factor analysis 2

7/31/2019 Factor Analysis 2

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Factor Analysis1

Factor Analysis (Optional Session)


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Factor Analysis2

What is Factor Analysis

Data Reduction Technique

A factor is a weighted sum of the variables

The goal is to summarize the information in a largernumber of correlated variables into a smaller number of

factors that are not correlated with each other.

In contrast to Regression, there is no dependentvariable. We just look at the correlations between

variables to summarize.


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Factor Analysis3

Graphical Intuition: Factor Analysis workswhen data are correlated

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2.0

2.1

2.2

2.3

X

Y

Figure 1

A

BC

D


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Factor Analysis4

Graphical Intuition: Factor Analysis will not workwhen variables are uncorrelated

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

X

Y

Figure 2


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Factor Analysis5

When to do Factor Analysis in businessresearch?

Applications

Eliminating Multicollinearity problems inRegression

Measuring managerially useful constructs

Intelligence, Leadership Skills, Customersatisfaction

Useful in constructing perceptual maps ofproducts that are useful in positioning studies


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Factor Analysis6

Perceptual Map Example

Perceptual Map for Cars

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5 2

Fashion

EconomyTaurus

VW Golf

Camry

Dodge Neon

Lexus ES 300

BMW325


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Factor Analysis7

Applying Factor Analysis: EvaluatingMBA Applications

Suppose school believes success offuture managers depends on

Intelligence

Teamwork and Leadership skills


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Factor Analysis8

Applying Factor Analysis:

Evaluating MBA Applications

Variables available

GPA

GMAT score

Scholarships, fellowships won Evidence of Communications skills

Prior Job Experience

Organizational Experience

Other extra curricular achievements

Which variables

do you believecorrelate withintelligence andteamwork and

leadership skills?


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Factor Analysis9

Data

Appli-

cant

GPA GMAT Scholar

ship

Commun

ication

Job Ex Org.

skills

Extracur

ricular

1 3.7 680 3.5 4.4 4 3 2

2

3

20


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Factor Analysis10

Quick and dirty sense of the data: Looking atthe correlation matrix

Attribute GPA GMAT Fellowship Comm Job Ex Org Ex ExtraCurr

GPA 1.00 0.97 0.96 0.43 0.05 -0.05 -0.12

GMAT 0.97 1.00 0.99 0.55 0.27 0.16 0.12Fellowsh 0.96 0.99 1.00 0.47 0.19 0.07 0.05

Comm 0.43 0.55 0.47 1.00 0.82 0.79 0.69

Job Ex 0.05 0.27 0.19 0.82 1.00 0.99 0.98Org Ex -0.05 0.16 0.07 0.79 0.99 1.00 0.97

Extra Cur -0.12 0.12 0.05 0.69 0.98 0.97 1.00

Even if data is not as neatly correlated ashere Factor analysis will be helpful


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Factor Analysis11

First Step: Do Principal Component Analysis(PCA) to select # of factors

PCA uses the correlation matrix of the data andconstructs factors

Factors

If there are n variables we will have n factors First factor will explain most variance, second next,

and so on

Variance Explained by Factors

With standardized variables each variable has avariance of 1, so the total variance in n variables is n

Each factor will have an associated eigen-valuewhich is the amount of variance explained by thatfactor


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Factor Analysis 12

SPSS Output of PCA: Eigen Analysis

85.9% of variance in 7 variablesexplained by just 2 factors

Total Variance Explained

3.744 53.480 53.480 3.744 53.480 53.480

2.268 32.398 85.878 2.268 32.398 85.878.425 6.069 91.948

.288 4.113 96.060

.140 1.994 98.054

.098 1.406 99.460

.038 .540 100.000

Component

1

23

4

5

6

7

Total % of Variance Cumulative % Total % of Variance Cumulative %

Init ia l Eigenvalues Extraction Sums of Squared Loadings

Extraction Method: Principal Component Analysis.


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Factor Analysis 13

SPSS Output of PCA: Scree Plot


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Factor Analysis 14

Second Step: Do Factor Analysis withnumber of factors selected from Step 1

First interpret resulting factors

Use factor loadings to interpret factors

If it is not interpretable use rotation optionsuntil we get something that can beinterpreted

Look at factor equations and factor

scores Score plots will be useful


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Factor Analysis 15

Why not Unrotated Factor Loadings?Variables correlation with the factors

Unrotated Factor Loadings and Communalities

Component Matrixa

.891 -.388

.766 -.586

.777 -.552

.883 .052

.683 .662

.518 .730

.493 .705

gmat

gpa

fellow

commjobex

organze

extra

1 2Component

Extraction Method: Principal Component Analysi

2 components extracted.a.


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Factor Analysis 16

Interpreting Factors: Looking atLoading Plot without Rotations

Loading Plot of GMAT-Extra without Rotations


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Factor Analysis 17

Rotated Factor Loadings and Communalities

Varimax Rotation

Rotated Factor Loadings: Variables

correlation with the factors

Rotated Component Matrixa

.954 .186

.963 -.048

.953 -.014

.698 .543

.187 .933

.013 .895

.007 .860

gmatgpa

fellow

comm

jobex

organze

extra

1 2

Component


Rotation Method: Varimax with Kaiser Normalizatio

Rotation converged in 3 iterations.a.


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Factor Analysis 18

Interpreting Factors: Looking atLoading Plot with Rotation

Loading Plot of GMAT-Extra with Rotations


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Factor Analysis 19

Naming Factors

Apriori, theory based selection ofvariables

Should be easy to name factors

Otherwise use managerial intuition


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Factor Analysis 20

How did applicants score on Intelligence andLeadership Factors

Intelligence=0.293 GMAT + 0.315 GPA + 0.309 Fellowships +

0.181 Communications - 0.015 Job Ex

- 0.068 Organizational Skills

- 0.068 ExtraCurricular

Leadership= - 0.006 GMAT - 0.097 GPA - 0.083 Fellowships

+ 0.153 Communications + 0.344 Job Ex

+ 0.343 Organizational Skills

+ 0.331 ExtraCurricular

Component Score Coefficient Matrix

.293 -.006

.315 -.097

.309 -.083

.181 .153

-.015 .344

-.068 .343

-.068 .331

gmat

gpa

fellow

comm

jobex

organze

extra

1 2

Component


Rotation Method: Varimax with Kaiser Normalizati

Com onent Scores.


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Factor Analysis 21

Which Applicants to select for Haas: TheScore Plot

Bookworms

Successful Applicants

No Good

Too Risky

-2

-1

0

1

-2 -1 0 1 2

F1Score

Too risky

Successfulapplicants

Book worms

Surerejects


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Factor Analysis 22

Step 1: Choosing number of factors toextract from data

Do Factor Analysis

In SPSS select Analyze>Data

Reduction>Factor

Select Extraction, select PrincipleComponent Analysis

Select the variables you want to factor analyze inVariables box

Select Correlation as the data that will be analyzed; this

will mean that the data will be standardized and thereforeeach variable will have equal effect.

Ask for Scree Plot (using Graphs button) which graphs theamount of variance explained by each factor


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Factor Analysis 23

Step 2: Performing Factor Analysis with #of factors from Step 1

Do Factor Analysis

Number of Factors to extract should be fromStep 1

Try None rotation for a start (else try

Varimax or others if it doesnt work)

In Graphs: select loading plot and score plot

In Storage: in the scores box store the factorscores by selecting 2 variables

factor analysis 2

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