Kuliah 3 - Factor Analysis1

8/7/2019 Kuliah 3 - Factor Analysis1

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Factor Analysis

PLG 701

Lecture 3


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Factor analysis examines the interrelationships among a largenumber of variables and then attempts to explain them in

terms of their common underlying dimensions.

These common underlying dimensions are referred to as

factors.

Factor analysis is a summarization and data reduction

technique that does not have independent and dependent

variables, but an interdependence technique in which allvariables are considered simultaneously.

Hair et al. (2006)

WHAT IS ?WHAT IS ?


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What is FA?

FA is a statistical technique used to identify a

relatively small number of factors that can be

used to represent relationship among sets of

many interrelated variables


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WHAT IS ?

Statistical techniques for identifying

interrelationships between items with the

goal of identifying items that group or cluster

together.


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The basic assumption of FA is that underlying

dimensions or factors can be used to explain

complex phenomena. Observed correlations

between variables result from their sharing of

these factors. For example, correlation

between test scores might be attributable to

such shared factors as general intelligence,abstract reasoning skills, and reading

comprehension.

What is FA?


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The Purpose of Factor Analysis

The purpose of factor analysis is to discover

simple patterns in the pattern of relationships

among the variables. In particular, it seeks todiscover if the observed variables can be

explained largely or entirely in terms of a

much smaller number of variables called

factors.


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Factor Analysis....(cont.)

Many statistical methods are used to study the relation

between independent and dependent variables.

Factor analysis is different; it is used to study the patterns of

relationship among many dependent variables, with the goal

of discovering something about the nature of the

independent variables that affect them, even though those

independent variables were not measured directly.

Thus answers obtained by factor analysis are necessarily more

hypothetical and tentative than is true when independentvariables are observed directly. The inferred independent

variables are called factors.


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Factor Analysis....(cont.)


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Analysis of Interdepence

Data Reduction Identification of

Structures: Is Analysis Exploratory or

Confirmatory?

Grouping Target: Variable or Case?

R-Type Factor

Analysis

Structural

EquationModeling

Q-type Factor

Analysis or

Cluster Analysis

Confirmatory

Cases

Exploratory

Variables


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HistoryHistory

The method of factor analysis originated with

C. Spearman

and G.H. Thomson

L.L. Thurstone

J. B. Carroll K. Joreskog

have contributed to further developments


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There are two approaches to calculate the correlation matrix

that determine the type of factor analysis performed:

R-type factor analysis: input data matrix is computed from

correlations between variables.

Q-type factor analysis: input data matrix is computed from

correlations between individual respondents.

Variables in factor analysis are generally metric.

Designing a Factor Analysis


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Qfactor analysis

If the data available involve a relatively small number of

persons with many measurements on these persons it

is possible to undertake a Qfactor analysis to cluster

persons

n

N

n

N

Ordinary factor analysis

Q factor analysis

N = number of personsn = number of variates


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Steps in a Factor Analysis

First, the correlation matrix for all variables is computed.

Variables that do not appear to be related to other variables

can be identified. The appropriateness of the factor model

can also be evaluated.

Second step, factor extraction - the number of factors

necessary to represent the data and the method of calculating

them must be determined.

The third step, rotation, focuses on transforming the factors to

make them more interpretable.

At the fourth step, scores for each factor can be computed for

each case. These scores can then be used in a variety of other

analyses.


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General Steps to FA (Hair et al., 2006)

Step 1: Selecting and Measuring a set of variablesin a given domain

Step 2: Data screening in order to prepare the

correlation matrix

Step 3: Factor Extraction

Step 4: Factor Rotation to increase

interpretability Step 5: Interpretation

Further Steps: Validation and Reliability of themeasures


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Rulesof ThumbRulesof Thumb 3311

Factor AnalysisDesignFactor AnalysisDesignyy Factoranalysisisperformedmostoftenonlyonmetricvariables,Factoranalysisisperformedmostoftenonlyonmetricvariables,

although specializedmethodsexistfortheuseofdummyalthough specializedmethodsexistfortheuseofdummy

variables. A smallnumberofdummyvariablescanbeincludedvariables. A smallnumberofdummyvariablescanbeincludedinasetofmetricvariablesthatarefactoranalyzed.inasetofmetricvariablesthatarefactoranalyzed.

yy Ifastudyisbeingdesignedtorevealfactorstructure,strivetoIfastudyisbeingdesignedtorevealfactorstructure,strivetohaveatleastfive variablesforeach proposedfactor.haveatleastfive variablesforeach proposedfactor.

yy Forsamplesize:Forsamplesize:oo thesamplemusthavemoreobservationsthanvariables.thesamplemusthavemoreobservationsthanvariables.

oo theminimumabsolutesamplesizeshouldbetheminimumabsolutesamplesizeshouldbe 5050observations.observations.

yy Maximizethenumberofobservationspervariable, with aMaximizethenumberofobservationspervariable, with aminimumoffiveandhopefullyatleasttenobservationsperminimumoffiveandhopefullyatleasttenobservationspervariable.variable.


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RulesofThumbRulesofThumb 3322

Testing AssumptionsofFactorAnalysisTesting AssumptionsofFactorAnalysis

yy TheremustbeastrongconceptualfoundationtosupporttheTheremustbeastrongconceptualfoundationtosupporttheassumptionthatastructuredoesexistbeforethefactorassumptionthatastructuredoesexistbeforethefactoranalysisisperformed.analysisisperformed.

yy A statisticallysignificantBartlettstestofsphericity(sig. >A statisticallysignificantBartlettstestofsphericity(sig. >.05) indicatesthatsufficientcorrelationsexistamongthe.05) indicatesthatsufficientcorrelationsexistamongthevariablesto proceed.variablesto proceed.

yy MeasureofSampling Adequacy(MSA) valuesmustexceed.50MeasureofSampling Adequacy(MSA) valuesmustexceed.50forboth theoveralltestandeach individualvariable. Variablesforboth theoveralltestandeach individualvariable. Variableswith valueslessthan.50 shouldbeomittedfromthefactorwith valueslessthan.50 shouldbeomittedfromthefactoranalysisoneatatime, with thesmallestonebeingomittedanalysisoneatatime, with thesmallestonebeingomittedeach time.each time.


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A principal component factor analysis requires:

The variables included must be metric level or dichotomous(dummy-coded) nominal level

The sample size must be greater than 50 (preferably 100)

The ratio of cases to variables must be 5 to 1 or larger

The correlation matrix for the variables must contain 2 or morecorrelations of 0.30 or greater

Variables with measures of sampling adequacy less than 0.50must be removed

The overall measure of sampling adequacy is 0.50 or higher

The Bartlett test of sphericity is statistically significant. The first phase of a principal component analysis is devoted to

verifying that we meet these requirements. If we do not meet theserequirements, factor analysis is not appropriate.


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The second phase of a principal component factor analysisfocuses on deriving a factor model, or pattern of relationshipsbetween variables and components, that satisfies the followingrequirements:

The derived components explain 50% or more of the variance in each of

the variables, i.e. have a communality greater than 0.50 None of the variables have loadings, or correlations, of 0.40 or higher for

more than one component, i.e. do not have complex structure

None of the components has only one variable in it

To meet these requirements, we remove problematic variablesfrom the analysis and repeat the principal component analysis.


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NumberofFactors?NumberofFactors?

A PrioriCriterion.A PrioriCriterion. LatentRootCriterion.LatentRootCriterion.

PercentageofVariance.PercentageofVariance.

Scree TestCriterion.Scree TestCriterion.


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NumberofFactors?NumberofFactors?

A PrioriCriterion.A PrioriCriterion. LatentRootCriterion.LatentRootCriterion.

PercentageofVariance.PercentageofVariance.

Scree TestCriterion.Scree TestCriterion.


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Eigenvalue PlotforScree TestCriterionEigenvalue PlotforScree TestCriterion


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Number of Factor Extracted

Eigenvalue > 1.0

Scree plot


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RotationofFactors

Factorrotation = thereferenceaxesoftheFactorrotation = thereferenceaxesofthefactorsaretunedabouttheoriginuntilsomefactorsaretunedabouttheoriginuntilsome

otherpositionhasbeenreached. Sinceunrotatedotherpositionhasbeenreached. Sinceunrotatedfactorsolutionsextractfactorsbasedonhowfactorsolutionsextractfactorsbasedonhowmuch variancetheyaccountfor, with eachmuch variancetheyaccountfor, with eachsubsequentfactoraccountingforlessvariance,subsequentfactoraccountingforlessvariance,theultimateeffectofrotatingthefactormatrixistheultimateeffectofrotatingthefactormatrixistoredistributethe variancefromearlierfactorstotoredistributethe variancefromearlierfactorstolateronestoachieveasimpler,theoreticallymorelateronestoachieveasimpler,theoreticallymoremeaningfulfactorpattern.meaningfulfactorpattern.


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Choosing Factor Models and Number of FactorsChoosing Factor Models and Number of Factors Although both component and common factor analysis models yield similarAlthough both component and common factor analysis models yield similar

results in common research settings (results in common research settings (3030 or more variables or communalitiesor more variables or communalities

of .of .6060 for most variables):for most variables): the component analysis model is most appropriate when data reduction isthe component analysis model is most appropriate when data reduction is

paramount.paramount. the common factor model is best in wellthe common factor model is best in well--specified theoretical applications.specified theoretical applications.

Any decision on the number of factors to be retained should be based onAny decision on the number of factors to be retained should be based onseveral considerations:several considerations: use of several stopping criteria to determine the initial number of factors to retain.use of several stopping criteria to determine the initial number of factors to retain.

Factors With Eigenvalues greater thanFactors With Eigenvalues greater than 11..00..

A preA pre--determined number of factors based on research objectives and/or priordetermined number of factors based on research objectives and/or prior

research.research.

Enough factors to meet a specified percentage of variance explained, usuallyEnough factors to meet a specified percentage of variance explained, usually 6060%%or higher.or higher.

Factors shown by the scree test to have substantial amounts of common varianceFactors shown by the scree test to have substantial amounts of common variance

(i.e., factors before inflection point).(i.e., factors before inflection point).

More factors when there is heterogeneity among sample subgroups.More factors when there is heterogeneity among sample subgroups.

Consideration of several alternative solutions (one more and one less factorConsideration of several alternative solutions (one more and one less factorthan the initial solution) to ensure the best structure is identified.than the initial solution) to ensure the best structure is identified.


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TwoRotationalApproaches:

1.1. Orthogonal = axesareOrthogonal = axesaremaintainedatmaintainedat9090 degrees.degrees.

2.2. Oblique = axesarenotOblique = axesarenotmaintainedatmaintainedat9090 degrees.degrees.


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OrthogonalFactorRotationOrthogonalFactorRotation

UnrotatedUnrotatedFactorIIFactorII

UnrotatedUnrotatedFactorIFactorI

RotatedRotatedFactorIFactorI

RotatedFactorIIRotatedFactorII

-1.0 -.50 0 +.50 +1.0

-.50

-1.0

+1.0

+.50

V1

V2

V3

V4

V5


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UnrotatedUnrotatedFactorIIFactorII

UnrotatedUnrotatedFactorIFactorI

ObliqueObliqueRotation:Rotation:FactorIFactorI

OrthogonalOrthogonal

Rotation: FactorIIRotation: FactorII

-1.0 -.50 0 +.50 +1.0

-.50

-1.0

+1.0

+.50

V1

V2

V3

V4

V5

OrthogonalOrthogonalRotation: FactorIRotation: FactorI

ObliqueRotation:ObliqueRotation:FactorIIFactorII

Oblique FactorRotationOblique FactorRotation


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Choosing Factor Rotation MethodsChoosing Factor Rotation Methods

yy Orthogonal rotation methods:Orthogonal rotation methods:

oo are the most widely used rotational methods.are the most widely used rotational methods.

oo are The preferred method when the research goal isare The preferred method when the research goal is

data reduction to either a smaller number of variables ordata reduction to either a smaller number of variables or

a set of uncorrelated measures for subsequent use ina set of uncorrelated measures for subsequent use in

other multivariate techniques.other multivariate techniques.

yy Oblique rotation methods:Oblique rotation methods:

oo best suited to the goal of obtaining several theoreticallybest suited to the goal of obtaining several theoretically

meaningful factors or constructs because, realistically,meaningful factors or constructs because, realistically,

very few constructs in the real world are uncorrelated.very few constructs in the real world are uncorrelated.


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RulesofThumb 3RulesofThumb 355

Assessing Factor LoadingsAssessing Factor Loadings

While factor loadings ofWhile factor loadings of ++..3030 toto ++..4040 are minimally acceptable,are minimally acceptable,values greater thanvalues greater than ++ ..5050 are considered necessary for practicalare considered necessary for practical

significance.significance.

To be considered significant:To be considered significant:oo A smaller loading is needed given either a larger sample size, orA smaller loading is needed given either a larger sample size, or

a larger number of variables being analyzed.a larger number of variables being analyzed.

oo A larger loading is needed given a factor solution with a largerA larger loading is needed given a factor solution with a larger

number of factors, especially in evaluating the loadings on laternumber of factors, especially in evaluating the loadings on later

factors.factors.

Statistical tests of significance for factor loadings are generallyStatistical tests of significance for factor loadings are generallyvery conservative and should be considered only as starting pointsvery conservative and should be considered only as starting points

needed for including a variable for further consideration.needed for including a variable for further consideration.


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Factor Loadings Interpretation

Factor Loading is the correlation of the variable/item and the factor

the squared loading = amount of the variables total variance

accounted for by the factor

Minimum Factor Loading Significance: s0.30 (} 10% variance

accounted by factor)

The larger the size, the more important the loading in interpreting

the factor matrix

Extremely large factor loadings (u0.80) are not typical

Therefore emphasis is on practical significance

The guidelines are applicable when N u 100


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Interpreting The FactorsInterpreting The Factors

yy An optimal structure exists when all variables have high loadingsAn optimal structure exists when all variables have high loadingsonly on a single factor.only on a single factor.

yy Variables that crossVariables that cross--load (load highly on two or more factors) areload (load highly on two or more factors) areusually deleted unless theoretically justified or the objective isusually deleted unless theoretically justified or the objective is

strictly data reductionstrictly data reduction..

yy Variables should generally have communalities of greater than .Variables should generally have communalities of greater than .5050to be retained in the analysis.to be retained in the analysis.

yy Respecification of a factor analysis can include options such as:Respecification of a factor analysis can include options such as:

oo deleting a variable(s),deleting a variable(s),

oo changing rotation methods, and/orchanging rotation methods, and/or

oo increasing or decreasing the number of factors.increasing or decreasing the number of factors.


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Goodness ofMeasures 43

Factor Analysis SPSS


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HATCO DataHATCO Data

Correlation Matrix

1.000 -.349 .509 .050 .612 .077 -.483

-.349 1.000 -.487 .272 .513 .186 .470

.509 -.487 1.000 -.116 .067 -.034 -.448

.050 .272 -.116 1.000 .299 .788 .200

.612 .513 .067 .299 1.000 .241 -.055

.077 .186 -.034 .788 .241 1.000 .177

-.483 .470 -.448 .200 -.055 .177 1.000

Delivery Speed

Price Level

Price Flexibility

Manufacturer Image

Service

Salesf rce Image

Pr duct Quality

Delivery

Speed

Price

Level

Price

Flexibility

Manufacturer

Image Service

Salesf rce

Image

Pr duct

Quality

KMO and Bartlett's Test

.446

567.541

21

.000

Kaiser-Meyer-Olkin Measure f Sampling

Adequacy.

Appr x. Chi-Square

df

Sig.

Bartlett's Test f

Sphericity

Low value indicating that

there might not be sufficient

number of significant

correlations to conduct

Factor analysis

Typically any correlation less

than 0.3 is insignificant

The above indicates that there might be some variables that should not

be included in the Factor analysis


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HATCO DataAn i-image Ma ices

2.797E-02 2.847E-02 2.380E-03 1.464E-02 -2.47E-02 -6.116E-03 -2.13E-03

2.847E-02 3.165E-02 2.152E-02 1.403E-02 -2.62E-02 -4.851E-03 -1.98E-02

2.380E-03 2.152E-02 .608 4.372E-02 -1.07E-02 -4.045E-02 8.592E-02

1.464E-02 1.403E-02 4.372E-02 .347 -1.54E-02 -.275 -1.81E-02

-2.47E-02 -2.621E-02 -1.07E-02 -1.538E-02 2.281E-02 4.756E-03 1.044E-02

-6.12E-03 -4.851E-03 -4.05E-02 -.275 4.756E-03 .371 -4.41E-02-2.13E-03 -1.981E-02 8.592E-02 -1.815E-02 1.044E-02 -4.414E-02 .623

.344a .957 1.825E-02 .149 -.978 -6.005E-02 -1.62E-02

.957 .330a .155 .134 -.975 -4.478E-02 -.141

1.825E-02 .155 .913a 9.514E-02 -9.13E-02 -8.520E-02 .140

.149 .134 9.514E-02 .558a -.173 -.766 -3.90E-02

-.978 -.975 -9.13E-02 -.173 .288a 5.171E-02 8.762E-02

-6.01E-02 -4.478E-02 -8.52E-02 -.766 5.171E-02 .552a -9.19E-02

-1.62E-02 -.141 .140 -3.903E-02 8.762E-02 -9.186E-02 .927a

Delivery Speed

Price Level

Price Flexibility

Manufacturer Image

Service

Salesforce ImageProduct Quality

Delivery Speed

Price Level

Price Flexibility

Manufacturer Image

Service

Salesforce Image

Product Quality

Anti-image Covariance

Anti-image Correlation

Delivery

Speed Price Level

Price

Flexibility

Manufacturer

Image Service

Salesforce

Image

Product

Quality

Measures of Sampling Adequacy(MSA)a.

MSA< 0.5 indicates that these variables that should not be included in the

Factor analysis. Delete one at a time pick the lowest value, and repeat the

analysis excluding X5


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Before deleting

Tot l V ri c E pl i d

2. 26 36. 2 36. 2 2. 26 36. 2 36. 2 2.379 33.984 33.984

2. 20 30.291 66.374 2.120 30.291 66.374 1.827 26.098 60.082

1.181 16.873 83.246 1.181 16.873 83.246 1.622 23.165 83.246

.541 7.731 90.977

.418 5.972 96.949

.204 2.920 99.869

.009 .131 100.000

Component

1

2

3

4

5

6

7

Tot l%of

V ri nce Cumul ti e% Tot l%of

V ri nce Cumul ti e% Tot l%of

V ri nce Cumul ti e%

Initi l Ei envalues E traction umsof uared oadings Rotation umsof uared oadings

E traction et od: rincipalComponent nal sis.


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CommunalityCommunality

Communal t es

1 000 884

1 000 895

1 000 649

1 000 885

1 000 995

1 000 901

1 000 618

x1 Delive y Speed

x2 P i e Level

x3 P i e Flexibili y

x4 Manu a u e age

x5 Se vi e

x6 Sale o e age

x7 P odu uali y

ni ial Ex a ion

Ex a ion Me hod: P in ipal Co ponen Analy i

Amount of shared, or

common variance, among

the variables

General guidelines should

be above 0.5


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Component Matrix

Component t x

-.528 .752 .202

.792 .093 .508

-.692 .374 -.173

.564 .602 -.452

.186 .779 .595

.492 .604 -.542

.739 -.270 -.005

x1 Delivery Speed

x2 Price Level

x3 Price Flexibility

x4 Manufacturer Image

x5 Service

x6 Salesforce Image

x7 Product Quality

1 2 3

Component

Extraction Method: Principal Component Analysis.

3 components extracted.a.

Rotated Component Matri a

-.752 .071 .560

.754 .108 .561

-.806 .006 .010

.117 .921 .153

-.062 .176 .980

.034 .945 .077

.760 .193 -.064

x1 Delivery Speed

x2 Price Level

x3 Price Flexibility

x4 Manufacturer Image

x5 Service

x6 Salesforce Image

x7 Product Quality

1 2 3

Component

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

Rotation converged in 5 iterations.a.

Unrotated Rotated


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Communality

Co unalities

1. .658

1. .580

1. .646

1. .8821. .872

1. .616

DeliverySpeed

Pri e evel

Pri e lexi ility

Manufacturer ImageSalesfor e Image

Product uality

Initial Extraction

ExtractionMethod Principal Component Analysis.

Amount of shared, or

common variance, among

the variables

General guidelines should

be above 0.5


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HATCO Data

Tot l Variance Explained

2. . 2 41. 2 2. .497 39.497

1.740 28.992 70.883 1.883 31.386 70.883

Co o t

1

2

Tot l of V ri c Cumul ti Tot l of V ri c Cumul ti

tr ctio ums of quar oadi s Rotatio ums of quar d oadi s

tractio t od: ri ci al Compo t A al sis.

These represents the percentage of variance in X1, X2, X3, X4, X6,

X7 captured by the two factors. Cumulatively the two factors

captured 71% of the variance, which is quite high

When reporting factor results you need to also include thepercentage of variance explained beneath each factor

The next issue: Which items to which factor? This assignment must

be done uniquely; I.e. one item can be loaded onto only one and

only one factor.


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HATCO DataComponent t x

-.627 .514

.759 -6.79E-02

-.730 .337.494 .798

.425 .832

.767 -.168

Delivery Speed

Price Level

Price FlexibilityManufacturer Image

Salesforce Image

Product Quality

1 2

Component

Extraction Method: Principal Component Analys

2 components extracted.a.

The purpose is to assign

uniquely each item to

only one factor.

We do this by looking at

the factor loadings They represent the

correlation between the

item and the factor.

Thus when an item has

significant ( > 0.3)loadings on more than

one factor, then we see

the problem ofcross-

loadings.

The above pattern of cross-loadings especially

for X1, X3, X4, X6 indicates that these 4 items

cannot be uniquely assigned to either one of

these two factors.

To get a clearer picture, we do a rotation


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HATCO Data After varimax rotation,

we can see a clearer

pattern of assignment

with minimal problem of

cross loadings

Factor 1 consists of X1,X2, X3, and X7 which

refers to physical quality

attributes. Thus it can be

labeled as Physical

QualityPerceptions

Factor 2 refers to X4 and

X6 and both are related

to Image. Thus we can

label it as Image

Rot tedComponent t x

-.787 .194

.714 .266

-.804 -1.06E-02

.102 .933

2.537E-02 .934

.764 .179

Delivery Speed

Price Level

Price Flexibility

Manufacturer Image

Salesforce Image

Product Quality

1 2

Component

Extraction Method: Principal Component Analysis.

otation Method: arimax ith Kaiser ormali ation.

otation converged in 3 iterations.a.

The next stage in the analysisis to test reliability of the 4

items for physical quality

perceptions are reliable.

Similarly for the 2 items of

image


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HATCO Data

Sorted by size of loadings

within each factor

Obtained by using the

OPTIONS

Rotate Com onent Matrixa

-.804 -1.06 -02

-.787 .194

.764 .179

.714 .266

2.537 -02 .934

.102 .933

Price Flexibility

Delivery Speed

Pr duct QualityPrice Level

Salesf rce Image

Manufacturer Image

1 2

mp nent

xtracti n Met d: Principal mp nent nalysis.R tati n Met d: arimax wit aiser N rmalizati

R tati n c nverged in 3 iterati ns.a.

Note: X3(Price Flexibility) and X1(Delivery Speed) has negative

loading. This implies that it has to be reverse scored (using

recoding) before X1, X2, X3, and X7 can be combined to form a

scale


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Sample Table 1

Compo tCommunalit

It ms 1 2

x1 li r peed -0.787 0.194 0.658

x2 rice evel 0.714 0.266 0.580

x3 rice lexi ilit -0.804 -0.011 0.646

x4 anufacturer Image 0.102 0.933 0.882

x6 ales force Image 0.025 0.934 0.872

x7 roduct ualit 0.764 0.179 0.616

Eigenvalue

Variance (70.883)2.370

39.497

1.883

31.386


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Reliability Analysis

When

Before forming a composite index to become avariable from a number of items

Command

Analyze Scale ReliabilityAnalysis (withoption forStatistics item, scale, scale if itemdeleted)

Interpretation

alpha value greater than 0.7 is good (differsfrom author to author); more than 0.5 isacceptable; delete some items if necessary


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Reliability - SPSS


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HATCO DataHATCO Data

Cronbach E very low < 0.6

Cronbach if the item is deleted:

For purpose of identifying which

item should be excluded in the

scale

Item-Total Stat st cs

28.058 11.382 .030 .325

29.209 10.897 .135 .251

23.679 13.487 -.198 .477

26.325 8.987 .460 .029

28.657 9.896 .620 .050

28.908 10.282 .509 .097

24.602 12.068 -.108 .451

x1 DeliverySpeed

x2 Pri eLevel

x3 Pri eFlexibili y

x4 Manufa urer Image

x5 Servi e

x6 Sale for e Image

x7 Produ uali y

S aleMeanif

Item Dele ed

S aleVarian eif

Item Dele ed

Corre edItem-To al

Correla ion

Cronba h'Alphaif Item

Dele ed

Rel ab l ty Stat st cs

.291 7

Cronba h'Alpha Nof Item


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HATCO Data

After reverse scoring the negatively loaded variables X1 and X3

Reliability Stati tics

.810 4

Cronbac 'sAlpha N of Items

Item-Total Statistics

15.8200 10.592 .769 .69519.9410 13.532 .453 .835

15.8200 10.592 .769 .695

15.3340 10.613 .564 .805

X1R Delivery Speedx2 Price Level

X3R Price Flexibility

x7 Product Quality

Scale Mean ifItem Deleted

ScaleVariance if

Item Deleted

CorrectedItem-Total

Correlation

Cronbach'sAlpha if Item

Deleted

Should be

0.3 and

above


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HATCO Data

Now we can form two factors to represent our independent

variables Quality Perceptions I.e Physical quality

perceptions (X1, X2, X3, X7) and Image (X4, X6) using compute

MEANS

Reliability Statistics

.846 2

Cronbach'sAlpha N of Items

Item-Total Statistics

2.665 .594 .788 .a

5.248 1.280 .788 .ax4 anufacturer Image

x6 alesforce Image

Scale ean ifItem eleted

ScaleVariance if

Item eleted

CorrectedItem-TotalCorrelation

Cronbach'sAlpha if Item

Deleted

Thevalue is negativedue toanegativeaverage covarianceamong items. Thisviolates reliabilit model assumptions. Youmay want to chec item codings.

a.

ll


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1. The most basic assumption is that the set of variables analyzed

are related.

Variables must be interrelatedin some way since factor analysis

seeks the underlying common dimensions among the variables. If the

variables are not related, then each variable will be its own factor.

Example: if you had 20 unrelated variables, you would have 20 different

factors. When the variables are unrelated, factor analysis has no common

dimensions with which to create factors. Thus, some underlying structure or

relationship among the variables must exist.

The variables should not correlate too highly (>0.9): that makes itdifficult to determine the unique contribution of the variables to a factor

(multicollinearity).

2. The sample should be homogenous with respect to some underlying

factor structure.

Assumptions in Factor AnalysisAssumptions in Factor Analysis


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2. Factor analysis assumes the use of metricdata.

Metric variables are assumed, although dummy

variables may be used (coded 0-1).

Factor analysis does not require multivariate

normality. Multivariate normality is necessary if

the researcher wishes to apply statistical tests for

significance of factors.



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3. The data matrix must have sufficient correlations tojustify the use of factor analysis.

Rule of thumb: a substantial number of correlations

greater than .30 are needed.

Tests of appropriateness: anti-image correlation

matrix of partial correlations, the Bartlett test of

sphericity, and the measure of sampling adequacy

(MSA greater than .50).



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MSA

MSA Comment

0.80andabove Meritorious

0.70 0.80M

iddling0.60 0.70 Mediocre

0.50 0.60 Miserable

Below 0.50 Unacceptable

Oth I


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Other Issues

Sample size of 100 or more

There are 1101 subjects in the sample who have valid

data for all questions and which will be included in the

factor analysis. This requirement is met.

Ratio of subjects to variables should be 5 to 1

There are 1101 subjects and 10 variables in the analysisfor a ratio of cases to variables of 110 to 1. This

requirement is met.

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Kuliah 3 - Factor Analysis1

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