Exploratory &
Confirmatory Factor
Analysis
Alan C. Acock
OSU Summer Institute, 2009
1
EFA — One
Dimension
• Latent
variables appear in
(Depression)
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
2
•
•
•
•
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
3
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?
4
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
5
•
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
6
•
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
7
Algebra
yi    i   i


•
Cov yi , y j  i  j
rµij  i  j
  
yi is an indicator
 is the intercept, nu
 is a matrix containing the lambdas
 is the name of the latent variable
(depression), eta
 is the vector of errors, epsilon
 is the variance of eta, their covariance
with multiple latent variables, psi
 is the covariance of all the yi ' s
8
EFA with 2 Factors
9
CFA--with 2 Factors
10
•
•
•
•
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
•
ry1 ry4  1I4 I  1E 4 E  1I4 E
11
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)
12
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
13
How are coefficients
estimated?
• With 7 indicators, Σ has (7*8/2 = 56
variances and covariances
• We could write 56 equations.
• r = λ ⋄λ
• r = λ ⋄ϕ⋄λ
y21
1I
y41
1I
2I
4E
14
•
•
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.
15
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
16