1. Applied Statistics Using SAS and SPSS
Topic: Factor Analysis
By Prof Kelly Fan, Cal State Univ, East Bay

2. Outline Introduction Principal component analysis Rotations
Using communalities other than one

3. Introduction Reduce data
Summarize many ordinal categorical factors by a few combinations of them (new factors)

4. Example. 6 Questions
Goal: a measure of depression and a measure of happiness (how pleasant)
6 questions with response using number 1 to 7. The smaller the number is, the stronger the subject agrees. 4: no opinion

5. Example. 6 Questions I usually feel blue. People often stare at me.
I think that people are following me.
I am usually happy.
Someone is trying to hurt me.
I enjoy going to parties.
Q. Which questions will a depressed person likely agree with? A happy person?

6. Data Set:
Subj 1 2 3 4 5 6 7 8 9 Q u e s t i o n

7. Data Set:
Subj 10 11 12 13 14 15 Q u e s t i o n 1 6 3 5 2 7 4

8. Principal Component Analysis
Analyze >> Data Reduction >> Factor…
The bigger the eigenvalue is,
the more information this factor (component) carries.

9. A Visual Tool: Scree Plot

11. Communalities
Communalities represent how much variance in the original variables is explained by all of the factors kept in the analysis.

12. SPSS Output

13. SPSS Output

14. A Visual Tool: Component Plot (Loading plots in SPSS)

15. Discussion
Q4 & Q6 are highly and positively correlated and so should be at the same direction of any factor (here component 1 & 2)
Similarly, the other questions should be at the same direction of factor 1 & 2 (component 1 & 2)
Need a rotation!!

16. Rotation: Varimax Method

17. Varimax rotation

18. Before and After Rotation

20. Rotation: Promax Method (optional)
Used when factors (depression/happiness) are allowed to be correlated (non-orthogonal)

21. Using Communalities Other Than One
When the original questions are not equally important
Different methods of “extraction”

22. Un-weighted Least Squares
Initial communality of a question is the R^2 (squared multiple correlation) of regressing all others against this question

23. Before and After Varimax Rotation

24. Varimax rotation