CJT 765: Structural CJT 765: Structural Equation ModelingEquation Modeling

Class 8: Confirmatory Factory Class 8: Confirmatory Factory AnalysisAnalysis

Outline of ClassOutline of Class Finishing up Model Testing IssuesFinishing up Model Testing Issues Confirmatory Factor AnalysisConfirmatory Factor Analysis Recent ReadingsRecent Readings

Comparison of Models Comparison of Models

Hierarchical Models: Hierarchical Models: • Difference of Difference of 22 test test

Non-hierarchical Models:Non-hierarchical Models:• Compare model fit indicesCompare model fit indices

Model RespecificationModel Respecification

Model trimming and buildingModel trimming and building Empirical vs. theoretical Empirical vs. theoretical

respecificationrespecification Consider equivalent modelsConsider equivalent models

Sample Size GuidelinesSample Size Guidelines Small (under 100), Medium (100-200), Large Small (under 100), Medium (100-200), Large

(200+) [try for medium, large better](200+) [try for medium, large better] Models with 1-2 df may require samples of Models with 1-2 df may require samples of

thousands for model-level power of .8.thousands for model-level power of .8. When df=10 may only need n of 300-400 for When df=10 may only need n of 300-400 for

model level power of .8.model level power of .8. When df > 20 may only need n of 200 for power When df > 20 may only need n of 200 for power

of .8of .8 20:1 is ideal ratio for # cases/# free parameters, 20:1 is ideal ratio for # cases/# free parameters,

10:1 is ok, less than 5:1 is almost certainly 10:1 is ok, less than 5:1 is almost certainly problematicproblematic

For regression, N For regression, N >> 50 + 8m for overall R 50 + 8m for overall R22, with m , with m = # IVs and N = # IVs and N > > 104 + m for individual predictors104 + m for individual predictors

Statistical PowerStatistical Power Use power analysis tables from Cohen to Use power analysis tables from Cohen to

assess power of specific detecting path assess power of specific detecting path coefficient.coefficient.

Saris & Satorra: use Saris & Satorra: use 22 difference test using difference test using predicted covariance matrix compared to predicted covariance matrix compared to one with that path = 0one with that path = 0

McCallum et al. (1996) based on RMSEA McCallum et al. (1996) based on RMSEA and chi-square distribution for close fit, not and chi-square distribution for close fit, not close fit and exact fitclose fit and exact fit

Small number of computer programs that Small number of computer programs that calculate power for SEM at this pointcalculate power for SEM at this point

Factor analysisFactor analysis

Indicators : Indicators : continuouscontinuous Measurement error are independent Measurement error are independent

of each other and of the factorsof each other and of the factors All associations between the factors All associations between the factors

are unanalyzedare unanalyzed

Identification of CFAIdentification of CFA

Can estimate v*(v+1)/2 of Can estimate v*(v+1)/2 of parametersparameters

Necessary Necessary • # of free parameters <= # of # of free parameters <= # of

observationsobservations• Every latent variable should be Every latent variable should be scaledscaled

Additional: fix the unstandardized residual path of the error to 1. (assign a scale of the unique variance of its indicator)

Scaling factor: constrain one of the factor loadings to 1

( that variables called – reference variable, the factor has a scale related to the explained variance of the reference variable)

OR

fix factor variance to a constant ( ex. 1), so all factor loadings are free parameters

Both methods of scaling result in the same overall

fit of the model

Identification of CFAIdentification of CFA

Sufficient :Sufficient :• At least three (3) indicators per factor to At least three (3) indicators per factor to

make the model identifiedmake the model identified• Two-indicator rule – prone to estimation Two-indicator rule – prone to estimation

problems (esp. with small sample size)problems (esp. with small sample size)

Interpretation of the estimatesInterpretation of the estimatesUnstandardized solutionUnstandardized solution

Factor loadings =unstandardized regression Factor loadings =unstandardized regression coefficientcoefficient

Unanalyzed association between factors or errors= Unanalyzed association between factors or errors= covariances covariances

•Standardized solutionStandardized solution Unanalyzed association between factors or errors= Unanalyzed association between factors or errors=

correlationscorrelations Factor loadings =standardized regression coefficient Factor loadings =standardized regression coefficient

( structure coefficient).( structure coefficient). The square of the factor loadings = the proportion of The square of the factor loadings = the proportion of

the explained ( common) indicator variance, the explained ( common) indicator variance, RR22(squared multiple correlation)(squared multiple correlation)

Problems in estimation of CFAProblems in estimation of CFA

Heywood cases – negative variance estimated or Heywood cases – negative variance estimated or correlations > 1.correlations > 1.

Ratio of the sample size to the free parameters – Ratio of the sample size to the free parameters – 10:1 ( better 20:1)10:1 ( better 20:1)

Nonnormality – affects ML estimationNonnormality – affects ML estimation

Suggestions by March and Hau(1999)when Suggestions by March and Hau(1999)when sample size is small: sample size is small:

indicators with high standardized loadings( >0.6) indicators with high standardized loadings( >0.6) constrain the factor loadingsconstrain the factor loadings

Testing CFA modelsTesting CFA models

Test for a single factor with the theory or notTest for a single factor with the theory or not If reject HIf reject H0 0 of good fit - try two-factor of good fit - try two-factor

model…model… Since one-factor model is restricted version Since one-factor model is restricted version

of the two -factor model , then Compare one-of the two -factor model , then Compare one-factor model to two-factor model using Chi-factor model to two-factor model using Chi-square test . If the Chi-square is significant – square test . If the Chi-square is significant – then the 2-factor model is better than 1-then the 2-factor model is better than 1-factor model.factor model.

Check RCheck R22 of the unexplained variance of the of the unexplained variance of the indicators..indicators..

Respecification of CFARespecification of CFA

IFIF lower factor loadings lower factor loadings

of the indicator of the indicator (standardized<=0.2) (standardized<=0.2)

High loading on High loading on more than one factormore than one factor

High correlation High correlation residuals residuals

High factor High factor correlationcorrelation

THENTHEN Specify that indicator on Specify that indicator on

a different factora different factor

Allow to load on one Allow to load on one more than one factor more than one factor ( might be a problem)( might be a problem)

Allow error Allow error mmeasurements easurements to covary to covary

Too many factors Too many factors specifiedspecified

Other testsOther tests

Indicators:Indicators:• congeneric – measure the same constructcongeneric – measure the same construct

if model fits , then if model fits , then

-tau-equivalent – constrain all unstandardized -tau-equivalent – constrain all unstandardized loadings to 1loadings to 1

if model fit, thenif model fit, then

- parallelism – equality of error variances- parallelism – equality of error variances

Constraint interaction of CFAConstraint interaction of CFA

Factors with 2 indicators and Factors with 2 indicators and loadings on different factors are loadings on different factors are constrained to be equal.constrained to be equal.

- depends how factors are scaled- depends how factors are scaled

Nonnormal distributionsNonnormal distributions

Normalize with transformationsNormalize with transformations Use Use corrected normal theory method, corrected normal theory method, e.g. e.g.

use robust standard errors and corrected test use robust standard errors and corrected test statistics, ( Satorra-Bentler statistics)statistics, ( Satorra-Bentler statistics)

Use Asymptotic distribution free or arbitrary Use Asymptotic distribution free or arbitrary distribution function (ADF) - no distribution distribution function (ADF) - no distribution assumption - Need large sampleassumption - Need large sample

Use elliptical distribution theory – need only Use elliptical distribution theory – need only symmetric distributionsymmetric distribution

Mean-adjusted weighted least squares (MLSW) Mean-adjusted weighted least squares (MLSW) and variance-adjusted weighted least square and variance-adjusted weighted least square (VLSW) - MPLUS with categorical indicators(VLSW) - MPLUS with categorical indicators

Use normal theory with nonparametric Use normal theory with nonparametric bootstrappingbootstrapping

Remedies to nonnormalityRemedies to nonnormality

Use a parcel which is a linear Use a parcel which is a linear composite of the discrete scores, as composite of the discrete scores, as continuous indicators continuous indicators

Use parceling ,when underlying Use parceling ,when underlying factor is unidimentional.factor is unidimentional.

Hayduk et al. Hayduk et al.

Pearl’s D-SeparationPearl’s D-Separation Better ways of controlling for Better ways of controlling for

extraneous variablesextraneous variables

Holbert & StephensonHolbert & Stephenson

Indirect Effects in Media ResearchIndirect Effects in Media Research Viewing of Presidential Debates as Viewing of Presidential Debates as

ExampleExample

NoarNoar

Use of CFA in scale developmentUse of CFA in scale development Test of multiple factor modelsTest of multiple factor models

LanceLance

Multi-Trait, Multi-MethodMulti-Trait, Multi-Method Comparison of Correlated Trait-Comparison of Correlated Trait-

Correlated Method versusCorrelated Method versus Correlated Uniqueness ModelsCorrelated Uniqueness Models