Anon

Factor Analysis and Principal Components Analysis 249

introduce the concept of the Frobenius norm of a matrix. Given any matrix
Aa={}ij , the Frobenius norm of A, written as AF, is the square root of the
sum of the squares of the absolute value of its terms:

AaF ij

ij

= ∑

2

,

The Frobenius norm is a measure of the average magnitude of the terms of
a matrix.
We can now compute the ratio between the Frobenius norm of the dif-
ference ΣΨX−+('BB ) and that of ΣX. We obtain the value

ΣΨ

Σ

X F

XF

−+BB

=

(')

.0 0089

which shows that the magnitude of the difference ΣΨX−+('BB ) is less than
1% of the magnitude of ΣX itself; hence, it is a good approximation.

estimation of Factors

Having determined the factor loadings with MLE, we can now estimate the
factors. As observed above, factors are not unique and cannot be estimated
with MLE methods. The estimated factors are called factor scores or simply
scores. In the literature and in statistical software, estimates of factor scores
might be referred to as predictions so that estimated factors are called pre-
dicted scores. There are several methods to estimate factor scores.
The most commonly used method is to look at equation (12.5) as a mul-
tiple regression equation with x as regressors. After estimating the matrix B
in equation (12.5), we can look at the variables xt, ft as variables with a joint
normal distribution. Recall that from equation (12.7),

Ex

Ef

fEf

t

t

tt tt

()=

()=

= ()=

0

0

cov( ,)εε''' 0

while from equation (12.9),

cov'

cov

xBB

fI

t

t

()=+

()=

Ψ

and therefore cov,()xftt=+cov,()BfttfBt =ε.