Factor Analysis1

Factor Analysis (Optional Session)

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 larger

number of correlated variables into a smaller number of factors that are not correlated with each other.

In contrast to Regression, there is no dependent variable. We just look at the correlations between variables to summarize.

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Graphical Intuition: Factor Analysis works when data are correlated

A

BC

D

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Graphical Intuition: Factor Analysis will not work when 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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When to do Factor Analysis in business research?

Applications Eliminating Multicollinearity problems in

Regression Measuring managerially useful constructs

Intelligence, Leadership Skills, Customer satisfaction

Useful in constructing perceptual maps of products that are useful in positioning studies

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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

Economy Taurus

VW Golf

Camry

Dodge Neon

Lexus ES 300

BMW325

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Applying Factor Analysis: Evaluating MBA Applications

Suppose school believes success of future managers depends on Intelligence Teamwork and Leadership skills

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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 believe correlate with intelligence and teamwork and leadership skills?

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Data…

Appli-cant

GPA GMAT Scholar ship

Communication

Job Ex Org. skills

Extracurricular

1 3.7 680 3.5 4.4 4 3 2 2 3 20

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

Attribute GPA GMAT Fellowship Comm Job Ex Org Ex Extra Curr

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 0.12

Fellowsh 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 0.98 Org 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 as here… Factor analysis will be helpful

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

PCA uses the correlation matrix of the data and constructs 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 a variance of 1, so the total variance in n variables is n

Each factor will have an associated eigen-value which is the amount of variance explained by that factor

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SPSS Output of PCA: Eigen Analysis

85.9% of variance in 7 variables explained 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

Component1

2

3

4

5

6

7

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

Initial Eigenvalues Extraction Sums of Squared Loadings

Extraction Method: Principal Component Analysis.

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SPSS Output of PCA: Scree Plot

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Second Step: Do Factor Analysis with number of factors selected from Step 1

First interpret resulting factors Use factor loadings to interpret factors If it is not interpretable use rotation options

until we get something that can be interpreted

Look at factor equations and factor scores Score plots will be useful

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Why not Unrotated Factor Loadings? Variable’s 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

comm

jobex

organze

extra

1 2

Component

Extraction Method: Principal Component Analysis.

2 components extracted.a.

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Interpreting Factors: Looking at Loading Plot without Rotations

Loading Plot of GMAT-Extra without Rotations

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Rotated Factor Loadings and CommunalitiesVarimax Rotation 

Rotated Factor Loadings: Variable’s correlation with the factors

Rotated Component Matrixa

.954 .186

.963 -.048

.953 -.014

.698 .543

.187 .933

.013 .895

.007 .860

gmat

gpa

fellow

comm

jobex

organze

extra

1 2

Component

Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.

Rotation converged in 3 iterations.a.

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Interpreting Factors: Looking at Loading Plot with Rotation

Loading Plot of GMAT-Extra with Rotations

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Naming Factors

Apriori, theory based selection of variables Should be easy to name factors

Otherwise use managerial intuition

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How did applicants score on Intelligence and Leadership 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

Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization. Component Scores.

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Which Applicants to select for Haas: The Score Plot

Bookworms

Successful Applicants

No Good

Too Risky

-2-1

01

F2S

core

-2 -1 0 1 2F1Score

Too risky

Successfulapplicants

Book wormsSure rejects

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Step 1: Choosing number of factors to extract from data

Do Factor Analysis In SPSS select Analyze>Data

Reduction>Factor… Select “Extraction”, select “Principle

Component Analysis” Select the variables you want to factor analyze in

Variables box Select “Correlation” as the data that will be analyzed; this

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

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

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

Do Factor Analysis Number of Factors to extract should be from

Step 1 Try “None” rotation for a start (else try

Varimax or others if it doesn’t work) In Graphs: select loading plot and score plot In Storage: in the scores box store the factor

scores by selecting 2 variables