1

Exploratory & Confirmatory Factor

AnalysisAlan 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)• 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

4

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?

5

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

6

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

7

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

8

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

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

ry1ry4∑ =λ1Iλ4 I + λ1Eλ4E + λ1Iφλ4E

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.

•ry21 = λ1I⋄λ2I

•ry41 = λ1I⋄ϕ⋄λ4E

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 56-15 = 41 degrees of freedom from over identifying restrictions. These include our theoretical assumptions:

• λ4I = 0.0

• λ42 = 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