Measurement Models: Exploratory and Confirmatory Factor Analysis

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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 be consistent with
  certain predictions (e.g., moderately correlated
  with one another)

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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

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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 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)

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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 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

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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
  observed variables 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 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

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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.

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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 the
  technique 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 the
  technique 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 (N355)
• CFA (N375)
• 1-Item overall self-rating of stress
• SEM (N710)

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Table1 An oblique five factor pattern solution
(N170)
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Factors

• Factor 1 Workload
• Factor 2 Professional Recognition
• Factor 3 Student 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

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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 correlational relations
  among the latent variables
• Non-Recursive Model
• Mediated Model
• Regression Model

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Non-recursive model
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Regression Model
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Comparison of Fit Indices
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Results

• Two major contributors to teacher stress
• Work load
• Student Misbehavior