized = TRUE)

Examining the Results Used Total Number of observations 7828 7830

Number of missing patterns 92

Estimator ML Minimum Function Test Statistic 6068.046 Degrees of freedom 87

P-value (Chi-square) 0.000

Parameter estimates:

Information Observed Standard Errors Standard

Examining the Results Estimate Std.err Z-value P(>|z|)

Std.lv Std.allLatent variables: Competence =~ RESP_1 1.000 0.778 0.902 RESP_3 1.038 0.009 121.814 0.000 0.807 0.889 RESP_7 1.072 0.009 114.296 0.000 0.834 0.867 RESP_8 0.957 0.008 114.973 0.000 0.745 0.871 RESP_9 1.026 0.009 110.423 0.000 0.798 0.855 RESP_10 0.695 0.007 94.256 0.000 0.541 0.792 Course =~ RESP_11 1.000 0.853 0.869 RESP_12 0.971 0.009 110.946 0.000 0.829 0.891 RESP_13 0.947 0.009 107.388 0.000 0.808 0.879 RESP_14 0.766 0.008 90.252 0.000 0.654 0.805 RESP_15 0.829 0.009 90.857 0.000 0.707 0.808 RESP_16 0.890 0.010 88.775 0.000 0.760 0.795 Interaction =~ RESP_6 1.000 0.612 0.822 RESP_4 1.151 0.012 97.686 0.000 0.704 0.910 RESP_5 1.196 0.012 100.429 0.000 0.731 0.922

Regressions: Course ~ Interaction 0.075 0.019 4.059 0.000 0.054 0.054 Competence 0.929 0.016 56.843 0.000 0.847 0.847

Covariances: Competence ~~ Interaction 0.394 0.008 48.130 0.000 0.828 0.828

Plotting the SEM Path Diagram#SEM path diagram

Require(“semplot”)

# Plot input path diagram

semPaths(fit,title=FALSE, curvePivot = TRUE, exoVar = FALSE, exoCov = FALSE)

# Plot output path diagram with standardized parameters

semPaths(fit, "std", edge.label.cex = 1.0, curvePivot = TRUE)

Input Path Diagram

Output Path Diagram

Relating to the Results

Latent variables: Competence =~ RESP_1 1.000 0.778 0.902 RESP_3 1.038 0.009 121.814 0.000 0.807 0.889 RESP_7 1.072 0.009 114.296 0.000 0.834 0.867 RESP_8 0.957 0.008 114.973 0.000 0.745 0.871 RESP_9 1.026 0.009 110.423 0.000 0.798 0.855 RESP_10 0.695 0.007 94.256 0.000 0.541 0.792 Course =~ RESP_11 1.000 0.853 0.869 RESP_12 0.971 0.009 110.946 0.000 0.829 0.891 RESP_13 0.947 0.009 107.388 0.000 0.808 0.879 RESP_14 0.766 0.008 90.252 0.000 0.654 0.805 RESP_15 0.829 0.009 90.857 0.000 0.707 0.808 RESP_16 0.890 0.010 88.775 0.000 0.760 0.795 Interaction =~ RESP_6 1.000 0.612 0.822 RESP_4 1.151 0.012 97.686 0.000 0.704 0.910 RESP_5 1.196 0.012 100.429 0.000 0.731 0.922

Estimate Std.err Z-value P(>|z|) Std.lv Std.all


Intercepts: RESP_1 4.380 0.010 448.881 0.000 4.380 5.077 RESP_3 4.366 0.010 425.167 0.000 4.366 4.810 RESP_7 4.306 0.011 395.835 0.000 4.306 4.479 RESP_8 4.435 0.010 458.797 0.000 4.435 5.191 RESP_9 4.361 0.011 413.101 0.000 4.361 4.674 RESP_10 4.637 0.008 600.331 0.000 4.637 6.792 RESP_11 4.295 0.011 386.301 0.000 4.295 4.372 RESP_12 4.301 0.011 408.596 0.000 4.301 4.624 RESP_13 4.313 0.010 414.576 0.000 4.313 4.694 RESP_14 4.472 0.009 486.091 0.000 4.472 5.506 RESP_15 4.408 0.010 444.632 0.000 4.408 5.036 RESP_16 4.345 0.011 401.296 0.000 4.345 4.546 RESP_6 4.578 0.008 543.730 0.000 4.578 6.155 RESP_4 4.548 0.009 519.595 0.000 4.548 5.879 RESP_5 4.558 0.009 507.973 0.000 4.558 5.747 Competence 0.000 0.000 0.000 Course 0.000 0.000 0.000 Interaction 0.000 0.000 0.000



Variances: RESP_1 0.139 0.003 0.139 0.187 RESP_3 0.172 0.003 0.172 0.209 RESP_7 0.230 0.004 0.230 0.248 RESP_8 0.176 0.003 0.176 0.241 RESP_9 0.234 0.004 0.234 0.269 RESP_10 0.174 0.003 0.174 0.373 RESP_11 0.237 0.005 0.237 0.245 RESP_12 0.178 0.004 0.178 0.205 RESP_13 0.191 0.004 0.191 0.227 RESP_14 0.232 0.004 0.232 0.352 RESP_15 0.266 0.005 0.266 0.348 RESP_16 0.336 0.006 0.336 0.368 RESP_6 0.179 0.003 0.179 0.324 RESP_4 0.103 0.003 0.103 0.172 RESP_5 0.094 0.003 0.094 0.150 Competence 0.605 0.012 1.000 1.000 Course 0.148 0.004 0.204 0.204 Interaction 0.374 0.009 1.000 1.000


ReferencesBorsboom, D., Mellenbergh, G. J., & Van Heerden, J. (2003),

The theoretical status of latent variables, Psychological review, 110(2), 203.

Borsboom, D. (2008), Latent variable theory, Measurement 6, 25-53.

Grönroos, C. (1984), A service quality model and its marketing implications, European Journal of marketing, 18(4), 36-44.

Howell, R. D. (2014), course materials from MKT 6355 Theory Testing.

Hunt, S. D. (2010), Foundations of marketing theory: Toward a general theory of marketing, ME Sharpe.

Rosseel, Yves. "lavaan: An R package for structural equation modeling."Journal of Statistical Software 48.2 (2012): 1-36

Thank You

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Jiyoon An Kiran Pedada Structural Equation Modeling