Measurement Models:
Exploratory and Confirmatory
Factor Analysis
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
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)
Principles of Measurement
• Reliability is concerned with random error • Validity is concerned with random and
Measurement Reliability
• Test-Retest
• Alternate Forms
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)
Types of Measurement Models
• Exploratory (EFA) • Confirmatory (CFA)
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.
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
CFA Features
• The measurement error terms are
independent of each other and of the factors
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
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
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
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
EFA Procedures
• Decide which indicators to include in the analysis.
• Select the method to establish the factor model
– ML (assumes a multivariate normal distribution)
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
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
EFA Procedures
• Select the appropriate number of factors • Identify which indicators load on each
factor or latent variable
Uses of CFA
• Evaluation of test instruments • Construct validation
– Convergent validity – Discriminant validity
• Evaluation of methods effects
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
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 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
A Structural Model of the
Dimensions of Teacher Stress
• Survey of teacher stress, job satisfaction and career commitment
Methods
• 20-Item survey of teacher stress • EFA (N=355)
• CFA (N=375)
Factors
• Factor 1 – Workload
• Factor 2 – Professional Recognition • Factor 3 –Student Misbehavior
EFA Results
• 5 Factor solution • 4 Items deleted • Fit Statistics:
• Chi Square = 156.94 • df = 70
CFA Results
• 5 Factor solution • 2 Items deleted • Fit Statistics:
• Chi Square = 171.14 • df = 70
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
Results
• Two major contributors to teacher stress • Work load
