Measurement Models: Exploratory and Confirmatory Factor Analysis

1. Measurement Models: Exploratory and Confirmatory Factor Analysis James G. Anderson, Ph.D. Purdue University

2. 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)

3. 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

4. Principles of Measurement • Reliability is concerned with random error • Validity is concerned with random and systematic error

5. 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

6. 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)

7. Types of Measurement Models • Exploratory (EFA) • Confirmatory (CFA) • Multitrait-Multimethod (MTMM) • Hierarchical CFA

8. An Exploratory Factor Model

9. 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

10. A Confirmatory Factor Model

11. 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

12. CFA Features • The measurement error terms are independent of each other and of the factors • All associations between factors are unanalyzed

13. 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

14. 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

15. 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.

16. 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.

17. 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)

18. 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)

19. 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)

20. 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

21. 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.

22. 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.

23. 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.

24. 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

25. A Covariance Structure Model

26. 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.

27. Methods • 20-Item survey of teacher stress • EFA (N=355) • CFA (N=375) • 1-Item overall self-rating of stress • SEM (N=710)

28. Table1: An oblique five factor pattern solution (N=170)

29. Factors • Factor 1 – Workload • Factor 2 – Professional Recognition • Factor 3 –Student Misbehavior • Factor 4 - Time/Resource Difficulties • Factor 5 – Poor Colleague Relations

30. Factor Patterns

31. EFA Results • 5 Factor solution • 4 Items deleted • Fit Statistics: • Chi Square = 156.94 • df = 70 • AGFI = 0.906 • RMR = 0.053

32. Confirmatory Factor Analysis

33. Covariances between exogenous latent traits

34. CFA Results • 5 Factor solution • 2 Items deleted • Fit Statistics: • Chi Square = 171.14 • df = 70 • AGFI = 0.911 • RMR = 0.057

35. 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

36. Non-recursive model

37. Regression Model

38. Comparison of Fit Indices

39. Results • Two major contributors to teacher stress • Work load • Student Misbehavior