Factor Analysis
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Transcript of Factor Analysis
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.
Factor Analysis3
Graphical Intuition: Factor Analysis works when data are correlated
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Factor Analysis4
Graphical Intuition: Factor Analysis will not work when variables are uncorrelated
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Figure 2
Factor Analysis5
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
Factor Analysis6
Perceptual Map… Example
Perceptual Map for Cars
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Fashion
Economy Taurus
VW Golf
Camry
Dodge Neon
Lexus ES 300
BMW325
Factor Analysis7
Applying Factor Analysis: Evaluating MBA Applications
Suppose school believes success of future managers depends on Intelligence Teamwork and Leadership skills
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 believe correlate with intelligence and teamwork and leadership skills?
Factor Analysis9
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
Factor Analysis10
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
Factor Analysis11
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
Factor Analysis12
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
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Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings
Extraction Method: Principal Component Analysis.
Factor Analysis13
SPSS Output of PCA: Scree Plot
Factor Analysis14
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
Factor Analysis15
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.
Factor Analysis16
Interpreting Factors: Looking at Loading Plot without Rotations
Loading Plot of GMAT-Extra without Rotations
Factor Analysis17
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
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Component
Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.
Rotation converged in 3 iterations.a.
Factor Analysis18
Interpreting Factors: Looking at Loading Plot with Rotation
Loading Plot of GMAT-Extra with Rotations
Factor Analysis19
Naming Factors
Apriori, theory based selection of variables Should be easy to name factors
Otherwise use managerial intuition
Factor Analysis20
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.
Factor Analysis21
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
Factor Analysis22
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
Factor Analysis23
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