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Exploratory & Confirmatory Factor
AnalysisAlan C. Acock
OSU Summer Institute, 2009
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EFA — One Dimension
(Depression)• Latent variables appear in ovals
• Latent variables are not observed directly
• Latent variables represent the shared variances of a set of indicators
• In SEM, latent variables can be predictors or outcomes
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EFA — One Dimension
(Depression)• y1 - y7 are called indicators
of the latent variable
• y1 - y7 could be 7 observed scores
•Could be 7 individual items
•Could be 4 items, 2 scales, & 1 observer rating
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EFA — One Dimension
(Depression)• e1 - e7 are called errors or unique variances
• e1 - e7 sometimes labeled as δ’s or ε’s
• Arrow shows the errors explain part of the variances in the indicators
• How is this error variance? How is this unique variance?
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EFA — One Dimension
(Depression)• Your depression and your ei each explain how you score on the observed variable
• All arrows go to the observed indicators.
• Your score on y1 depends on your true level of depression and your error/unique variance
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EFA — One Dimension
(Depression)• Errors/Unique variances
may be correlated
• e1 and e6 might be measured the same method; hence a methods effect
• e4 and e5 might both deal with suicide ideation
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EFA• EFA seeks to explain relationships between the y’s based
on two sources
• variance yi explained by your true level of depression and error/unique variance
• covariance yi & yj, cov(yi,yj) explained by:
• loadings of yi on Depression
• Variance of Depression
• Loadings of yi on errors
•Correlated errors
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yi =ν + Ληi + ε i
Cov yi ,yj( )∑ =λiΨλ j
rijµ =λiλ j
Σ =ΛΨ ′Λyi is an indicator
ν is the intercept, nu
Λ is a matrix containing the lambdas
η is the name of the latent variable (depression), etaε is the vector of errors, epsilon
Ψ is the variance of eta, their covariance with multiple latent variables, psi
Σ is the covariance of all the yi 's
Algebra
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EFA with 2 Factors
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CFA--with 2 Factors
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EFA with 2 Factors• Internalizing loads strongly on first three y’s
• Externalizing loads strongly on last four y’s
• Internalizing and Externalizing are correlated, represented by ϕ
• Correlating errors adds another link, reducing lambdas
ry1ry4∑ =λ1Iλ4 I + λ1Eλ4E + λ1Iφλ4E
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How are coefficients estimated?
•The equation on the last slide has several parameters that form a vector:
•λ’s for the loadings,
•The variances of latent variables (1 in a standardized solution), and
•The covariances of the latent variables (r’s in the standardized solution)
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How are coefficients estimated?
•Mplus iteratively tries different values in the vector that try to reproduce the covariance matrix Σ
•In EFA there are mathematically convenient assumptions that let us identify the model
•In CFA there are theoretical restrictions that identify the model
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How are coefficients estimated?
•With 7 indicators, Σ has (7*8/2 = 56 variances and covariances
•We could write 56 equations.
•ry21 = λ1I⋄λ2I
•ry41 = λ1I⋄ϕ⋄λ4E
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How are coefficients estimated?
•We need to estimate 7 λ’s, 7 e’s, and ϕ for a total of 15 parameters.
•We have 56-15 = 41 degrees of freedom from over identifying restrictions. These include our theoretical assumptions:
• λ4I = 0.0
• λ42 = 0.0
•etc.
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Identification Rules of Thumb
•3 indicators of each latent variable and CFA is okay—4 would be even better
•2 indicators of some latent variables will be identified if there are 3 or more indicators of other latent variables
•1 or 2 indicators are okay if you can fix the error at some value, e.g. 0