1. Factor Analysis
An Alternative technique for studying correlation and covariance structure

2. have a p-variate Normal distribution with mean vector
Let
have a p-variate Normal distribution
with mean vector
The Factor Analysis Model:
Let F1, F2, … , Fk denote independent standard normal observations (the Factors)
Let e1, e2, … , ep denote independent normal random variables with mean 0 and var(ei) = yp
Suppose that there exists constants lij (the loadings) such that:
x1= m1 + l11F1+ l12F2+ … + l1k Fk + e1
x2= m2 + l21F1+ l22F2+ … + l2k Fk + e2 …
xp= mp + lp1F1+ lp2F2+ … + lpk Fk + ep

3. Factor Analysis Model
where
and

4. Note:
hence
and
i.e. the component of variance of xi that is due to the common factors F1, F2, … , Fk.
i.e. the component of variance of xi that is specific only to that observation.

5. F1, F2, … , Fk are called the common factors
e1, e2, … , ep are called the specific factors
= the correlation between xi and Fj. if

6. Rotating Factors
Recall the factor Analysis model
This gives rise to the vector
having covariance
matrix:
Let P be any orthogonal matrix, then
and

7. Hence if
with
is a Factor Analysis model
then so also is
with
where P is any orthogonal matrix.

8. Rotating the Factors

9. Rotating the Factors

10. There are many techniques for rotating the factors VARIMAX Quartimax
The process of exploring other models through orthogonal transformations of the factors is called rotating the factors
There are many techniques for rotating the factors
VARIMAX
Quartimax
Equimax
VARIMAX rotation attempts to have each individual variables load high on a subset of the factors

11. Extracting the Factors

12. Several Methods – we consider two
Principal Component Method
Maximum Likelihood Method

13. Principle Component Method
Recall
where
are eigenvectors of S of length 1 and
are eigenvalues of S.

14. Hence
Thus
This is the Principal Component Solution with p factors
Note: The specific variances, yi, are all zero.

15. The objective in Factor Analysis is to explain the correlation structure in the data vector with as few factors as necessary
It may happen that the latter eigenvalues of S are small.

16. In addition let
In this case
The equality will be exact along the diagonal
where

17. Maximum Likelihood Estimation
Let
denote a sample from
where
The joint density of is

18. The Likelihood function is
The maximum likelihood estimates
Are obtained by numerical maximization of

19. Example: Olympic decathlon Scores
Data was collected for n = 160 starts (139 athletes) for the ten decathlon events (100-m run, Long Jump, Shot Put, High Jump, 400-m run, 110-m hurdles, Discus, Pole Vault, Javelin, 1500-m run). The sample correlation matrix is given on the next slide

20. Correlation Matrix

22. Identification of the factors