1. Happiness comes not from material wealth but less desire.

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

3. Outline
Introduction
Principal component analysis
Rotations

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

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

6. 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?

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

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

9. Principal Component Analysis
Eigenvalues of the Correlation Matrix: Total = 6 Average = 1
Eigenvalue
Difference
Proportion
Cumulative 1
0.6114 2
0.2067
0.8180 3
0.0886
0.9066 4
0.0573
0.9639 5
0.0258
0.9897 6
0.0103
1.0000
The bigger the eigenvalue is,
the more information this factor (component) carries.

10. A Visual Tool: Scree Plot
Variance Explained by Each Factor
Factor1
Factor2

11. Two Summary Factors Factor Pattern Factor1 Factor2 QUES1 Feel Blue
QUES2
People Stare at Me
QUES3
People Follow Me
QUES4
Basically Happy
QUES5
People Want to Hurt Me
QUES6
Enjoy Going to Parties

12. Final Communality Estimates: Total = 4.908255
Communalities
Communalities represent how much variance in the original variables is explained by all of the factors kept in the analysis (here the two factors)
Final Communality Estimates: Total =
QUES1
QUES2
QUES3
QUES4
QUES5
QUES6

13. Discussion
Q4 & Q6 should be at the same direction of factor 1 & 2 (component 1 & 2)
The other questions should be at the same direction of factor 1 & 2 (component 1 & 2)
Need a rotation!!

14. Rotation: Varimax Rotation
Orthogonal Transformation Matrix 1 2
Rotated Factor Pattern
Factor1
Factor2
QUES1
Feel Blue
QUES2
People Stare at Me
QUES3
People Follow Me
QUES4
Basically Happy
QUES5
People Want to Hurt Me
QUES6
Enjoy Going to Parties

15. Component Plot after Rotation
Plot of Factor Pattern for Factor1 and Factor2
Factor1 1 D F .8 .7 .6 .5 .4 .3 .2 .1
-.1 B
-.2 C
-.3 E
-.4
-.5
-.6
-.7
-.8
-.9 A -1
QUES1=A QUES2=B QUES3=C QUES4=D
QUES5=E QUES6=F
Component Plot after Rotation
Variance Explained by Each Factor
Factor1
Factor2

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

17. SAS Code PROC FACTOR DATA=FACTOR PREPLOT PLOT ROTATE=VARIMAX
NFACTORS=2
OUT=FACT SCREE;
TITLE "Example of Factor Analysis";
VAR QUES1-QUES6;
RUN;