03a Measurement Models
Measurement Models:
Exploratory and Confirmatory
Factor Analysis
James G. Anderson, Ph.D.
Purdue University
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 be
consistent with certain predictions (e.g.,
moderately correlated with one another)
Multi-Indicator Approach
• A multiple-indicator approach reduces the
overall effect of measurement error of any
individual observed variable on the accuracy
of the results
• A distinction is made between observed
variables (indicators) and underlying latent
variables or factors (constructs)
• Together the observed variables and the
latent variables make up the measurement
model
Principles of Measurement
• Reliability is concerned with random error
• Validity is concerned with random and
systematic error
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
Measurement Validity
• Content ( (whether an indicator’s items are
representative of the domain of the construct)
• Criterion-Related (whether a measure relates to an
external standard against which it can be evaluated)
• Concurrent (when scores on the predictor and criterion
are collected at the same time)
• Predictive (when scores on the predictor and criterion
are collected at different times)
• Convergent (items that measure the same construct
are correlated with one another)
• Discriminant (items that measure different constructs
are not correlated highly with one another)
Types of Measurement Models
•
•
•
•
Exploratory (EFA)
Confirmatory (CFA)
Multitrait-Multimethod (MTMM)
Hierarchical CFA
An Exploratory Factor Model
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
A Confirmatory Factor Model
CFA Features
• The number of factors and the observed
variables (indicators) that load on each
construct (factor or latent variable) are
specified in advance of the analysis
• Generally indicators load on only one
construct (factor)
• Each indicator is represented as having two
causes, a single factor that it is suppose to
measure and all other unique sources of
variance represented by measurement error
CFA Features
• The measurement error terms are
independent of each other and of the
factors
• All associations between factors are
unanalyzed
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 observed
variables or indicators.
• Two types of analysis
– Exploratory Factor Analysis
– Confirmatory Factor Analysis
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 the
number 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
EFA vs CFA
• CFA is confirmatory. The number of
factors and the pattern of indicator factor
loadings are specified in advance.
• CFA analyzes the variance-covariance
matrix of unstandardized variables.
• The prespecified factor solution is
evaluated in terms of how well it
reproduces the sample covariance matrix
of measured variables.
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.
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)
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 the
technique to rotate the initial factor matrix
to simple structure
– Orthogonal rotation (Varimax)
– Oblique rotation (e.g., Promax)
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 the
technique to rotate the initial factor matrix
to simple structure
– Orthogonal rotation (varimax)
– Oblique rotation (e.g., oblimin)
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
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.
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.
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.
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
A Covariance Structure Model
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.
Methods
• 20-Item survey of teacher stress
•
EFA (N=355)
•
CFA (N=375)
• 1-Item overall self-rating of stress
•
SEM (N=710)
Table1: An oblique five factor pattern solution (N=170)
Factors
•
•
•
•
•
Factor 1 – Workload
Factor 2 – Professional Recognition
Factor 3 –Student Misbehavior
Factor 4 - Time/Resource Difficulties
Factor 5 – Poor Colleague Relations
Factor Patterns
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
Covariances between exogenous latent traits
CFA Results
• 5 Factor solution
• 2 Items deleted
• Fit Statistics:
•
Chi Square = 171.14
•
df = 70
•
AGFI = 0.911
•
RMR = 0.057
Structural Equation Models
• True Null Model - Hypothesizes no significant
covariances among the observed variables
• Structural Null Model - Hypothesizes no
significant structural or correlational
relations among the latent variables
• Non-Recursive Model
• Mediated Model
• Regression Model
Non-recursive model
Regression
Model
Comparison of Fit Indices
Results
• Two major contributors to teacher stress
•
Work load
•
Student Misbehavior
Exploratory and Confirmatory
Factor Analysis
James G. Anderson, Ph.D.
Purdue University
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 be
consistent with certain predictions (e.g.,
moderately correlated with one another)
Multi-Indicator Approach
• A multiple-indicator approach reduces the
overall effect of measurement error of any
individual observed variable on the accuracy
of the results
• A distinction is made between observed
variables (indicators) and underlying latent
variables or factors (constructs)
• Together the observed variables and the
latent variables make up the measurement
model
Principles of Measurement
• Reliability is concerned with random error
• Validity is concerned with random and
systematic error
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
Measurement Validity
• Content ( (whether an indicator’s items are
representative of the domain of the construct)
• Criterion-Related (whether a measure relates to an
external standard against which it can be evaluated)
• Concurrent (when scores on the predictor and criterion
are collected at the same time)
• Predictive (when scores on the predictor and criterion
are collected at different times)
• Convergent (items that measure the same construct
are correlated with one another)
• Discriminant (items that measure different constructs
are not correlated highly with one another)
Types of Measurement Models
•
•
•
•
Exploratory (EFA)
Confirmatory (CFA)
Multitrait-Multimethod (MTMM)
Hierarchical CFA
An Exploratory Factor Model
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
A Confirmatory Factor Model
CFA Features
• The number of factors and the observed
variables (indicators) that load on each
construct (factor or latent variable) are
specified in advance of the analysis
• Generally indicators load on only one
construct (factor)
• Each indicator is represented as having two
causes, a single factor that it is suppose to
measure and all other unique sources of
variance represented by measurement error
CFA Features
• The measurement error terms are
independent of each other and of the
factors
• All associations between factors are
unanalyzed
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 observed
variables or indicators.
• Two types of analysis
– Exploratory Factor Analysis
– Confirmatory Factor Analysis
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 the
number 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
EFA vs CFA
• CFA is confirmatory. The number of
factors and the pattern of indicator factor
loadings are specified in advance.
• CFA analyzes the variance-covariance
matrix of unstandardized variables.
• The prespecified factor solution is
evaluated in terms of how well it
reproduces the sample covariance matrix
of measured variables.
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.
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)
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 the
technique to rotate the initial factor matrix
to simple structure
– Orthogonal rotation (Varimax)
– Oblique rotation (e.g., Promax)
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 the
technique to rotate the initial factor matrix
to simple structure
– Orthogonal rotation (varimax)
– Oblique rotation (e.g., oblimin)
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
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.
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.
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.
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
A Covariance Structure Model
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.
Methods
• 20-Item survey of teacher stress
•
EFA (N=355)
•
CFA (N=375)
• 1-Item overall self-rating of stress
•
SEM (N=710)
Table1: An oblique five factor pattern solution (N=170)
Factors
•
•
•
•
•
Factor 1 – Workload
Factor 2 – Professional Recognition
Factor 3 –Student Misbehavior
Factor 4 - Time/Resource Difficulties
Factor 5 – Poor Colleague Relations
Factor Patterns
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
Covariances between exogenous latent traits
CFA Results
• 5 Factor solution
• 2 Items deleted
• Fit Statistics:
•
Chi Square = 171.14
•
df = 70
•
AGFI = 0.911
•
RMR = 0.057
Structural Equation Models
• True Null Model - Hypothesizes no significant
covariances among the observed variables
• Structural Null Model - Hypothesizes no
significant structural or correlational
relations among the latent variables
• Non-Recursive Model
• Mediated Model
• Regression Model
Non-recursive model
Regression
Model
Comparison of Fit Indices
Results
• Two major contributors to teacher stress
•
Work load
•
Student Misbehavior