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### Doing Quantitative Research26E02900, 6 ECTS Cr.

Olli-Pekka Kauppila

Daria Volchek

Lecture IV - May 21, 2014

Learning objectives – AM session

Get familiar with the basic idea of structural equation modeling

Identify special characteristics of SEM

Develop understanding of the latent variable constructs

Understand the process of SEM

Develop skills to interpret the quality of models and research results obtained with the application of SEM

Researchquestion

Organizational

commitment

Role

ambiguity

Emotional

exhaustion

Job

satisfaction

Intention

to leave

Role

conflict

Performance

Babakus, E., Cravens, D. W., Johnston, M., & Moncrief, W. C. (1999). The Role of Emotional Exhaustion in Sales Force Attitude and Behavior Relationships. Journal of the Academy of Marketing Science, 27(1), 58-70.

Whatare the antecedents and consequences of the emotionalexhaustion of the individualswhodo ”peoplework”?

How many regression analysesdoyouneed to estimatethismodel?

Organizational

commitment

Role

ambiguity

Emotional

exhaustion

Job

satisfaction

Intention

to leave

Role

conflict

Performance

Babakus, E., Cravens, D. W., Johnston, M., & Moncrief, W. C. (1999). The Role of Emotional Exhaustion in Sales Force Attitude and Behavior Relationships. Journal of the Academy of Marketing Science, 27(1), 58-70.

Generalization of regression

SEM is like regression:Y = B0 + B1X1 + B2X2 + e

With assumptions intact, regression is beautiful, but . . .

SEM, a generalization, helps us address some of regression’s limitations

What is SEM?

amultivariatestatisticaltechnique

whichcombines (confirmatory) factoranalysis and multipleregression modeling

- Cansimultaneouslytestmeasures and structuralrelationships

for the purpose of analyzinghypothesizedrelationships

- Testsmodelsthatareconceptuallyderived, a priori
- Testsif the theoryfits the data

amonglatent (i.e. unobservedortheoretical) variablesmeasuredbymanifestvariables (i.e. observedorempiricalindicators)

What is SEM?

SEMencompasses an entirefamily of modelsknownbymanynames, e.g. covariancestructureanalysis, latentvariableanalysis, confirmatoryfactoranalysis, LISRELanalysis

- LInearStructiralRELationships
- TechnicallyLISREL is a computerprogramdevelopedby Karl Jöreskog and Dan Sörbom to docovariancestructureanalysis
- Alsoothersoftwaresareavailable for SEM: AMOS, Mplus…

What is SEM?

SEMtypicallyconsists of twoparts (orsub-models):

The measurementmodel

- specifieshowlatentvariablesdependuponorareindicatedby the observedvariables
- describesthe measurementproperties (reliabilities and validities) of the observedvariables

The structuralequationmodel

- specifiescausalrelationshipsamong the latentvariables
- describes the causaleffects
- assigns the explained and unexplainedvariance

WhySEM?

Organizational

commitment

Managerial

support

Intention

to leave

Job

satisfaction

Environmental

perceptions

The keybenefits of SEMare:

- Estimation of multiple and interrelateddependencerelationships via series of separate, butinterdependent, multiple regression equations

SEMallows to accomodatecontinuous, dummy, and categoricalmeasures.

WhySEM?

The keybenefits of SEMare:

- Estimation of multiple and interrelateddependencerelationships via series of separate, butinterdependent, multiple regression equations
- Ability to modelbothobserved and unobserved (latent) variables and account for measurementerror in the estimationprocess (parameterestimatescloser to populationvalues)

WhySEM?

The keybenefits of SEMare:

- Estimation of multiple and interrelateddependencerelationships via series of separate, butinterdependent, multiple regression equations
- Ability to modelbothobserved and unobserved (latent) variables and account for measurementerror in the estimationprocess (parameterestimatescloser to populationvalues)
- Ability to define a model to explain the entire set of relationshipssimultaneously

What is a LatentVariable?

- A latentvariableis an unobservedconceptthatcanonlybeapproximatedbyorservableormeasurablevariables(i.e. happiness, satisfaction, emotionalexhaustion), oftencalledfactor.
- The observedvariables, whicharegatheredfromrespondentsthroughvarious data collectionmethods, areknown as indicatorsormanifestvariables.

Latentvariables as presumedcause of itemvalues (reflectivemeasure)

Observed

variable

Observed

variable

Latent

variable

Factor

Manifestvariables/

indicators

Observed

variable

The latentvariable is viewed as an underlyingconstructthatgivesrise to somethingthat is observed(i.e. an observedvariable).

Latentvariables as summary of the measurements (formativemeasure)

Observed

variable

Observed

variable

Latent

variable

Factor

Manifestvariables/

indicators

Observed

variable

The latentvariable is viewed as a summary (weights of the relativeimportance) of the observedvariables. Changes in the indicatorscausechange in the latentvariable.

Classroomexercise I

In yourresearchprojectyouareinterested in identifying the antecedents and consequences of firm international performance

Based on yourextensiveliteraturereview, youhavefound out thatbothreflective and formativemeasurescouldbeused for firmperformance

Thus, youdecide to operationalizeboth:

- Comeup with measureitemsthatyouthinkwouldcapturefirm international performance (1) in a reflective and (2) in formative manner

Firm international performance

- Formative:
- Percent of foreignsales in totalsales;
- N of countries a firmhasentered;
- Percent of foreignclients a firmhas.

Reflective:

Generally speaking, we are satisfied with our success in international markets;

We have achieved the turnover objectives we set for internationalization;

We have achieved the market-share objectives we set for internationalization;

Internationalization has had a positive effect on our company’s profitability;

Internationalization has had a positive effect on our company’s image;

Internationalization has had a positive effect on the development of our company’sexpertise;

The investments we have made in internationalization have paid off.

Measurementerror

No matter how concrete we think our variables are, they always contain some error when we try to measure them.

Measurement error is that proportion of the variable which our measure is unable to capture for variousreasons (systematicorrandom).

It is vital to consider the amount of error in our measurement, no matter how confident we are that we have’gotitright’.

However, in all other multivariate techniques we assume there isno error in variables.

Measurementerror

The impact of measurementerror:

βyx = βs* ρx

βyx – observed regression coefficient

βs – truestructuralcoeffients

ρx – reliability of the predictorvariable

Unless the reliability is 100%, the observedcorrelation (and resulting regression coefficient) willalwaysunderstate the ”true” relationship.

SEMcorrection for measurementerror

SEM ”accounts for” or ”corrects for” the amount of measurementerror in the variables (latentconstructs) and estimateswhat the relationshipwouldbeiftherewas no measurementerror.

βs= βyx / ρx

Dueto thiscorrection, SEM regression coefficientsaremoreaccurate (closer to populationvalue) and tend to belargerthancoefficientsobtained with multiple regression analysis.

Incorporatingerrors

Delta – an errortermassociated with an estimated, measured

x-variable

x1

δ1

x2

δ2

Role

ambiguity

x3

δ3

x4

δ4

x5

δ5

Types of relationships in a SEMmodel

δ1

x1

δ1

x1

Role

ambiguity

δ2

x2

Role

ambiguity

δ2

x2

δ3

x3

δ3

x3

Emotional

exhaustion

δ4

x4

δ4

x4

Role

conflict

y3

y2

y1

δ5

x5

Role

conflict

x5

δ5

δ6

x6

δ6

x6

ε1

ε2

ε3

(1) Correlation (2) Dependence

Types of variables in SEMmodel

y6

y5

y4

Exogenousvariables

Endogenousvariables

x1

Organizational

commitment

Role

ambiguity

x2

x3

Intention

to leave

Emotional

exhaustion

Job

satisfaction

x4

y12

y11

y9

y8

y10

y3

y7

y2

y1

Role

conflict

x5

x6

Notexplainedbyanyotherconstruct in the model

Determinedbyconstructswithin the model

Importance of theory

SEMmodelshouldnotbedevelopedwithoutunderlyingtheory!

SEManalysesshouldbedictatedfirst and foremostby a strongtheoreticalbase.

Theoryimpliesconsequences, some of whicharetested vs. data. Refutinganyconsequencesrefutes the theory (i.e. SEM is primarily a confirmatorymethod)

Testingtheory-basedmodels

Modelimplies a pattern in the covariancematrix

Undermultiple regression assumptionsintact, wecancomparemodel-impliedcovariancematrix with empiricalcovariancematrix (i.e. the onebased on the collected data)

If the difference in covariancematrices is nonsignificant, weconfirm the hypothesizedtheoreticalrelationships (χ2 test)

Theory-drivenmodelingstrategy

Confirmatorymodelingstrategy – the researcherspecifies a single modelcomposed of a set of the relationships and usesSEM to assesshowwellitfits the data (iteitherworksoritdoesn’t)

Competingmodelsstrategy – estimatedmodel is compared with alternativemodels (e.g. a test of competingtheories)

- Equivalentmodels– modelshave the samenumber of parameters with differentrelationshipsbetweenthem, and the alternativemodel(s) fits at least as well as the proposedmodel

Modeldevelopmentstrategy– althoughbasicframework is proposed, a purpose of modeling is to improvethisframeworkthroughSEMmodeifications

Causationevidence in SEM

Covariation – SEMcandeterminesignificantcovariationbetween the cause and effectconstructs;

Sequence – causation in the temporalsequancecouldbeprovidedthroughexperimentalorlongitudinalresearch design;

Nonspuriouscovariation – the size and nature of the relationshipbetween the cause and the effectshouldnotbeaffectedbyincludingotherconstructs (variables) in the model (Support=>Job satisfaction and Work environment)

Theoreticalsupport – compellingtheoreticalrationale to support a cause-and-effectrelationship

Input matrix: Covariancematrix

SEMdiffers from other multivariate techniques in that it uses only the variance-covariance or correlation matrix as its input data.

Individual observations can be input into the programs, but they are converted into one of these two types of matrices before estimation.

The focus of SEM is on the pattern of relationships acrossrespondents.

Number of observedindicators

How many indicators should be used per construct?

The minimum number of indicators for a construct is one

− but the use of only a single indicator requires the researcher to provide estimates of reliability

A construct can be represented with two indicators, but three is the preferred minimum number of indicators

− because using only two indicators increases the chances of reaching an infeasible solution

There is no upper limit in terms of the number of indicators.

− In practice, 5-7 indicators should represent most constructs

What is CFA?

Confirmatory factor analysis – tests the extend to which a researcher’sa-priori, theoreticalpattern of factorloadings on prespecifiedconstructsrepresents the actual data (i.e. confirmsorrejectsourpreconceivedtheory)

The factorsareassignedbased on the researcher’spriortheoreticalknowledge (statisticaltechniquedoesnotassignvariables to factorslike in ExploratoryFactor Analysis)

Eachmeasuredvariableloadsonly on onepre-definedfactor

Cross-loadingsarenotassigned

CFA providesinformationabout the validities and reliabilities of the observedindicators

Stages in CFA

1. Developing a theoretically based model

2. Constructing a path diagram of causal relationships

3. Converting the path diagram into a set of measurement model

4. Choosing the input matrix type

5. Assessing the identification

6. Evaluatinggoodness-of-fitcriteria

7. Interpreting and modifying the model (if theoretically justified)

Assumptions of pathdiagram

First, all causal relationships are indicated.

Theory is the basis for inclusion or omission of any relationship.

It is just as important to justify why a causal relationshipdoes not exist between two constructs as it is to justify the existence of another relationship.

Yet it is important to remember that the objective is to model the relationships among constructs with the smallest number of causal paths or correlations among constructs that can be theoretically justified (parsimonious).

Assumptions of pathdiagram

Second, all causal relationships are assumed to be linear.

Nonlinear relationships cannot be directly estimated in structural equation modeling, but modified structural models can approximate nonlinear relationships.

Assumption of linearity of the relationships requires all other assumptions for multivariate analysis to hold true

Example of measurement (CFA) model

Role

ambiguity

Role

conflict

Intention

to leave

Emotional

exhaustion

Job

satisfaction

y9

y8

y6

y5

y7

y3

y4

y2

y1

x3

x6

x2

x5

x1

x4

ε1

ε2

ε3

ε4

ε5

ε6

ε7

ε9

ε8

δ5

δ1

δ2

δ3

δ4

δ6

- x1-x3correlate, butcorrelation is zeroifweidentify a common cause of x1-x3, i.e. ”Roleambiguity”
- Indicatorsareunidimensional, errortermsshouldnotbecorrelated
- First, wetest the measures (wecanfixit)
- Second, wetest the theorybetween the constructs

Translating the picture into SIMPLIS

Role

ambiguity

Role

conflict

Intention

to leave

Emotional

exhaustion

Job

satisfaction

y9

y8

y6

y5

y7

y3

y4

y2

y1

x3

x6

x2

x5

x1

x4

ε1

ε2

ε3

ε4

ε5

ε6

ε7

ε9

ε8

δ5

δ1

δ2

δ3

δ4

δ6

ROLCONF1 = 1*Rolconf

ROLCONF2= Rolconf

ROLCONF3 = Rolconf

EMEXH1 = 1*Emexh

EMEXH2= Emexh

EMEXH3= Emexh

JOBSAT1 = 1*Jobsat

JOBSAT2= Jobsat

JOBSAT3= Jobsat

LEAV1 = 1*Leav

LEAV2= Leav

LEAV3= Leav

e.g. Rolam = W1(ROLAM1) + W2(ROLAM2) + W3(ROLAM3)

ROLAM1 = 1*Rolam

ROLAM2 = Rolam

ROLAM3 = Rolam

How does CFA look like?

Errorterm

Observed

variable

Factorloading

Latent

construct

Correlation

between

latentconstructs

Evaluatinggoodness-of-fitcriteria(assessment of measurementmodel)

In evaluating the measurement part of the model, focus on the relationships between the latent variables and theirindicators (i.e. manifestvariables).

The evaluation of the measurement part of the model should precede the detailed evaluation of the structural partof the model.

The aim is to determine the validity and reliability of the measures used to represent the constructs of interest.

− validity reflects the extent to which an indicator actually measures what it is supposed to measure

− reliability refers to the consistency of measurement (i.e. the extent to which an indicator is free of random error)

Evaluatinggoodness-of-fitcriteria:Loadings

The next step is to examine the estimated loadings and to assess the statistical significance of each one.

- critical t-values:

when α=.01, criticalt-value=2.33

when α=.05, criticalt-value=1.645

when α=.10, criticalt-value=1.282

If statistical significance is not achieved, the researcher may wish to eliminate the indicator or attempt to transform it for better fit with the construct.

Evaluatinggoodness-of-fitcriteria:Loadings

One problem with relying on unstandardizedloadingsand associatedt-values is thatitmaybedifficult to compare the validity of differentindicatorsmeasuring a particularconstruct.

- Indicators of the sameconstructmaybemeasures in verydifferentscales, ifthis is the case, thendirectcomparisons of the magnitudes of the loadingsareclearlyinappropriate
- It is recommendedthat the magnitudes of the standardizedloadingsarealsoinspected (completelystandardizedsolution)

Evaluatinggoodness-of-fitcriteria:R2

Reliability of the indicatorscanbeexaminedbylooking at the squaredmultiplecorrelations (R2) of the indicators

- they show the proportion of variance in an indicatorthat is explainedbyitsunderlyinglatentvariable
- the rest is due to measurementerror
- highmultiplesquaredcorrelationsvaluedenoteshighreliability for the indicator

Evaluatinggoodness-of-fitcriteria: Compositereliability (CR)

In addition to assessing the reliability of the individualindicators, it is possible to calculate a compositereliabilityvalue for eachlatentvariable

- alsoknown as constructreliability
- to dothisuseinformation on the indicatorloadings and errorvariancesfrom the CompletelyStandardizedSolution
- with single itemmeasures, it is notpossible to empiricallyestimate the reliability (couldbefixed at 1.0=noerrororestimatedby the researcher)

Compositereliability (CR)

Recommendedthresholdvalue is .60

The reliabilityfor the latentconstructmustbecomputedseparately for eachmultipleindicatorconstruct in the model.

LISRELdoesnotcomputethemdirectlybutprovidesallnecessaryinformation.

Averagevarianceextracted (AVE)

Anothermeasure of reliability is the varianceextractedmeasure

Reflects the overallamount of variance in the indicatorsaccounted for by the latentconstruct

Highervarianceextractedoccurwhen the indicatorsaretrulyrepresentative of the latentconstruct

The AVEmeasure is a complementarymeasure to the CRvalue

Averagevarianceextracted (AVE)

AVE is quitesimilar to the CRmeasurebutdiffers in that the standardizedloadingsaresquaredbeforesummingthem

Guidelinesuggestthat the varianceextractedvalueshouldexceed .50 for a construct

Classroomexercise II

Calculate the reliabilitymeasures (CR and AVE) for the twolatentvariablesbelow:

Classroomexercise II

Calculate the reliabilitymeasures (CR and AVE) for the twolatentvariablesbelow:

Latentvariable 1:

- CR = 0.770602 => exceeds the common threshold of 0.60
- AVE = 0.403557 => lower than 0.50

Latentvariable 2:

- CR = 0.744912 => exceeds the common threshold of 0.60
- AVE = 0.423463 => lower than 0.50

Evaluatinggoodness-of-fitcriteria

Next step is to assess overall model fit with one or more goodness-of-fitmeasures

Goodness-of-fit measures reflect correspondence of the actual or observed input (covariance or correlation) matrix with that predicted from the proposed model.

There are three types of Goodness-of-fit measures:

- Absolute fit measures
- Incremental fit measures
- Parsimonious fit measures

Evaluatinggoodness-of-fitcriteria

Absoluteindices:

- Chi-square:provides a test of perfectfit in which the nullhypothesis is that the modelfits the population data perfectly (i.e. wewant the chi-squaretest to benon-significant p>.05)
- RMSEA (RootMean Square Error of Approximation)

Focuses on the discrepancybetween a predicted and observedcovariance per degree of freedom

Takes into account the modelcomplexity

Showshowwellwould the model, with unknownbutoptimallychosenparametervalues, fit the populationcovariancematrixifitwasavailable

Ideallybelow 0.05

Evaluatinggoodness-of-fitcriteria

Incrementalindices:

- NormedFit Index (NFI)
- Non-normedFit Index (NNFI)
- ComparativeFit Index (CFI)
- Goodness of Fit Index (GFI)

Parsimonyindices:

- Adjustedgoodness of Fit Index (AGFI)
- Allideallyabove 0.90

Frommeasurement to structuralmodel

ε4

ε5

ε6

y6

y5

y4

ζ2

δ1

x1

Organizational

commitment

Role

ambiguity

δ2

x2

ζ4

ζ1

ζ3

δ3

x3

Intention

to leave

Emotional

exhaustion

Job

satisfaction

δ4

x4

y12

y11

y9

y8

y10

y3

y7

y2

y1

Role

conflict

x5

δ5

δ6

x6

ε7

ε8

ε9

ε11

ε12

ε1

ε2

ε3

ε10

δn – measurementerror in exogenous (independent) variables

εn– measurementerrorin endogenous (dependent) variables

ζn – covariationbetween the endogenousvariableserrors

Testing the hypotheses

y6

y5

y4

x1

Organizational

commitment

-.223*

-.164*

Role

ambiguity

x2

.514*

.495*

x3

Intention

to leave

Emotional

exhaustion

Job

satisfaction

-.322*

-.252 *

x4

y12

y11

y9

y8

y10

y3

y7

y2

y1

Role

conflict

x5

.392*

x6

Fitindices for structuralmodel

Sameindices as for a measurementmodelapply:

- Chi-square, p-value
- RMSEA
- Allotherincrementalindices (NFI, NNFI, CFA, GFI)
- Parsimonyindices (AGFI)

Let’ssumup

SEM…

Providesintegrativefunction

Helpsresearchers to bemoreprecise in theirspecification of hypotheses and operationalization of constructs

Takes into accountreliability of measures in tests of hypotheses in waysgoingbeyond the averaging of multi-itemmeasures of constructs

Guidesconfirmatoryresearch in a manner of combiningself-insight with theory

Oftensuggestsnovelhypothesesoriginallynotconsidered and opensup new avenues for research

Is useful in experimental and surveyresearch, cross-sectional and longitudinalstudies, measurementorhypothesistesting, withinoracrossgroupsand institutionalorculturalcontexts

Is easy to use

Is fun

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