Ch. 13: Factor Analysis
I. Situation
A. For a given set of correlated observed variables, we try to
extract hidden (latent) common factors which can explain the
correlated variables.
B. A variable-reduction technique.
C. An interdependence model.
II. Terminology
A. Factor Loading (Factor Pattern): Standardized regression
coefficient to predict original (observed) variables using
common factors.
zy1 = λ11f1 + λ12f2 + . . + λ1mfm + ε1
zy2 = λ21f1 + λ22f2 + . . + λ2mfm + ε2
.
.
zyp = λp1f1 + λp2f2 + . . + λpmfm + εp
* λ = factor loadings, not eigenvalues.
m = # of factors
p = # of variables (m<p).
B. Eigenvalue:
1. The sum of squared factor loadings between a factor and
all variables.
2. The amount of variance of standardized variables
accounted for by a factor.
C. Communality
1. The sum of squared factor loadings between a variable and
all factors.
2. The amount of variance of a standardized variable
accounted for by all factors.
3. Theoretically, it should be smaller than 1, but in
reality, it can be 1 (Heywood case) or greater than 1
(Ultra Heywood case).
4. Reasons for Heywood or Ultra Heywood cases.
a) Too many or too few common factors.
b) Bad prior communality estimates.
c) Not enough subjects for stable estimates (n>5v).
d) Inappropriate models.
D. Factor score: individual subjects’ score on each factor.
III. Mathematical Model
A. Spearman (1904) model (single g-factor model)
1. y = λg + ε
where
y (pX1) = a vector of random variable y,
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