Factor Analysis

Istijanto MM, MCom

Definition

Factor analysis

Data reduction technique and summarization

Identifying the underlying factors/ dimensions

Based on interdependence relationships

Example:

Store image consists of six dimensions (article: “The alternative modes of measuring store image”)

StatisticsBartlett’s test of sphericity to test H0 that the variables are uncorrelated

Correlation matrix the correlation between all possible pairs of variables included in analysis

Eigenvalue total variance explained by each factor

Factor loadings correlation between the variables and the factors

Kaiser-Meyer-Olkin measure of sampling adequacy

Conducting Factor AnalysisFig. 19.3

Construction of the Correlation Matrix

Method of Factor Analysis

Determination of Number of Factors

Determination of Model Fit

Problem Formulation

Calculation ofFactor Scores

Interpretation of Factors

Rotation of Factors

Selection ofSurrogate Variables

Example: Malhotra, p.639

30 respondents were asked with 6 questions:

V1: It is important to buy a toothpaste that prevents cavities

V2: I like a toothpaste that gives shiny teeth

V3: A toothpaste should strengthen your gums

V4: I prefer a toothpaste that freshness breadth

V5: Preventions of tooth decay is not an important benefit offered by a toothpaste

V6: The most considerations in buying a toothpaste is attractive teeth

RQ: What are the underlying dimensions of a preferable toothpaste?

Using a seven-point scale (1=STS, 7=SS)

Procedures

Construct correlation matrix

Prerequisites:

Correlation matrix between variables: must be correlated

Barlett test of sphericity, H0: variables are uncorrelated, should be rejected

KMO, measure sampling adequacy, should be > 0,5

Method of factor analysis

Principal component analysis the total variance in the data is considered. (recommended)

Procedures (cont’d) Determine the number of factors

a priori determination (researcher’s knowledge)

based on eigenvalues: >1.0 retained

based on Scree Plot look at distinct break

based on percentage of variance: at least 60%

retained

Rotate factors

Factor matrix is transformed into simpler one

Use varimax procedure to enhance the interpretability of the factors

Intepret factor

Example: Malhotra, p.563

30 respondents were asked with 6 questions:

V1: It is important to buy a toothpaste that prevents cavities

V2: I like a toothpaste that gives shiny teeth

V3: A toothpaste should strengthen your gums

V4: I prefer a toothpaste that freshness breadth

V5: Preventions of tooth decay is not an important benefit offered by a toothpaste

V6: The most considerations in buying a toothpaste is attractive teeth

RQ: What are the underlying dimensions of a preferable toothpaste?

Using a seven-point scale (1=STS, 7=SS)

Conducting Factor AnalysisRESPONDENT

NUMBER V1 V2 V3 V4 V5 V61 7.00 3.00 6.00 4.00 2.00 4.002 1.00 3.00 2.00 4.00 5.00 4.003 6.00 2.00 7.00 4.00 1.00 3.004 4.00 5.00 4.00 6.00 2.00 5.005 1.00 2.00 2.00 3.00 6.00 2.006 6.00 3.00 6.00 4.00 2.00 4.007 5.00 3.00 6.00 3.00 4.00 3.008 6.00 4.00 7.00 4.00 1.00 4.009 3.00 4.00 2.00 3.00 6.00 3.00

10 2.00 6.00 2.00 6.00 7.00 6.0011 6.00 4.00 7.00 3.00 2.00 3.0012 2.00 3.00 1.00 4.00 5.00 4.0013 7.00 2.00 6.00 4.00 1.00 3.0014 4.00 6.00 4.00 5.00 3.00 6.0015 1.00 3.00 2.00 2.00 6.00 4.0016 6.00 4.00 6.00 3.00 3.00 4.0017 5.00 3.00 6.00 3.00 3.00 4.0018 7.00 3.00 7.00 4.00 1.00 4.0019 2.00 4.00 3.00 3.00 6.00 3.0020 3.00 5.00 3.00 6.00 4.00 6.0021 1.00 3.00 2.00 3.00 5.00 3.0022 5.00 4.00 5.00 4.00 2.00 4.0023 2.00 2.00 1.00 5.00 4.00 4.0024 4.00 6.00 4.00 6.00 4.00 7.0025 6.00 5.00 4.00 2.00 1.00 4.0026 3.00 5.00 4.00 6.00 4.00 7.0027 4.00 4.00 7.00 2.00 2.00 5.0028 3.00 7.00 2.00 6.00 4.00 3.0029 4.00 6.00 3.00 7.00 2.00 7.0030 2.00 3.00 2.00 4.00 7.00 2.00

Table 19.1

Correlation Matrix

1.000 -.053 .873 -.086 -.858 .004

-.053 1.000 -.155 .572 .020 .640

.873 -.155 1.000 -.248 -.778 -.018

-.086 .572 -.248 1.000 -.007 .640

-.858 .020 -.778 -.007 1.000 -.136

.004 .640 -.018 .640 -.136 1.000

V1

V2

V3

V4

V5

V6

CorrelationV1 V2 V3 V4 V5 V6

Construct correlation matrix

KMO and Bartlett's Test

.660

111.314

15

.000

Kaiser-Meyer-Olkin Measure of SamplingAdequacy.

Approx. Chi-Square

df

Sig.

Bartlett's Test ofSphericity

> 0.5

Ho tolak

Method of Factor analysisCommunalities

1.000 .926

1.000 .723

1.000 .894

1.000 .739

1.000 .878

1.000 .790

V1

V2

V3

V4

V5

V6

Initial Extraction

Extraction Method: Principal Component Analysis.Total Variance Explained

2.731 45.520 45.520 2.731

2.218 36.969 82.488 2.218

.442 7.360 89.848

.341 5.688 95.536

.183 3.044 98.580

8.521E-02 1.420 100.000

Component1

2

3

4

5

6

Total % of Variance Cumulative % Total

Initial Eigenvalues Extraction Sums of Squared Loadings

Extraction Method: Principal Component Analysis.

Determine the number of factorsTotal Variance Explained

2.731 45.520 45.520 2.731

2.218 36.969 82.488 2.218

.442 7.360 89.848

.341 5.688 95.536

.183 3.044 98.580

8.521E-02 1.420 100.000

Component1

2

3

4

5

6

Total % of Variance Cumulative % Total

Initial Eigenvalues Extraction Sums of Squared Loadings

Extraction Method: Principal Component Analysis.

retained

Rotate factorsComponent Matrixa

.928 .253

-.301 .795

.936 .131

-.342 .789

-.869 -.351

-.177 .871

V1

V2

V3

V4

V5

V6

1 2

Component

Extraction Method: Principal Component Analysis.

2 components extracted.a.

Rotated Component Matrixa

.962 -2.66E-02

-5.72E-02 .848

.934 -.146

-9.83E-02 .854

-.933 -8.40E-02

8.337E-02 .885

V1

V2

V3

V4

V5

V6

1 2

Component

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

Rotation converged in 3 iterations.a.

Sometimes, difficult to interpret

Easier and simpler to interpret

Interpret factors

Component Plot in Rotated Space

Component 1

1.0.50.0-.5-1.0

Co

mp

on

en

t 2

1.0

.5

0.0

-.5

-1.0

v6

v5

v4

v3

v2

v1

Factor 1: V1, V3, V5 health benefit factor/ dimension

Factor 2: V2, V4, V6 social benefit factor/ dimension

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