1 Exploratory & Confirmatory Factor Analysis Alan C. Acock OSU Summer Institute, 2009

2 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

3 EFA — One Dimension (Depression) y 1 - y 7 are called indicators of the latent variable y 1 - y 7 could be 7 observed scores Could be 7 individual items Could be 4 items, 2 scales, & 1 observer rating

4 EFA — One Dimension (Depression) e 1 - e 7 are called errors or unique variances e 1 - e 7 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?

5 EFA — One Dimension (Depression) Your depression and your e i each explain how you score on the observed variable All arrows go to the observed indicators. Your score on y 1 depends on your true level of depression and your error/unique variance

6 EFA — One Dimension (Depression) Errors/Unique variances may be correlated e 1 and e 6 might be measured the same method; hence a methods effect e 4 and e 5 might both deal with suicide ideation

7 EFA EFA seeks to explain relationships between the y’s based on two sources variance y i explained by your true level of depression and error/unique variance covariance y i & y j, cov(y i,y j ) explained by: loadings of y i on Depression Variance of Depression Loadings of y i on errors Correlated errors

8 Algebra

9 EFA with 2 Factors

10 CFA--with 2 Factors

11 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

12 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)

13 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

14 How are coefficients estimated? With 7 indicators, Σ has (7*8/2 = 56 variances and covariances We could write 56 equations. r y21 = λ 1I ⋄ λ 2I r y41 = λ 1I ⋄ϕ⋄ λ 4E

15 How are coefficients estimated? We need to estimate 7 λ’s, 7 e’s, and ϕ for a total of 15 parameters. We have = 41 degrees of freedom from over identifying restrictions. These include our theoretical assumptions: λ 4I = 0.0 λ 42 = 0.0 etc.

16 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