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Joe F. Hair, Jr.

Kennesaw State University

Arthur Money

Henley Business School

For more details, see MultivariateData

Analysis, 7e, 2010.

Confirmatory Factor Analysis

and Structural Equations

Modeling: An Introduction

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What is Structural Equations Modeling (SEM)?

Applying SEM is a two- step process:

1. Confirm Measurement Model (CFA)

2. Evaluate Hypothesized Relationships (SEM)

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Theoretically-Based HBAT

Employee Retention SEM Model

JS

OC

SI

EP

AC

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

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

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

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What is Structural Equations Modeling (SEM)?

Two Steps:

1. Confirm measurement model (CFA) = CFA determines the

reliability and validity of the models constructs and evaluates

the fitbetween observed and estimated covariance matrices.

2. Evaluate structural model (SEM) = SEM determines whether

hypothesized relationships existbetween the constructsandenables you to accept or reject your theory.

In developing models to test using CFA/SEM,

researchers draw upon theory, prior experience, and

research objectives to identify and develop hypotheses

about relationships between multiple independent and

dependent variables.

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What is the Difference Between EFA and CFA?

o In EFA (Exploratory Factor Analysis) the data determines thefactorstructure.

Orthogonal rotation is default; oblique is option

Cross loadings

Statistical objective = extract variance

o In CFA (Confirmatory Factor Analysis) researcher specifies the

factor structure on the basis of a good theory and then uses

CFA to determine whether there is empiricalsupport for the

proposed theoretical factor structure.

Oblique rotation

No cross loadings Statistical objective = reproduce

covariance matrix

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Graphical Display of HBAT 5 Construct CFA Model

Attitudes

toward

Coworkers

JS4

JS3

JS5

JS2

JS1

OC1OC2 OC3

OC4

AC3

AC2

AC4

AC1

SI2

SI3

SI1

SI4

EP2

EP1

EP3

Note: Measured variables are shown as a box with labels corresponding to those shown in the HBAT

questionnaire. Latent constructs are an oval. Each measured variable has an error term, but the error terms

are not shown. Two headed connections indicate covariance between constructs. One headed connectors

indicate a causal path from a construct to an indicator (measured) variable. In CFAall connectors between

constructs are two-headedcovariances / correlations.

EP4

Organizational

Commitment

Staying

Intentions

Job

Satisfaction

Environmental

Perceptions

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What is the Difference Between

CB-SEM and PLS-SEM?

o In CB-SEM the objective is to reproducethe observed

covariance matrix.

Weaknesswhat population does the sample covariance

matrix represent?

o In PLS-SEM the objective is to maximize the explained

varianceof the dependent (endogenous) variables.

Weaknessless well know and therefore less accepted

by reviewers.

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HBAT CFA/SEM Case Study

HBAT employs thousands of workers in different operations around the world. Like

many firms, one of their biggest management problems is attracting and keeping productive

employees. The cost to replace and retrain employees is high. Yet the average new person

hired works for HBAT less than three years. In most jobs, the first year is not productive,meaning the employee is not contributing as much as the costs associated with employing

him/her. After the first year, most employees become productive. HBAT management would

like to understand the factors that contribute to employee retention. A better understanding

can be obtained if the key constructs are measured accurately. Thus, HBAT is interested in

developing and testing a measurement model made up of constructs that impact employees

attitudes and opinions about remaining with HBAT.

HBAT initiated a research project to study the employee retention/turnover problem.Preliminary research discovered that a large number of employees are exploring job options

with the intention of leaving HBAT should an acceptable offer be obtained from another firm.

Based on published literature and some preliminary interviews with employees, an employee

retention/turnover study was designed focusing on five key constructs. The five constructs

are defined as:

Job Satisfaction (JS)reactions / beliefs about onesjob situation.

Organizational Commitment (OC)the extent to which an employee identifies and

feels part of HBAT. Staying Intentions (SI)the extent to which an employee intends to continue

working for HBAT and is not participating in activities that make quitting more likely.

Environmental Perceptions (EP)beliefs an employee has about their day-to-day,

physical working conditions.

Employee Attitudes toward Coworkers (AC)attitudes an employee has toward

the coworkers he/she interacts with on a regular basis.

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Theoretically-Based HBATEmployee Retention SEM Model

JS

OC

SI

EP

AC


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


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HBAT CFA/SEM Constructs and Indicator VariablesOrganizational Commitment

OC1 = My work at HBAT gives me a sense of accomplishment.OC2 = I am willing to put in a great deal of effort beyond that normally expected to help HBAT

be successful.OC3 = I have a sense of loyalty to HBAT.OC4 = I am proud to tell others that I work for HBAT.

Staying IntentionsSI1 = I am not actively searching for another job.SI2 = I seldom look at the job listings on monster.com.SI3 = I have no interest in searching for a job in the next year.SI4 = How likely is it that you will be working at HBAT one year from today?

Attitudes Towards Co-Workers

AC1 = How happy are you with the work of your coworkers?AC2 = How do you feel about your coworkers?AC3 = How often do you do things with your coworkers on your days off?AC4 = Generally, how similar are your coworkers to you?

Environmental PerceptionsEP1 = I am very comfortable with my physical work environment at HBAT.EP2 = The place I work in is designed to help me do my job better.EP3 = There are few obstacles to make me less productive in my workplace.

EP4 = What term best describes your work environment at HBAT?Job Satisfaction

JS1 = All things considered, I feel very satisfied when I think about my job.JS2 = When you think of your job, how satisfied do you feel?JS3 = How satisfied are you with your current job at HBAT?JS4 = How satisfied are you with HBAT as an employer?JS5 = Please indicate your satisfaction with your current job with HBAT by placing a percentage in

the blank, with 0% = not satisfied at all and 100% = highly satisfied.

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Variable Description Variable Type

JS1 I feel satisfied when I think about my job. (0-10, Agree-Disagree) Metric

OC1 My work at HBAT give me a sense of accomplishment (0-10, Agree-Disagree). Metric

OC2 I am willing to put in a great deal of effort . . to help HBAT(0-10, Agree-Disagree).

MetricEP1 I am . . . comfortable with my . . . work environment at HBAT (0-10, Agree-Disagree). Metric

OC3 I have a sense of loyalty to HBAT (0-10, Agree-Disagree). Metric

OC4 I am proud to tell others that I work for HBAT (0-10, Agree-Disagree). Metric

EP2 The place I work in is designed to help me do my job better (0-10, Agree-Disagree). Metric

EP3 There are few obstacles to make me less productive in my workplace (0-10, Ag-Disa). Metric

AC1 How happy are you with the work of your coworkers? (5-pt. Happy-Unhappy) Metric

EP4 What term best describes your work environment? (7-pt. Hectic-Soothing?) Metric

JS2 When you think of your job, how satisfied do you feel? (7-pt) MetricJS3 How satisfied are you with your current job with HBAT? (7-pt) Metric

AC2 How do you feel about your coworkers? (7-pt.Unfavorable-Favorable) Metric

SI1 I am not actively searching for another. (5-pt. Agree/Disagree) Metric

JS4 How satisfied are you with HBAT as an employer? (5-pt. Not vs. Very Much) Metric

SI2 I seldom look at the job listings on Monster.com. (5-pt. Agree-Disagree) Metric

JS5 Please indicate your satisfaction with your current job. (0-100% Satisfied) Metric

AC3 How often do you do things with your coworkers on your days off? (5-pt. Never-Often) Metric

SI3 I have no interest in searching for a job in the next year. (5-pt. Agree-Disagree) Metric

AC4 Generally, how similar are your coworkers to you? (6-pt. Different-Similar) Metric

SI4 How likely is it that you will be working at HBAT one year from today? (5-pt) Metric

X22 Your work typefull time or part time? (0 = Full Time/1 = Part Time) Nonmetric

X23 Your gendermale or female? (0 = Female/1 = Male) Nonmetric

X24 Your geographic locationin USA or outside USA? (0 = Outside/1 = USA) Nonmetric

X25 Your age in years ___? Metric

X26 How long have you worked for HBATyears and months? MetricX27 Performanceas measured by their supervisor. Metric

Description of HBAT CFA-SEM Database Variables

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Basic Elements of CFA-SEM continued . . .

Constructs

o Exogenous = variable or construct that acts as a predictor forother constructs or variables in the model only have arrowsleading out of them and none leading into them.

o Endogenous = variable or construct that is the outcomevariable in at least one causal relationship has one or more

arrows leading into them.

Relationshipso Recursive = arrow goes one way.

o Nonrecursive = arrows go both ways.

o Correlational = arrow is curved with points on both ends.

Indicatorso Formative = arrows go from observed indicator variables to

unobserved construct.

o Reflective = arrows go from unobserved construct to observedindicator variables.

C

C C = construct

C V V = Indicator variable

C V

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Basic Elements of CFA-SEM Models continued

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 variablesthey 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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Exogenous

Construct

X1 X2 X3 X4

Endogenous

Construct

Y1 Y2 Y3 Y4

Two Latent Constructs and the Measured

Variables that Represent Them

Loadings (AMOS = standardized regression weights) represent therelationships from constructs to variables as in factor analysis.

Path estimates represent the relationships between constructs,

similar to beta weights in regression analysis.

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Construct

X1 X2 X3 X4

CFA Assumes No Cross-Loadings

and Unidimensionality

Cross-Loadings = when indicator variables in 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

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Attitudes

toward

Coworkers

JS4

JS3

JS5

JS2

JS1

OC1OC2 OC3

OC4

AC3

AC2

AC4

AC1

SI2

SI3

SI1

SI4

EP2

EP1

EP3




indicate a causal path from a construct to an indicator (measured) variable. In CFA all connectors between

constructs are two-headed covariances / correlations.

EP4

Organizational

Commitment

Staying

Intentions

Job

Satisfaction

Environmental

Perceptions

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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 = boss, homekids, spouse,work evaluations, debt, medical conditioncancer,heart problems, job changes, moving, etc. Theseactually 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 straightline or stumbling, slurred speech, talking loud,laughing, etc.

Formative measures = alcohol/drugs combinedwith lack of sleep, how much you have eaten, howfast and how much you drink, etc.

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Theoretically-Based HBATEmployee Retention SEM Model

JS

OC

SI

EP

AC


H4: AC +

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

Endogeneous

Variable

Exogeneous

Variable

Endogeneous

Variable

Note: all causal

relationships

are recursive.


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Attitudes

toward

Coworkers

JS4

JS3

JS5

JS2

JS1

OC1OC2 OC3

OC4

AC3

AC2

AC4

AC1

SI2

SI3

SI1

SI4

EP2

EP1

EP3




indicate a causal path from a construct to an indicator (measured) variable. In CFA all connectors between

constructs are two-headed covariances / correlations.

EP4

Organizational

Commitment

Staying

Intentions

Job

Satisfaction

Environmental

Perceptions

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AMOS Software:

o Analysis of Moment Structures

o Examples of Moments are:

Means (for population; x for thesample)

Variances (population 2; sample s2) Covariances (population xy; sample

sxy)

o Easy to use graphical interface.

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HBAT Five Construct CFA Model drawn with AMOS software

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HBAT ThreeConstructCFA Model

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Exercise: Three Construct Model

For the HBAT example, using the AMOS 16software, we will do the following:

Draw the diagram for the measurement model

for a three-construct HBAT CFA. Perform a CFA on the HBAT data. Analyze and assess the reliability and validity of

the HBAT measurement model constructs.

Perform SEM after we have confirmed the CFA.

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This is the AMOS 16

screen where you drawyour theoretical model to

do CFA and SEM.

The icons to draw or

modify your theoretical

model are on the left side of

the screen.

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Click on each of the icons to identify

their function. Now draw the three

construct HBAT model.

This is the icon to draw a latent

(unobserved) construct. Click on this

icon and then move it to the blank screenand draw your construct.

This is the icon to draw the observed

indicator variables for the latent

constructs. Click on this icon and then go

to the blank screen and draw your

indicators after you have first drawn thelatent constructs.

This is the icon to draw the correlations between

constructs. Click on this icon and then go to the

blank screen and draw your path.

1

2

3

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How to label constructs?

How to label constructs??

1. Place cursor over a construct.

2. Right-click mouse.

3. Select Object Properties.

4. The dialog box at the left willappear. Type the name of theconstruct in the Variable namespace.

5. Adjust font size if needed.

HBAT Three

Construct CFA

Model

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HBAT Three

Construct

CFA Model

Drawing the Model

1. Select objects first.

2. Click on Plugins3. Then click on Draw

Covariances.

4. Next click Name

Unobserved Variables.

How to draw covariances and name

unobserved variables?

Use this icon to

select objects.

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After your model is drawn, use the File pull down menu andclick on data files to access the HBAT_SEM data.

Click here toselect thedata file.

After selecting

the data fileclick OK.

How to find

your data?

1

2

Icon to select

data files.

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Setting the scale for

latent constructs.

Because they areunobserved, latentconstructs have nometric scale = norange of values. Tosolve this:

1. The value of one ofthe factor loadings isset (fixed) at 1.

2. The variance of

individual indicatorvariables is set to 1.

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HBAT Three

Construct CFA

Modelno data Right click on an objectto give it a name, such as

EnvironmentalPerceptions.

1

Name all the error

terms by clicking on

plugins, and name

unobserved variables.

Click on this icon to

get the variables box

below.

Drag the observed

indicator variables to

the appropriate

boxes in the model.

4

5

3

Draw covariances by

selecting constructs,

clicking on plugins, and

draw covariances.

2

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HBAT Three

Construct CFA

Modelwith data

Click on this

icon to select

desired output.

Click on this

icon to run the

model.

Click on this

icon to see the

output.

1

2

3

How to run

CFA/SEM?

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Maximum likelihood

estimates are the default

option for most SEM

programsincluding AMOS

and LISREL.

How to select the

CFA/SEM output?

The default is only

Minimization history.

You also want to

select Standardized

estimates and

Squared multiple

correlations.

These are the Estimation and Outputboxes where you choose your output options.

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This is the Analysis Summary portion of

the output. Other sections of the outputare shown below. Click on each of themto see that part of the output.

These are the Notes forGroup and Variable

Summary portions of theoutput.

AMOS CFA/SEM output

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HBAT Three Construct

CFA Modelwith

unstandardized

estimates.Click on this icon to see

the calculatedestimatesshown onthe model.

Click hereto display the

standardizedestimates.

Variance

Covariancebetweenconstructs

UnstandardizedRegressionWeights

Variance

HBAT Th C t t

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HBAT Three ConstructCFA Modelwithstandardized

estimates.

StandardizedRegressionWeights, alsocalled FactorLoadings.

Standardized

Regression

Weights, also

called Factor

Loadings.

Click here

to display the

standardized

estimates.

Correlation

betweenconstructs

Squared

Multiple

Correlations,

also called

communality.

Squared

Multiple

Correlations

Squared MultipleCorrelations, alsocalled communality.

D fi iti

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Definitions

Communality = the total amount of variance a measured variable has in common with the

construct upon which it loads. Good measurement practice suggests that each measured

variable should load on only one construct. So it can be thought of as the variance explained in

a 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 constructs

with all cross-loadings assumed to be zero. Also, there is no covariance for between- or

within-construct error variances, meaning they are all fixed at zero.

Estimated covariance matrix = a covariance matrix comprised of the predicted covariances

between 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 the

value is specified as zero, indicating no relationship, although there are instances in which an

actual value (e.g., 1.0 or such) can be specified.

Free parameter = a parameter estimated by the structural equation program to represent the

strength 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 the

covariance matrix among the indicator variables.

Maximum likelihood estimation (MLE) = an estimation method commonly employed in

structural equation models. An alternative to ordinary least squares used in multiple

regression, MLE is a procedure that iteratively improves parameter estimates to minimize a

specified fit function.

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Definitions continued . . .

Observed sample covariance matrix = the typical input matrix for SEM estimation comprised

of the observed variances and covariances for each measured variable. Typically

abbreviated 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 the

theoretical 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 or

share 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 construct

definition, based solely on the researchersjudgment.

Nomological validity = is tested by examining whether or not the correlations between the

constructs in the measurement theory make sense. The covariance matrix Phi () of

construct 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 generate

estimates for. Parameters are numerical characteristics of the SEM relationships,

comparable to regression coefficients in multiple regression.

Variance extracted (AVE) = a summary measure of convergence among a set of items

representing a construct. It is the average percent of variation explained among the items.

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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 0a negative error variancetypically because of small sample

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

Solution = convert negative error variance to positivee.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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Making Sense of the AMOS Output

Analysis Summary Notes for Group Variable Summary Parameter Summary Sample Moments Notes for Model Estimates Minimization History

Model Fit Execution Time

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This is the VariableSummary portion of

the output.

This is the AnalysisSummary portion of

the output.

Notes for Model

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Notes for Model

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

Degrees of freedom (df) = the number of bits of information available to estimate the

sampling distribution of the data after all model parameters have been estimated.

You get this screen by clicking on the probability level.

The 2 goodness of fit statistic indicates

that the observed covariance matrix does

not match the estimated covariance matrix

within sampling variance.

Note that researchers seldom have

CFA/SEM models that are not significantly

different and routinely overlook the Chi-

square and rely on other measures to

assess their models.

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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 the model

you specify reproduces the observed data.

2. Incremental Fit Measures = indicate how well the

model you specify 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 you

specify is parsimonious; i.e., 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 developed

different 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 minimumR2for a regression equationthere 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 size

and 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 thesame 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

2. Validity and Reliability of MeasurementModel?

3. Significant and Meaningful StructuralRelationships?

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Criteria One: Goodness of Fit (GOF)

. . . indicates how well the specified modelreproduces the covariance matrix among theindicator variables that is, it examines thesimilarity of the observed and estimated covariance

matrices.

The initial measure of GOF is the Chi-squarestatistic. The null hypothesis is No difference in

the two covariance matrices. Since you do notwant the matrices to be different, you hope for aninsignificant Chi-square (>.05) so you can accept thenull hypothesis.

AMOS Data Input = observed sample covariances

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AMOS Data Input = observed sample covariances

for HBAT 3-Construct model

Covariances calculated for the

samplerequest Sample

moments and look in Output

under that subheading.

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Covariancesestimated by AMOS

softwarerequest Implied moments

and look in Output under Estimates.

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Residuals = difference between observed

and estimated covariancesrequest

Residual moments.

A negative sign indicates the

observed covariance (2.137)

is smaller than the estimated

covariance (2.229) by -.093.

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Standardized Residualsyou look for

patterns of larger residuals, generally => 4.0

HBAT 5 Construct SEM Model: Model Fit diagnostics

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CMIN/DF a value below 2 is preferred but

between 2 and 5 is considered acceptable.

TheGFI is .938 above the .90

recommended minimum.

The AGFIis .921

above the .90 minimum.

HBAT 5-Construct SEM Model: Model Fit diagnostics

The CFIis 0.976it exceeds the

minimum (>0.90) for a model of this

complexity and sample size.

CMIN= minimum discrepancythe discrepancy

between the unrestricted sample covariance matrix

and the restricted (estimated) covariance matrix.

NPAR= number of parameters in the model.

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What does CFA/SEM actually test?

Three Criteria:

1. Goodness of Fit? Does estimated covariance matrix =observed covariance matrix? If X2significant then not equal, butoften will examine other fit indices.

2. Validity and Reliability of Measurement Model?

3. Significant and Meaningful Structural Relationships?

----------------

GFI= a measure of the amount of covariance in the samplecovariance matrix explained by the estimated covariance matrix.

AGFI= differs from the GFI only in the fact that it adjust for the

number of degrees of freedom (DF) in the specified model.

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2. Assessing the Measurement Model

Construct Validityo Face

o Convergent

o Discriminant

o Nomological

Construct Reliability3. Assessing the Structural Model

Significant and Meaningful Structural Relationships

Second & Third Criteria

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Guidelines for Establishing Acceptable Fit

Use multiple indices of differing types, not just X2. 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 testing atrue model is not a good trade-off.

CMIN/DF a value below 2 is preferred but

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This is theModel Fit

portion 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

between 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 Approximationa value

of 0.10 or less is consideredacceptable (6e, p. 748).

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.

RMSEArepresents 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 numbersfor the default model.

HBAT Three Construct Results

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The GFI, an absolute fit index, is .965.

This value is above the .90 guideline

for this model . Higher values

indicate better fit (6e, p. 747).

The AGFI, an incremental fit index,

is .946. This value is above the .90

guideline for this model . Attempts

to adjust for model complexity, but

penalizes more complex models.

The CFI, an incremental fit index, is0.984, which exceeds the guidelines

(>0.90) for a model of this complexity

and sample size (7e, p. 650).

HBAT Three Construct Results

CFI (Comparative Fit Index)represents theimprovement of fit of the specified model over abaseline model in which all variables are constrained tobe uncorrelated. It is a revised version of NFI thattakes sample size into consideration.

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Other Indices

The NFI, RFI and IFI are other indices. Our

guidelines indicate the NFI should be

>0.90 for a model of this complexity and

sample size. For the RFI and IFI weindicate that larger values (01.0) are

better.

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The RMSEA, an absolute

fit index, is 0.043. This value

is quite low and well below

the .08 guideline for a model

with 12 measured variablesand a sample size of 400.

This also is called a Badness-

Of-Fit index.

The 90 percent confidence

interval for the RMSEA is

between a LO of .028 and a

HI of 0.058. Thus, even theupper bound is not close to

.08.

Using the RMSEA(Root Mean Square Error of

Approximation) and the CFI(Comparative Fit Index) satisfies

our rule of thumb that both a badness-of-fit index and a

goodness-of-fit index be evaluated. In addition, other index

values also are supportive. For example, the GFI is 0.95, and

the AGFI is 0.93.

We therefore now move on to examine the construct validity

of the model.

PCLOSEis a closeness of fit

measure. It tests the hypothesis

that RMSEA is good in the

population. The .767 is the

probability of getting a RMSEA

as large as .043

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

is its ability to 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 SEM 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 to

empirically estimate validity using more rigorous

approaches; e.g., construct validity.

C t t lidit

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Construct validity

. . . . is made up of four components:

Face validity = the extent to which the content of the items is

consistent with the construct definition, based solely on theresearchersjudgment.

Convergent validity = the extent to which indicators of a specific

construct converge or share a high proportion of variance in

common. To assess we examine construct loadings, variance

extracted and reliability.Discriminant validity = the extent to which a construct is truly

distinct from other constructs (i.e., unidimensional).

Nomological validity = examines whether the correlations

between 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 the

observed indicator variables.

Face Validity HBAT Constructs and Indicator Variables

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y

Organizational CommitmentOC1 = My work at HBAT gives me a sense of accomplishment.OC2 = I am willing to put in a great deal of effort beyond that normally

expected to help HBAT be successful.OC3 = I have a sense of loyalty to HBAT.

OC4 = I am proud to tell others that I work for HBAT.Staying Intentions

SI1 = I am not actively searching for another job.SI2 = I seldom look at the job listings on monster.com.SI3 = I have no interest in searching for a job in the next year.SI4 = How likely is it that you will be working at HBAT one year from today?

Attitudes Towards Co-Workers

AC1 = How happy are you with the work of your coworkers?AC2 = How do you feel about your coworkers?AC3 = How often do you do things with your coworkers on your days off?AC4 = Generally, how similar are your coworkers to you?

Environmental PerceptionsEP1 = I am very comfortable with my physical work environment at HBAT.EP2 = The place I work in is designed to help me do my job better.EP3 = There are few obstacles to make me less productive in my workplace.

EP4 = What term best describes your work environment at HBAT?Job Satisfaction

JS1 = All things considered, I feel very satisfied when I think about my job.JS2 = When you think of your job, how satisfied do you feel?JS3 = How satisfied are you with your current job at HBAT?JS4 = How satisfied are you with HBAT as an employer?JS5 = Please indicate your satisfaction with your current job with HBAT by placing a percentage in

the blank, with 0% = not satisfied at all and 100% = highly satisfied.

Construct validity

tells us if the

indicator variablesaccurately measure

the latent constructs.

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Convergent Validity

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

Rules of Thumb: Convergent Validity Standardized loadings estimates should be .5 or higher, and

ideally .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 provideevidence of discriminant validity.

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

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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) 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.

HBAT CFA Three Factor Completely Standardized This is the same

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

Reliability Estimates

OC EP AC

Item

Reliabilities Error

OC1 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 0.32AC1 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

Average

Variance

Extracted 56. 61% 60. 25% 67. 86%

Construct

Reliability 0.84 0.86 0.89 The error is calculated as 1 minus the itemreliability, e.g., the AC4 delta is 1.666 = .33

The error is also referred to as the standardized

error variance.

Factor Loadings

This is the same

as the eigenvalue

in exploratory

factor analysis

2.264/4 = 56.61

Squared Factor Loadings

(communalities)

l f i d

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nVE

n

i

i

12

Formula for Variance Extracted(AVE)

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 a 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 the

latent 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

Formula for Construct Reliability

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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 theconvergent 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 variance-extracted estimates (AVE) 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 aconstruct is truly distinct from other constructs.

Rule of Thumb: all construct variance extracted

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

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

indicator variables have more in common with the

construct they are associated with than they do with

the 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.

y

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Discriminant validity compares the varianceextracted (AVE) estimates for each factor withthe squared interconstruct correlations (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 Interconstruct

Correlations) from the IC (Innerconstruct

Correlations) obtained from the

correlations table on the AMOS printout

(see previous slide):

IC SIC

EPAC .254 .0645

EPOC .500 .2500

ACOC .303 .0918


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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.

(In LISREL these are called Phi = )

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 positiveand significantsee 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.

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These are the R-squaredvalues (Squared Standardized

Loadings in Congeneric CFA).

So subtract these from 1 to

get (the standardized error

term estimate).

Error Variances(Unstandardized)

To get the

standardized errorvariances, subtract thesquared standardizedloadings shown belowfrom 1 for each item.

The Squared Multiple Correlations are also

referred to as the squared loadings, i.e., they are

calculated by squaring the standardized regression

weights (loadings).

The squared loadings are used in calculating the

variance extracted (AVE) for each construct.

Diagnosing Meas rement Model Problems

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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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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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With the AMOSsoftware you

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software youmust add anerror term on

your endogenousvariable.

This shows thechange from the

two-headedarrow to a single

headed arrow.


SEM Modelno

estimates.

Squared multiplecorrelation for

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SEM Modelwith

standardized estimates.

StandardizedRegression Weights forindicator variables,also called FactorLoadings.

endogenous variableOrganizational

Commitment.

Can be interpretedlike the R2in 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 unstandardized

regression 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 atthe top are for the two

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pnew causal paths to thenew endogenousvariable Organizational

Commitment.

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

Interpretation:When Environmental

Perceptions go up by 1standard deviation,OrganizationalCommitment goes up by.452 standarddeviations.

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These measures are

the 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

6 - CFA-SEM Intro_4-18-11

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