Happiness comes not from material wealth but less desire.

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Applied Statistics Using SAS

Topic: Factor Analysis

By Prof Kelly Fan, Cal State Univ, East Bay

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Outline

Introduction

Principal component analysis

Rotations

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Introduction

Reduce data

Summarize many ordinal categorical factors by a few combinations of them (new factors)

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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

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Example. 6 Questions

1. I usually feel blue.2. People often stare at me.3. I think that people are following me.4. I am usually happy.5. Someone is trying to hurt me.6. I enjoy going to parties.Q. Which questions will a depressed person

likely agree with? A happy person?

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Data Set:

Subj 1 2 3 4 5 6 7 8 9

Question

1 7 6 3 2 3 6 1 3 2

2 2 3 6 2 4 3 2 3 1

3 3 2 7 2 2 4 3 2 1

4 4 1 3 5 4 2 7 3 6

5 5 3 6 3 2 3 2 4 2

6 6 2 3 4 3 2 2 3 5

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Data Set:

Subj 10 11 12 13 14 15

Question

1 6 3 6 5 2 1

2 2 5 7 1 1 2

3 3 4 6 1 1 1

4 2 2 2 2 6 7

5 2 3 6 6 1 1

6 2 3 2 2 5 7

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Principal Component Analysis

The bigger the eigenvalue is, the more information this factor (component) carries.

Eigenvalues of the Correlation Matrix: Total = 6 Average = 1

Eigenvalue Difference Proportion Cumulative1 3.66827888 2.42830320 0.6114 0.6114

2 1.23997569 0.70866159 0.2067 0.8180

3 0.53131410 0.18729333 0.0886 0.9066

4 0.34402077 0.18927986 0.0573 0.9639

5 0.15474091 0.09307124 0.0258 0.9897

6 0.06166967 0.0103 1.0000

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A Visual Tool: Scree Plot

Scree Plot of Eigenvalues ‚ ‚ ‚ ‚ 4.0 ˆ ‚ ‚ ‚ 1 ‚ 3.5 ˆ ‚ ‚ ‚ ‚ 3.0 ˆ ‚ ‚ ‚ E ‚ i 2.5 ˆ g ‚ e ‚ n ‚ v ‚ a 2.0 ˆ l ‚ u ‚ e ‚ s ‚ 1.5 ˆ ‚ ‚ ‚ 2 ‚ 1.0 ˆ ‚ ‚ ‚ ‚ 0.5 ˆ 3 ‚ ‚ 4 ‚ 5 ‚ 6 0.0 ˆ ‚ ‚ ‚ Šƒƒƒƒˆƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒˆƒƒƒƒ 0 1 2 3 4 5 6 Number

Variance Explained by Each Factor

Factor1 Factor23.6682789 1.2399757

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Two Summary Factors

Factor Pattern

Factor1 Factor2QUES1 Feel Blue 0.76843 -0.54767

QUES2 People Stare at Me 0.72985 0.59840

QUES3 People Follow Me 0.77904 0.50692

QUES4 Basically Happy -0.87354 0.36879

QUES5 People Want to Hurt Me 0.72583 0.26237

QUES6 Enjoy Going to Parties -0.80519 0.34660

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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 = 4.908255

QUES1 QUES2 QUES3 QUES4 QUES5 QUES60.89042545 0.89075763 0.86386768 0.89907825 0.59566231 0.76846325

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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!!

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Rotation: Varimax Rotation

Orthogonal Transformation Matrix

1 21 -0.73625 0.67671

2 0.67671 0.73625

Rotated Factor Pattern

Factor1 Factor2QUES1 Feel Blue -0.93637 0.11677

QUES2 People Stare at Me -0.13241 0.93446

QUES3 People Follow Me -0.23053 0.90040

QUES4 Basically Happy 0.89271 -0.31960

QUES5 People Want to Hurt Me -0.35684 0.68434

QUES6 Enjoy Going to Parties 0.82737 -0.28969

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Component Plot after Rotation

Plot of Factor Pattern for Factor1 and Factor2 Factor1 1 D .9 F .8 .7 .6 .5 .4 .3 .2 .1 -1 -.9-.8-.7-.6-.5-.4-.3-.2-.1 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

-.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

Variance Explained by Each Factor

Factor1 Factor22.5562772 2.3519773

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Using Communalities Other Than One

When the original factors are not equally important

Different methods of “extraction”

SAS Code

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PROC FACTOR DATA=FACTOR PREPLOT PLOT ROTATE=VARIMAX

NFACTORS=2 OUT=FACT SCREE;

TITLE "Example of Factor Analysis"; VAR QUES1-QUES6;RUN;