N. Kumar, Asst. Professor of Marketing Database Marketing Factor Analysis

N. Kumar, Asst. Professor of Marketing

Database Marketing

Factor Analysis


Web Advertising

Objective: Identify the profile of customers who visit your website

Important information for advertisers who may wish to use your advertising services


Repositioning your Web Site

You may wish to learn of features that consumers value when browsing thro’ websitesAnalysis of consumer data may help uncover different facets (dimensions) of customers’ preferencesCan make a perceptual map to help form the basis of your strategy


How can Factor Analysis Help?

Often Factor Analysis can help summarize the information in many variables into a few underlying constructs/dimensions

Reduces the number of variables that you have to deal with little loss of information


Why Reduce Data?

Census Bureau – each zip code has more than 200 pieces of information

Typical customer survey on attitudes, lifestyles, opinions will probably have responses to more than 100 questions


Why Reduce Data … contd.

Too much data can be hard to absorb and comprehend

Difficult to work with too much data

Even if you can get it to work results will be distorted (multicollinearity problem) – regression example


What is Factor Analysis?

What is Factor Analysis?Factor analysis is a MV technique which analyzes the structure of the interrelationships among a large number of variables

Can identify the separate dimensions of the structure and can also determine the extent to which each variable is explained by each dimension


Factor Analysis: Intuitive Description

Factor Analysis summarizes information in Data by reducing original set of “items”/attributes to a smaller set of “factors”/“dimensions”/“constructs”

A Factor can be viewed as an “Index”:

Dow Jones Index -- summarizes the movement of stock market

Consumer Price Index -- reflects prices of consumer products and indicator of inflation

How to create such an “index” that appropriately summarizes the data with the minimum loss of information?


Factor Analysis: Intuitive Description (cont.)How does Factor Analysis work?

Factor Analysis “constructs” factors/axes by including original attributes with different weights

If the responses are rated almost identically for an attribute, Factor Analysis gives much lower weight

If two attributes, say attributes #3 and #4, are highly correlated i.e. stores which rate highly on attribute #3 are also rated high on #4, Factor Analysis treats #3 and #4 as measurements of the same underlying construct


Factor Analysis: e-admission

Data: Students’ scores on different subjects – say Physics, Chemistry, Math, History, English and French

Task at hand: to make an assessment about the student’s ability to succeed in school given these scores

Do we need to look at the scores on all subjects or can we use a simplified heuristic?


Single Factor Model

Suppose we could get something like this:

M = 0.8 I + Am P = 0.7 I + Ap

C = 0.9 I + Ac E = 0.6 I + Ae

H = 0.5 I + Ah F = 0.65 I + Af

A’s denote aptitude specific to the subject


Factor Analysis vs. Regression

RegressionHave data on I

Objective is to work out the weight on I

Factor Analysis I is the underlying construct that we are trying to work out


Some Terminology

Communality – that which is common with the variable and the underlying factor.

Formally, the square of the pattern loading

Unique/Specific Variance – that which is unexplained by the factor(s)


Input: Correlations

M P C E H F

M 1

P 0.56 1

C 0.72 0.63 1

E 0.48 0.42 0.54 1

H 0.40 0.35 0.45 0.30 1

F 0.52 0.46 0.59 0.39 0.33 1


Results

Variable Communality Unique Variance

Pattern Loading

M 0.640 0.360 0.8

P 0.490 0.510 0.7

C 0.810 0.190 0.9

E 0.360 0.640 0.6

H 0.250 0.750 0.5

F 0.423 0.577 0.65

Total 2.973 3.027


Two-Factor Model

Suppose we could get something like this:

M = 0.8 Q + 0.2 V +Am P = 0.7 Q + 0.3 V + Ap

C = 0.6 Q + 0.3 V +Ac E = 0.2 Q + 0.8 V + Ae

H = 0.15 Q + 0.82 V +Ah F = 0.25 Q + 0.85 V + Af

A’s denote aptitude specific to the subject


Results:

Variable Q V Unique

Variance

Q V

M 0.640 0.040 0.321 0.800 0.200

P 0.490 0.090 0.420 0.700 0.300

C 0.360 0.090 0.551 0.600 0.300

E 0.040 0.640 0.321 0.200 0.800

H 0.023 0.672 0.304 0.150 0.820

F 0.063 0.723 0.215 0.250 0.850

Total 1.616 2.255 2.132


Results: 2

Variable Q V Unique

Variance

Q V

M 0.445 0.234 0.321 0.667 -0.484

P 0.462 0.118 0.420 0.680 -0.343

C 0.378 0.071 0.551 0.615 -0.267

E 0.549 0.130 0.321 0.741 0.361

H 0.526 0.170 0.304 0.725 0.412

F 0.659 0.126 0.215 0.812 0.355

Total 3.019 0.849 2.132


Factor Analysis: Basic Concepts

Each original item (variable) is expressed as a linear combination of the underlying factors

X1 X4X2 X3OriginalItems

F1 F2UnderlyingFactors


Factor Analysis: Basic Concepts (cont.)

Each Factor can be expressed as a linear combination of the original items (variables)

X1 X4X2 X3OriginalItems

F1 F2UnderlyingFactors



Mathematical Model

Common Factors, F1, …, FM, can be expressed as linear combinations of the original variables, X1, …, XN

F1 = r11X1 + r12X2 + … + r1NXN

…………………………………………….. …………………………………………….. FM = rM1X1 + rM2X2 + … + rMNXN

rij = factor loading coefficient of the ith variable on the jth factor


Key Words

Factor Loading: Correlation of a factor with the original variable.

Communality: Variance of a variable summarized by the underlying factors

Eigenvalue (latent root): Sum of squares of loadings of each factor – just a measure of variance

e.g. the eigenvalue of factor 1, 1,

1 = r112 + r12

2 + … + r1M2




What does a Factor Analysis program do?

finds the factor loadings, ri1, ri2, … , riN, for each of the underlying factors , F1, …, FM, to “best explain” the pattern of interdependence among the original variables, X1, …, XN

How are Factor Loadings determined?

select the factor loadings, r11, r12, … , r1N, for the first factor so that Factor 1 “explains” the largest portion of the total variance

select the factor loadings, r21, r22, … , r2N, for the second factor so that Factor 2 “explains” the largest portion of the “residual” variance, subject to Factor 2 being orthogonal to Factor 1

so on ...


How many Factors do you Choose?

Look at the Eigen Values of the Factors

If K of P factors have an eigen value > 1 then K factors will do a pretty good job

Scree plot helpful


Scree Plot: Selection of # of Factors

6

5

4

3

2

1

2 4 6 8 10

“elbow”


Factor Analysis:Geometric Interpretation

ErrorF1

F2

x1


Illustrative Example: Measurement of Department Store Image

Description of the Research Study:

To compare the images of 5 department stores in Chicago area -- Marshal Fields, Lord & Taylor, J.C. Penny, T.J. Maxx and Filene’s Basement

Focus Group studies revealed several words used by respondents to describe a department store

e.g. spacious/cluttered, convenient, decor, etc.

Survey questionnaire used to rate the department stores using 7 point scale


Portion of Items Used to Measure Department Store Image

1. Convenient place to shop q q q q q q Inconvenient place to shop

2. Fast check out q q q q q q q Slow checkout

3. Store is clean q q q q q q q Store is dirty

4. Store is not well organized q q q q q q q Store is well organized

5. Store is messy, cluttered q q q q q q q Store is neat, uncluttered

6. Convenient store hours q q q q q q q Inconvenient store hours

7. Store is far from home, school or work

q q q q q q q Store is close to home, school, orhome

8. Store has bad atmosphere q q q q q q q Store has good atmosphere

9. Attractive decor inside q q q q q q q Unattractive decor inside

10. Store is spacious q q q q q q q Store is crowded


Department Store Image Measurement:Input Data

Store 1

Store 2

Store 3

Store 4

Store 5

Attribute 1 … Attribute 10

Respondents

… … …

… … …


Pair-wise Correlations among the Items Used to Measure Department Store Image

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10

X1 1.00 0.79 0.41 0.26 0.12 0.89 0.87 0.37 0.32 0.18

X2 1.00 0.32 0.21 0.20 0.90 0.83 0.31 0.35 0.23

X3 1.00 0.80 0.76 0.34 0.40 0.82 0.78 0.72

X4 1.00 0.75 0.30 0.28 0.78 0.81 0.80

X5 1.00 0.11 0.23 0.74 0.77 0.83

X6 1.00 0.78 0.30 0.39 0.16

X7 1.00 0.29 0.26 0.17

X8 1.00 0.82 0.78

X9 1.00 0.77

X10 1.00


Principal Components Analysis for the Department Store Image Data : Variance Explained by Each Factor

Factor Variance(Latent Root) Explained

Factor 1 5.725Factor 2 2.761 Factor 3 0.366Factor 4 0.357Factor 5 0.243Factor 6 0.212Factor 7 0.132Factor 8 0.123Factor 9 0.079Factor 10 0.001


Scree Plot: Selection of # of Factors

6

5

4

3

2

1

2 4 6 8 10

“elbow”


Unrotated Factor Loading Matrix for Department Store Image Data Using Two Factors

Factor

Variable 1 2Achieved

Communality123456789

10

0.6330.6210.8720.8330.7740.6260.6190.8590.8650.790

0.7070.695-0.241-0.366-0.4690.7190.683-0.303-0.293-0.454

.9000.8690.8190.8280.8180.9080.8500.8290.8350.831

Eigenvalue or latent root 5.725 2.761©1991 The Dryden Press. All rights reserved


Factor Loading Matrix for Department Store Image Data after Rotation of the Two Using Varimax

Factor

Variable 1 2Achieved

Communality123456789

10

0.1500.1470.8640.8990.9040.1380.1510.8860.8870.910

0.9370.9200.2690.1420.0240.9420.9090.2090.2210.045

0.9000.8690.8190.8280.8180.9080.8500.8290.8650.831

Eigenvalue or latent root 4.859 3.628©1991 The Dryden Press. All rights reserved.


Procedure for Conducting a Factor Analysis

Data Collection Step 1

Run Factor Analysis Step 2

Determine the Number of Factors Step 3


Rotate Factors Step 4

Interpret Factors Step 5

Calculate FactorScore Step 6

Do Other Stuff

Procedure for Conducting a Factor Analysis

Step 7


Product Differentiation & Positioning Strategy

Product Differentiation: creation of tangible or intangible differences on one or two key dimensions between a brand/product and its main competitors

Example: Toyota Corolla and Chevy Prizm are physically nearly identical cars and yet the Corolla is perceived to be superior to the Prizm

Product Positioning: set of strategies that firms develop and implement to ensure that these perceptual differences occupy a distinct and important position in customers’ minds

Example: KFC differentiates its chicken meal by using its unique blend of spices and cooking processes


Product Positioning & Perceptual Maps

Information Needed for Positioning Strategy: Understanding of the dimensions along which target customers

perceive brands in a category and how these customers perceive our offering relative to competition

How do our customers (current or potential) view our brand?Which brands do those customers perceive to be our closest competitors?What product and company attributes seem to be most responsible for these perceived differences?

Competitive Market Structure Assessment of how well or poorly our offerings are

positioned in the market


Product Positioning & Perceptual Maps (cont.)

Managerial Decisions & Action: Critical elements of a differential strategy/action plan

What should we do to get our target customer segment(s) to perceive our offering as different?Based on customer perceptions, which target segment(s) are most attractive?How should we position our new product with respect to our existing products?What product name is most closely associated with attributes our target segment perceives to be desirable

Perceptual Map facilitate differentiation & positioning decisions


Application Summary: Data Reduction

Identifying underlying dimensions, or FACTORS, that explain the correlation among a set of variables

e.g. a set of lifestyle statements may be used to measure the psychographic profiles of consumers

Statement 1…………….…………….Statement N

Life-style Statements

M < N PsychographicFactors

PsychographicProfiles


Understanding customer preferencesWhat dimensions to differentiate on to be successful – implications for repositioning or introduction strategy

Application Summary: Product Positioning/Introduction


Web Advertising

Understanding the profile of customers Conduct a survey

Analyze the data – extract the factors

Interpret the factors – score the customers

Can even draw a perceptual map of customers in the factor space


Repositioning your Web Site

To learn of features that consumers value when browsing thro’ websites – conduct a survey

Factor analyze the data to uncover the underlying factors that influence customers’ preferences – interpret the factors

How score on these dimensions relative to your competition - perceptual map to help form the basis of your strategy

Documents

N. Kumar, Asst. Professor of Marketing Database Marketing Factor Analysis