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Confirmatory Factor Analysis and

Structural Equations Modeling:

An Introduction

Audhesh Paswan, Ph.D.Associate Professor

Dept. of Marketing and Logistics, COB

University of North Texas, USApaswana@unt.edu

May 2009

mailto:paswana@unt.edumailto:paswana@unt.edu

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What is Structural Equations Modeling (SEM)?

Two components:

1. Measurement model (CFA) = a visual representation that

specifies the models constructs, indicator variables, and

interrelationships. CFA provides quantitative measures of the

reliability and validity of the constructs.

2. Structural model (SEM) = a set of dependence relationshipslinking the hypothesized models constructs. SEM determines

whether relationships exist between the constructs and

along with CFA enables you to accept or reject your theory.

In developing models to test using CFA/SEM, theory, prior

experience, and the research objectives is used to identify and

develop hypotheses about which independent variables predict each

dependent variable.

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o In EFA (Exploratory Factor Analysis) we use the

data to determine the underlying structure. Also,

typically use an orthogonal rotation and cross-

loadings are permitted, as long as they are

relatively small.

o In CFA (Confirmatory Factor Analysis) we specify

the factor structure on the basis of a good theory

and then use CFA to determine whether there is

empirical support for the proposed theoretical

factor structure. Also, assumes oblique rotation

and no (zero) cross-loadings.

What is the difference between EFA and CFA?

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1. Do the indicator variables measure the same concept?

o Convergent validity measured by shared variance

o No (zero) cross-loadings unidimensional

o Uncorrelated errors2. Are the constructs measuring distinctly different concepts?

o Discriminant validity average shared variance (AVE)

must be larger than interconstruct correlations

3. Are the constructs reliable? Measured based on internalconsistency (similar to Cronbach Alpha).

4. Are the interconstruct correlations consistent with your

theory?

o Nomological validity

CFA provides quantitative measures that assess thevalidity and reliability of theoretical model . . .

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Cross-Loadings = when indicator variables on one construct areassumed to be related to another construct.

Congeneric measurement model = all cross-loadings are assumed tobe 0.

The assumption of no cross-loadings is based on the fact that theexistence of significant cross-loadings is evidence of a lack of

unidimensionality and therefore a lack of construct validity, i.e.

discriminant validity.

Construct

X1 X2 X3 X4

CFA Assumes No Cross-Loadings

and Unidimensionality

Construct

X1 X2 X3 X4

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Basic Elements ofBasic Elements of CFACFA--SEMSEM

ConstructsConstructs

oo ExogenousExogenous = variable or construct that acts as a predictor for= variable or construct that acts as a predictor for

other constructs or variables in the modelother constructs or variables in the model only have arrowsonly have arrows

leading out of them and none leading into themleading out of them and none leading into them..

oo EndogenousEndogenous = variable or construct that is the outcome= variable or construct that is the outcome

variable in at least one causal relationshipvariable in at least one causal relationship has one or morehas one or more

arrows leading into themarrows leading into them.

RelationshipsRelationships

oo Recursive = arrow goes one way.Recursive = arrow goes one way.

oo NonrecursiveNonrecursive = arrows go both ways.= arrows go both ways.

oo CorrelationalCorrelational = arrow is curved with points on both ends.= arrow is curved with points on both ends.

IndicatorsIndicatorsoo Formative = arrows go from observed indicator variables toFormative = arrows go from observed indicator variables to

unobserved construct.unobserved construct.

oo Reflective = arrows go from unobserved construct to observedReflective = arrows go from unobserved construct to observedindicator variables.indicator variables.

C

C C = construct

C

V V = Indicator variable

C V

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Basic Elements of CFA-SEM Models

Exogenous constructs: latent, multi-item equivalent of

independent variables that are not influenced by other variablesin the model. They use a variate (linear combination) of

measures to represent the construct, which acts as an

independent variable in the model.

Endogenous constructs: latent, multi-item equivalent to

dependent variables they are affected by other variables in thetheoretical model.

Unobserved variable: a hypothesized, latent construct (concept)

that can only be approximated by observable or measurable

indicator variables.

Observed variable: known as manifest or indicator variables, this

type of data is collected from respondents through various data

collection methods such as surveys, interviews or observations.

These are measurable variables that are used to represent the

latent constructs.

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ExogenousConstruct

X1 X2 X3 X4

EndogenousConstruct

Y1 Y2 Y3 Y4

Loadings represent the relationships from constructs to variables asin factor analysis.

Path estimates represent the relationships between constructs,

similar to beta weights in regression analysis.

Two Latent Constructs and the Measured

Variables that Represent Them

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Definitions

Communality = the total amount of variance a measured variable has in common with theconstruct upon which it loads. Good measurement practice suggests that each measuredvariable should load on only one construct. So it can be thought of as the variance explained ina measured variable by the construct. In CFA, the communality is referred to as the squared

multiple correlation for a measured variable. It is similar to the idea of communality from EFA.Factor loadings are squared to get the communality of an indicator variable.

Congeneric measurement model = a model consisting of several unidimensional constructswith all cross-loadings assumed to be zero. Also, there is no covariance for between- orwithin-construct error variances, meaning they are all fixed at zero.

Estimated covariance matrix = a covariance matrix comprised of the predicted covariancesbetween all indicator variables involved in a SEM based on the equations that represent the

hypothesized model. Typically abbreviated with k.

Fixed parameter = a parameter that has a value specified by the researcher. Most often thevalue is specified as zero, indicating no relationship, although there are instances in which anactual value (e.g., 1.0 or such) can be specified.

Free parameter = a parameter estimated by the structural equation program to represent thestrength of a specified relationship. These parameters may occur in the measurement model

(most often denoting loadings of indicators to constructs) as well as the structural model(relationships among constructs).

Goodness-of-fit (GOF) = a measure indicating how well a specified model reproduces thecovariance matrix among the indicator variables.

Maximum likelihood estimation (MLE) = an estimation method commonly employed instructural equation models. An alternative to ordinary least squares used in multipleregression, MLE is a procedure that improves parameter estimates in a way that minimizes the

differences between the observed and estimated covariance matrices.

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Definitions continued . . .

Observed sample covariance matrix = the typical input matrix for SEM estimation comprisedof the observed variances and covariances for each measured variable. Typicallyabbreviated with a bold, capital letter S (S).

Construct reliability (CR) = a measure of reliability and internal consistency based on thesquare of the total of factor loadings for a construct.

Construct validity = is the extent to which a set of measured variables actually represent thetheoretical latent construct they are designed to measure. It is made up of four components:convergent validity, discriminant validity, nomological validity and face validity.

Convergent validity = the extent to which indicators of a specific construct converge orshare a high proportion of variance in common.

Discriminant validity = the extent to which a construct is truly distinct from other constructs.

Face validity = the extent to which the content of the items is consistent with the constructdefinition, based solely on the researchers judgment.

Nomological validity = is tested by examining whether or not the correlations between theconstructs in the measurement theory make sense. The covariance matrix Phi () ofconstruct correlations is useful in this assessment.

Parameter = a numerical representation of some characteristic of a population. In CFA/SEM,relationships are the characteristic of interest that the modeling procedures will generateestimates for. Parameters are numerical characteristics of the SEM relationships,comparable to regression coefficients in multiple regression.

Average Variance extracted (AVE) = a summary measure of convergence among a set ofitems representing a construct. It is the average percent of variation explained among theitems.

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Theoretically-Based SEM Model

JS

OC

SI

EP

AC

Hypotheses:H1: EP + JSH2: EP + OCH3: AC +JS

H4: AC +OCH5: JS + OCH6: JS + SIH7: OC +SI

EndogeneousVariable

ExogeneousVariable

EndogeneousVariable

Note: all causalrelationshipsare recursive.

Note: observable indicator variables are not shown to simplify the model.

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Measurement Theories and CFA-SEM

Reflective Measurement Theory: assumes the latentconstructs cause the measured indicator variables and

that the error is a result of the inability of the latent

constructs to fully explain the indicators. Thus, arrows

are drawn from the latent constructs to the measured

indicators.

Formative Measurement Theory: assumes the

measured indicator variables cause the construct and

that the error is a result of the inability of the measuredindicators to fully explain the construct. Therefore, the

arrows are drawn from the measured indicators to the

constructs. In short, formative constructs are not

considered latent.

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Example: Formative vs. Reflective Constructs

Construct: Stress

Reflective measures: blood pressure, perspiration,nervousness, figidty, etc. These are caused by stress,or a reflection of it.

Formative measures: difficult boss, troubled homekids, spouse, poor work evaluations, debt, medicalcondition (cancer, heart problems, job changes,moving, etc). These actually cause stress instead of

stress causing them.

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Example: Formative vs. Reflective Constructs

Construct: Intoxicated/Drunk

Reflective measures = unable to walk in straight lineor stumbling, slurred speech, talking loud, laughing,

etc.

Formative measures = alcohol/drugs combined withlack of sleep, how much you have eaten, how fastand how much you drink, etc.

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So Your Model Doesnt Run Diagnosing Problems

Identification one parameter can be estimated for each unique

variance and covariance between measured items. Each time a

parameter is estimated you lose one degree of freedom. An

unidentified model is one with more parameters to be estimated than

there are item variances and covariances. The software will tell you if

this is a problem. Solution = constructs with 3+ indicators.

Heywood case the CFA solution produces an error variance lessthan 0 a negative error variance typically because of small sample

size or less than 3 indicators per construct. Software will tell you.

Solution = convert negative error variance to positive e.g., .005, or

you may just decide to delete the offending variable.

Software AMOS sometimes fails to set the scale on paths. Ifmodel does not run check this. Also, in drawing the model

sometimes constructs, paths, etc. are drawn on the screen and you

cannot see them. If model does not run, and you cannot find

problem, start over.

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Assessing Measurement Model Validity

Two Broad Approaches . . .

1. Examine the Goodness of Fit (GOF) indices.

2. Evaluate the construct validity and reliability

of the specified measurement model.

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Types of Fit Measures

Three Types:

1. Absolute Fit Measures = indicate how well your

estimated model reproduces the observed data.

2. Incremental Fit Measures = indicate how well your

estimated model fits relative to some alternative

baseline model. The most common baseline model isone that assumes all observed variables are

uncorrelated, which means you have all single item

scales.

3. Parsimony Fit Measures = indicate if the model youspecify is parsimonious. That is, whether your model

can be improved by specifying fewer estimated

parameter paths (specifying a simpler model).

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SEM GOF Rules of Thumb

SEM has no single statistical test that best describes the strength of

the models predictions. Instead, researchers have developeddifferent types of measures that in combination assess the results.

Multiple fit indices should be used to assess goodness of fit.For example:

o The 2and the 2/ df (normed Chi-square)

o One goodness of fit index (e.g., GFI, CFI, NFI, TLI)o One badness of fit index (e.g., RMSEA, RMSR)

Selecting a rigid cut-off for the fit indices is like selecting a minimumR2 for a regression equation there is no single magic value for

the fit indices that separates good from poor models. The quality of

fit depends heavily on model characteristics including sample sizeand model complexity.

Simple models with small samples should be held to very strict fitstandards.

More complex models with larger samples should not be held to the

same strict standards.

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What does SEM actually test?

Can your hypothesized theoretical model beconfirmed?

Three Criteria:1. Goodness of Fit?

Does the estimated covariance matrix

= observed covariance matrix (absolute fit)

2. Validity and Reliability of Measurement Model?

3. Significant and Meaningful StructuralRelationships?

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Criteria One: Goodness of Fit (GOF)

Indicates how well the specified model reproducesthe covariance matrix among the indicator variables that is, it examines the similarity of the observed

and estimated covariance matrices (absolute fit).

The initial measure of GOF is the Chi-square statistic.The null hypothesis is No difference in the twocovariance matrices. Since you do not want the

matrices to be different, you hope for aninsignificant Chi-square (>.05) so you can accept thenull hypothesis.

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Observed sample covariances for 3-Construct model

Covariances calculated for the

sample request Sample

moments and look in Outputunder that subheading.

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Covariances estimated by AMOS

software request Implied moments

and look in Output under Estimates.

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Residuals = difference between

observed and estimated covariances

request Residual moments.

A negative sign indicates theobserved covariance (2.137)

is smaller than the estimated

covariance (2.229) by -.093.

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Standardized Residuals you look for

patterns of larger residuals, generally => 4.0

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Three Construct Results Model Fit Diagnostics

CMIN/DF a value below 2 is preferred but

between 2 and 5 is considered acceptable.

The AGFI is .946 above

the .90 minimum.

The CFI is 0.984 it exceeds the

minimum (>0.90) for a model of this

complexity and sample size.

The GFI is .965 above the .90

recommended minimum.

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This is the

Model Fitportion of theoutput.

GFI = Goodness ofFit Index

AGFI = AdjustedGoodness of Fit Index

PGFI = Parsimonious

Goodness of Fit Index

TLI = Tucker- Lewis

CFI = ComparativeFit Index

PNFI = ParsimoniousNormed Fit Index

NFI = NormedFit Index

Chi-square (X2) =likelihood ratio chi-square

CMIN/DF a value below 2 is preferred butbetween 2 and 5 is considered acceptable.

Note: If you click on any of the Fit Indices it will give guidelines forinterpretation and references supporting the guidelines.

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RMSEA = Root Mean SquaredError of Approximation a valueof 0.10 or less is consideredacceptable (7e, p. 649).

Three Types of Models:

1. Default = your model, therelationships you propose andare testing.

2. Saturated model = a model

that hypothesizes thateverything is related toeverything (just-identified).

3. Independence model =hypothesizes that nothing isrelated to anything.

RMSEA represents thedegree to which lack of fit isdue to misspecification ofthe model tested versusbeing due to sampling error.

Note that when we

evaluate the measures

we use the numbers forthe default model.

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The GFI, an absolute fit index, is .965.This value is above the .90 guidelinefor this model . Higher valuesindicate better fit (7e, p. 649).

The AGFI, a parsimony fit index, is.946. This value is above the .90guideline for this model . Attemptsto adjust for model complexity, butpenalizes more complex models.

The CFI, an incremental fit index, is0.984, which exceeds the guidelines(>0.90) for a model of this complexityand sample size (7e, p. 650).

HBAT Three Construct Results

CFI represents the improvement of fit of thespecified model over a baseline model in which allvariables are constrained to be uncorrelated.

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Other Indices

The NFI, RFI and IFI areother incremental fitindices. Our guidelinesindicate the NFI should be>0.90 for a model of this

complexity and samplesize. For the RFI and IFIwe indicate that largervalues (0 1.0) are better.

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The RMSEA, an absolutefit index, is 0.043. This valueis quite low and well below

the .08 guideline for a modelwith 12 measured variablesand a sample size of 400.This also is called a Badness-Of-Fit index.

The 90 percent confidenceinterval for the RMSEA isbetween a LO of .028 and a HIof 0.058. Thus, even theupper bound is not close to.08.

Using the RMSEA and the CFI satisfies our ruleof thumb that both a badness-of-fit index and agoodness-of-fit index be evaluated. In addition,other index values also are supportive. Forexample, the GFI is 0.95, and the AGFI is 0.93.

We therefore now move on to examine theconstruct validity of the model.

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Guidelines for Establishing Acceptable Fit

Use multiple indices of differing types. Adjust the index cutoff values based on model

characteristics, e.g., number of constructs and

indicators, sample size. Simpler models and

smaller samples sizes require stricter evaluation.

Remove indicator variables that do not meetestablished criteria.

Use GOF indices to compare models.

The pursuit of better fit at the expense of testinga true model is not a good trade-off.

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What does SEM actually test?

Three Criteria:

1. Goodness of Fit?estimated covariance matrix = observed covariance matrix

2. Validity and Reliability of Measurement Model?

3. Significant and Meaningful Structural Relationships?

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2. Assessing the Measurement Model

Construct Validity

o Faceo Convergent

o Discriminant

o Nomological

Construct Reliability3. Assessing the Structural Model

Significant and Meaningful Structural Relationships

Second & Third Criteria

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CFA and Construct ValidityOne of the biggest advantages of CFA/SEM

is its ability to quantitatively assess the

construct validity of a proposed measurement

theory.

Construct validity . . . is the extent to which

a set of measured items actually reflect the

theoretical latent construct they are designed to

measure.

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Validity

Before running the CFA model, assessments of

validity are based on:

Face validity. Published results from previous studies. Pre-test or pilot study findings.

A major objective of applying CFA is toempirically estimate validity using more rigorous

approaches; e.g., construct validity.

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Construct validity is made up of four components:

Face validity = the extent to which the content of the items isconsistent with the construct definition, based solely on the

researchers judgment.Convergent validity = the extent to which indicators of a specificconstruct converge or share a high proportion of variance incommon. To assess we examine construct loadings and averagevariance extracted (AVE).

Discriminant validity = the extent to which a construct is trulydistinct from other constructs (i.e., unidimensional).

Nomological validity = examines whether the correlationsbetween the constructs in the measurement theory make sense.

We also look at the reliability of the constructs.

Reliability = a measure of the internal consistency of theobserved indicator variables.

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Convergent Validity

Convergent validity there are three measures:1. Factor loadings2. Average Variance extracted (AVE)3. Reliability

Rules of Thumb: Convergent Validity

Standardized loadings estimates should be .5 or higher, andideally .7 or higher.

AVE should be .5 or greater to suggest adequate convergentvalidity.

AVE estimates also should be greater than the square of thecorrelation between that factor and other factors to provide

evidence of discriminant validity.

Reliability should be .7 or higher to indicate adequateconvergence or internal consistency.

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Graphical Display of 5 Construct CFA Model

Attitudestoward

Coworkers

JS4

JS3

JS5

JS2

JS1

OC1 OC2 OC3 OC4

AC3AC2 AC4AC1

SI2

SI3

SI1

SI4

EP2EP1 EP3

Note: Measured variables are shown as a box with labels corresponding to those shown in the HBATquestionnaire. Latent constructs are an oval. Each measured variable has an error term, but the error termsare not shown. Two headed connections indicate covariance between constructs. One headed connectorsindicate a causal path from a construct to an indicator (measured) variable. In CFA all connectors between

constructs are two-headed covariances / correlations.

EP4

OrganizationalCommitment

Staying

Intentions

JobSatisfaction

EnvironmentalPerceptions

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This is the Estimatesportion of the output.

These are unstandardized

regression weights.

The asterisks indicate statisticalsignificance

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Factor Loadings Convergent Validity . . .

These are factor loadings but inAMOS they are called standardized

regression weights.

Factor loadings are the first thing tolook at in examining convergent validity.Our guidelines are that all loadings

should be at least .5, and preferably .7 orhigher. All loadings are significant asrequired for convergent validity. Thelowest is .592 (OC1) and there are onlytwo below .70 (EP1 & OC3).

When examining convergent validity, we look at two additional measures:

(1) Average Variance Extracted (AVE) by each construct.

(2) Construct Reliabilities (CR).

The AVE and CR are not provided by AMOS software so they have to be calculated.

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HBAT CFA Three Factor Completely StandardizedFactor Loadings, Variance Extracted, and

Reliability Estimates

OC EP AC

Item

Reliabilities deltaOC1 0.59 0.349 0.65

OC2 0.87 0.759 0.24

OC3 0.67 0.448 0.55

OC4 0.84 0.709 2.264 0.29

EP1 0.69 0.477 0.52

EP2 0.81 0.658 0.34

EP3 0.77 0.596 0.40

EP4 0.82 0.679 2.410 0.32

AC1 0.82 0.676 0.32

AC2 0.82 0.674 0.33

AC3 0.84 0.699 0.30

AC4 0.82 0.666 2.714 0.33

Variance

Extracted 56.61% 60.25% 67.86%

Construct

Reliability 0.84 0.86 0.89

The delta is calculated as 1 minus the itemreliability, e.g., the AC4 delta is 1 .666 = .33

The delta is also referred to as the standardizederror variance.

Factor Loadings

This is the sameas the eigenvalue

in exploratoryfactor analysis

2.264/4 = 56.61

Squared Factor Loadings(communalities)

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nVE

n

i

i 12

Formula for Variance Extracted

In the formula above the represents the standardized factor loading and i is the

number of items. So, for n items, AVE is computed as the sum of the squared

standardized factor loadings divided by the number of items, as shown above.

A good rule of thumb is an AVE of .5 or higher indicates adequate

convergent validity. An AVE of less than .5 indicates that on average, there ismore error remaining in the items than there is variance explained by thelatent factor structure you have imposed on the measure.

An AVE estimate should be computed for each latent construct in a

measurement model.

Calculated Variance Extracted (AVE):

OC Construct = .349 + .759 + .448 + .709 = 2.264 / 4 = .5661

EP Construct = .477 + .658 + .596 + .679 = 2.410 / 4 = .6025

AC Construct = .676 + .674 + .699 + .666 = 2.714 / 4 = .6786

The sum of the

squared loadings

This is the squared

loading for OC4

.842

= .709

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n

i

n

i

ii

n

i

i

CR

1 1

2

1

2

)()(

)(

Formula for Construct Reliability

Construct reliability is computed from the sum of factor loadings (i), squared

for each construct and the sum of the error variance terms for a construct (i) usingthe above formula. Note: error variance is also referred to as delta.

The rule of thumb for a construct reliability estimate is that .7 or higher suggestsgood reliability. Reliability between .6 and .7 may be acceptable provided that otherindicators of a models construct validity are good. A high construct reliabilityindicates that internal consistency exists. This means the measures all areconsistently representing something.

CR (OC) = (.59 +.87 +.67 +.84)2 / [(.59 +.87 +.67 +.84)2 + (.65 +.24 +.55 +.29)] = 0.84

CR (EP) = (.69 +.81 +.77 +.82)2 / [(.69 +.82 +.84 +.82)2 + (.52 +.34 +.40 +.32)] = 0.86

CR (AC) = (.82 +.82 +.84 +.82)2 / [(.82 +.82 +.84 +.82)2 + (.32 +.33 +.30 +.33)] = 0.89

The sum of the loadings, squared

Computation of Construct Reliability (CR)

The sum of the errorvariance (delta)

The sum of the loadings, squared

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Evaluation of HBAT Three-Construct Model

Convergent Validity

Taken together, the evidence provides initial support for the

convergent validity of the three construct HBAT measurement

model. Although three loading estimates are below .7, two of these

are just below the .7 and do not appear to be significantly harming

model fit or internal consistency.

The average variance extracted (AVE) estimates all exceed .5

and the construct reliability estimates all exceed .7. In addition, the

model fits relatively well based on the GOF measures. Therefore,

all the indicator items are retained at this point and adequate

evidence of convergent validity is provided.

We now move on to examine:

(1) Discriminant validity

(2) Nomological validity

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Discriminant Validity

Discriminant validity = the extent to which a

construct is truly distinct from other constructs.

Rule of Thumb: all construct average variance

extracted (AVE) estimates should be larger than thecorresponding squared interconstruct correlation

estimates (SIC). If they are, this indicates the

measured variables have more in common with the

construct they are associated with than they do withthe other constructs.

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Correlations between the EP,AC and OC constructs. These arestandardized covariances.

These are used in calculatingdiscriminant validity.

Covariancesbetween the EP,

AC and OCconstructs.

Discriminant Validity

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Discriminant validity compares the averagevariance extracted (AVE) estimates for eachfactor with the squared interconstructcorrelations (SIC) associated with that factor,as shown below:

AVE SIC

OC Construct .5661 .2500, .0918

EP Construct .6025 .0645, .2500

AC Construct .6786 .0645, .0918

All variance extracted (AVE) estimates in the above table are larger than thecorresponding squared interconstruct correlation estimates (SIC). This means theindicators have more in common with the construct they are associated with thanthey do with other constructs. Therefore, the HBAT three construct CFA modeldemonstrates discriminant validity.

In the columns below we calculatethe SIC (Squared InterconstructCorrelations) from the IC (InnerconstructCorrelations) obtained from thecorrelations table on the AMOS printout(see previous slide):

IC SIC

EP AC .254 .0645

EP OC .500 .2500

AC OC .303 .0918

Discriminant Validity

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Nomological Validity

Nomological validity . . . is tested byexamining whether the correlations between the

constructs in the measurement model make sense.

The construct correlations are used to assess this.

To demonstrate nomological validity in the

HBAT model . . . the constructs must be positively

related based on our HBAT theory. For the HBAT

three construct model all correlations are positive

and significant see next slide.

HBAT 3 Construct Nomological Validity

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The interconstruct

correlations are all positiveand significant (see aboveCovariances table).

The asterisksindicate that all

correlations aresignificant.

HBAT 3-Construct Nomological Validity

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These are the R-squaredvalues (Squared StandardizedLoadings).

So subtract these from 1 toget the standardized error termestimate.

Error Variances(Unstandardized)

To get thestandardized errorvariances, subtract thesquared standardizedloadings shown belowfrom 1 for each item.

The Squared Multiple Correlations are also referredto as the squared loadings, i.e., they are calculated bysquaring the standardized regression weights(loadings).

The squared loadings are used in calculating the

average variance extracted (AVE) for each construct.

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Diagnosing Measurement Model Problems

In addition to evaluating goodness-of-fit statistics, the following

diagnostic measures for CFA should be checked:

Path estimates the completely standardized loadings (AMOS =standardized regression weights) that link the individual indicators

to a particular construct. The recommended minimum = .7; but

.5 is acceptable. Variables with insignificant or low loadings

should be considered for deletion.

Standardized residuals the individual differences betweenobserved covariance terms and fitted covariance terms. The

better the fit the smaller the residual these should not exceed

|4.0|. Modification indices the amount the overall Chi-square valuewould be reduced by freeing (estimating) any single particular path

that is not currently estimated. That is, if you add or delete any

path what is the impact on the Chi-square.

Modifying the Measurement Model

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Modifying the Measurement Model

Examining Residuals . . .

The largest residual is-2.0659 (EP3 & OC1) sono residuals exceed ourguideline of |> 4.0|.

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Is the Measurement Model Valid?

No refine measures and design a newstudy.

Yes proceed to test the structural modelwith stages 5 and 6.

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Assessing the Structural Model Validity

To do so . . .

Assess the goodness of fit (GOF) of thestructural model. Should be essentially the

same as with the CFA model. Evaluate the significance, direction, and size

of the structural parameter estimates.

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AMOS Practice: Drawing a Three

Construct HBAT SEM Model

Constructs:

Exogenous Environmental Perceptions (EP)

Attitudes towards Coworkers (AC)

Endogenous Organizational Commitment (OC)

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Chapter 1

Introduction

Copyright 2007Prentice-Hall, Inc.

HBAT Three

Construct SEMModel

Note: this model isdrawn by simplychanging therelationships betweenEP OC and AC OCto straight arrows.

But the model doesnot run!

With th AMOS

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With the AMOSsoftware youmust add anerror term on

your endogenousvariable.

This shows thechange from the

two-headedarrow to a single

headed arrow.

HBAT Three Construct

SEM Model no

estimates.

Squared multiple

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HBAT Three ConstructSEM Model with

standardized estimates.

StandardizedRegression Weights forindicator variables,also called FactorLoadings.

q pcorrelation forendogenous variableOrganizational

Commitment.

Can be interpretedlike the R2 in multipleregression.

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This showsthe new

endogenousvariable.

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These results are

the same as withthe CFA model.

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The unstandardizedregression weights for theindicator variables are thesame as with the CFA model.

Interpretation is shown. Toget this click on the estimate.

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The twohypothesized

paths aresignificant basedon a two-tailed

test.

All loadingsare highlysignificant.

The new weights at

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The new weights atthe top are for the twonew causal paths to thenew endogenousvariable Organizational

Commitment.

The standardizedregression weights forthe indicator variablesare the same as with theCFA model.

Interpretation:When EnvironmentalPerceptions go up by 1standard deviation,OrganizationalCommitment goes up by.452 standarddeviations.

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Estimate of SquaredMultiple Correlation

It is estimated that thepredictors of OrganizationalCommitment (constructs AC

and EP) explain 28.3percent of its variance (i.e.,

71.7% of variance isunexplained).

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These measures arethe same as with the

CFA model.

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Where Do We Go From Here?

More AMOS Practice: Drawing the

5-Construct HBAT SEM Model