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8/10/2019 6 - CFA-SEM Intro_4-18-11
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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
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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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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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 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
Hypotheses:H1: EP +JSH2: EP +OCH3: AC +JS
H4: AC +
OCH5: JS +OCH6: JS + SIH7: OC +SI
Endogeneous
Variable
Exogeneous
Variable
Endogeneous
Variable
Note: all causal
relationships
are recursive.
Note: observable indicator variables are not shown to simplify the model.
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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 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.
Discriminant Validity
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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
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.
(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.
HBAT Three Construct
SEM Modelno
estimates.
Squared multiplecorrelation for
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HBAT Three Construct
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