03a Measurement Models
Transcript of 03a Measurement Models
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Measurement Models:
Exploratory and Confirmatory
Factor Analysis
James G. Anderson, Ph.D.Purdue University
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Conceptual Nature of Latent
Variables
Latent variables correspond to some type
of hypothetical construct
Require a specific operational definition
Indicators of the construct need to be
selected
Data from the indicators must beconsistent with certain predictions (e.g.,
moderately correlated with one another)
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Multi-Indicator Approach
A multiple-indicator approach reduces theoverall effect of measurement error of anyindividual observed variable on the accuracyof the results
A distinction is made between observedvariables (indicators) and underlying latentvariables or factors (constructs)
Together the observed variables and thelatent variables make up the measurementmodel
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Principles of Measurement
Reliability is concerned with random error
Validity is concerned with random and
systematic error
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Measurement Reliability
Test-Retest
Alternate Forms
Split-Half/Internal Consistency
Inter-rater Coefficient
0.90 Excellent
0.80 Very Good 0.70 Adequate
0.50 Poor
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Measurement Validity
Content ( (whether an indicators items arerepresentative of the domain of the construct)
Criterion-Related (whether a measure relates to anexternal standard against which it can be evaluated)
Concurrent (when scores on the predictor and criterionare collected at the same time)
Predictive (when scores on the predictor and criterionare collected at different times)
Convergent (items that measure the same constructare correlated with one another)
Discriminant (items that measure different constructsare not correlated highly with one another)
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Types of Measurement Models
Exploratory (EFA)
Confirmatory (CFA)
Multitrait-Multimethod (MTMM) Hierarchical CFA
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An Exploratory Factor Model
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EFA Features
The potential number of factors ranges from
one up to the number of observed variables
All of the observed variables in EFA are
allowed to correlate with every factor An EFA solution usually requires rotation to
make the factors more interpretable.
Rotation changes the correlations between
the factors and the indicators so the pattern
of values is more distinct
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A Confirmatory Factor Model
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CFA Features
The number of factors and the observedvariables (indicators) that load on eachconstruct (factor or latent variable) arespecified in advance of the analysis
Generally indicators load on only oneconstruct (factor)
Each indicator is represented as having two
causes, a single factor that it is suppose tomeasure and all other unique sources ofvariance represented by measurement error
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CFA Features
The measurement error terms are
independent of each other and of the
factors
All associations between factors are
unanalyzed
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EFA vs CFA
The purpose is to determine the number
and nature of latent variables or factors
that account for the variation and
covariation among a set of observedvariables or indicators.
Two types of analysis
Exploratory Factor Analysis
Confirmatory Factor Analysis
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EFA vs CFA
Both types of analysis try to reproduce the
observed relationships among a set of indicators
with a smaller set of latent variables.
EFA is data driven and used to determine thenumber of factors and which observed variables
are indicators of each latent variable.
In EFA all the observed variables are
standardized and the correlation matrix is
analyzed
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EFA vs CFA
CFA is confirmatory. The number offactors and the pattern of indicator factorloadings are specified in advance.
CFA analyzes the variance-covariancematrix of unstandardized variables.
The prespecified factor solution is
evaluated in terms of how well itreproduces the sample covariance matrixof measured variables.
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EFA vs CFA
CFA models fix cross-loadings to zero.
EFA models may involve cross-loadings of
indicators.
In EFA models errors are assumed to be
uncorrelated
In CFA models errors may be correlated.
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EFA Procedures
Decide which indicators to include in the
analysis.
Select the method to establish the factor
model
ML (assumes a multivariate normal
distribution)
Principle Factors (Distribution Free)
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EFA Procedures
Select the appropriate number of factors
Eigenvalues greater than one
Scree test
Goodness of fit of the model
If there is more than one factor, select thetechnique to rotate the initial factor matrix
to simple structure Orthogonal rotation (Varimax) Oblique rotation (e.g., Promax)
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EFA Procedures
Select the appropriate number of factors
Eigenvalues greater than one
Scree test
Goodness of fit of the model
If there is more than one factor, select thetechnique to rotate the initial factor matrix
to simple structure Orthogonal rotation (varimax) Oblique rotation (e.g., oblimin)
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EFA Procedures
Select the appropriate number of factors
Identify which indicators load on each
factor or latent variable
You can calculate factor scores to serve
as latent variables
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Uses of CFA
Evaluation of test instruments
Construct validation
Convergent validity
Discriminant validity
Evaluation of methods effects
Evaluation of measurement invariance Development and testing of the
measurement model for a SEM.
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Advantages of CFA
Test nested models
Test relationships among error variables
or constraints on factor loadings (e.g.,
equality)
Test equivalent measurement models in
two or more groups or at two or more
times.
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Advantages of CFA
The fit of the measurement model can be
determined before estimating the SEM
model.
In SEM models you can establish
relationships among variables adjusting for
measurement error.
CFA can be used to analyze mean
structures.
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CFA Model Identification
Identification pertains to the difference between
the number of estimated model parameters and
the number of pieces of information in the
variance/covariance matrix. Every latent variable needs to have its scale
identified.
Fix one loading of an observed variable on the latent
variable to one
Fix the variance of the latent variable to one
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A Covariance Structure Model
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A Structural Model of the
Dimensions of Teacher Stress Survey of teacher stress, job satisfaction
and career commitment
710 primary school teachers in the U.K.
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Methods
20-Item survey of teacher stress
EFA (N=355)
CFA (N=375) 1-Item overall self-rating of stress
SEM (N=710)
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Table1: An oblique five factor pattern solution (N=170)
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Factors
Factor 1 Workload
Factor 2 Professional Recognition
Factor 3Student Misbehavior Factor 4 - Time/Resource Difficulties
Factor 5 Poor Colleague Relations
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Factor Patterns
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EFA Results
5 Factor solution
4 Items deleted
Fit Statistics: Chi Square = 156.94
df = 70
AGFI = 0.906 RMR = 0.053
Confirmatory Factor Analysis
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Confirmatory Factor Analysis
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Covariances between exogenous latent traits
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CFA Results
5 Factor solution
2 Items deleted
Fit Statistics: Chi Square = 171.14
df = 70
AGFI = 0.911 RMR = 0.057
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Structural Equation Models
True Null Model - Hypothesizes no significant
covariances among the observed variables
Structural Null Model - Hypothesizes no
significant structural or correlationalrelations among the latent variables
Non-Recursive Model
Mediated Model Regression Model
Non-recursive model
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Non-recursive model
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Regression
Model
Comparison of Fit Indices
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Comparison of Fit Indices
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Results
Two major contributors to teacher stress
Work load
Student Misbehavior