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