1 Factor Analysis and Inference for Structured Covariance Matrices Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication

11

Factor Analysis and Inference for Factor Analysis and Inference for Structured Covariance MatricesStructured Covariance Matrices

Shyh-Kang JengShyh-Kang JengDepartment of Electrical Engineering/Department of Electrical Engineering/Graduate Institute of Communication/Graduate Institute of Communication/

Graduate Institute of Networking and MultiGraduate Institute of Networking and Multimediamedia

22

HistoryHistory

Early 20Early 20thth-century attempt to define -century attempt to define and measure intelligenceand measure intelligence

Developed primarily by scientists Developed primarily by scientists interested in psychometricsinterested in psychometrics

Advent of computers generated a Advent of computers generated a renewed interestrenewed interest

Each application must be examined Each application must be examined on its own meritson its own merits

33

Essence of Factor AnalysisEssence of Factor AnalysisDescribe the covariance among Describe the covariance among many variables in terms of a few many variables in terms of a few underlying, but unobservable, underlying, but unobservable, random random factorsfactors..A group of variables highly correlated A group of variables highly correlated among themselves, but having among themselves, but having relatively small correlations with relatively small correlations with variables in different groups variables in different groups represent a single underlying represent a single underlying factorfactor

44

Example 9.8Example 9.8Examination ScoresExamination Scores

55

Orthogonal Factor ModelOrthogonal Factor Model

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Example 9.3Example 9.3Consumer Preference DataConsumer Preference Data

1818

Example 9.3Example 9.3Determination of Determination of mm

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Example 9.3Example 9.3Principal Component SolutionPrincipal Component Solution

2020

Example 9.3Example 9.3FactorizationFactorization

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2121

Example 9.4Example 9.4Stock Price DataStock Price Data

Weekly rates of return for five stocksWeekly rates of return for five stocks– XX11: Allied Chemical: Allied Chemical– XX22: du Pont: du Pont– XX33: Union Carbide: Union Carbide– XX44: Exxon: Exxon– XX55: Texaco: Texaco

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Example 9.4Example 9.4Principal Component SolutionPrincipal Component Solution

2424

Example 9.4Example 9.4Residual Matrix for Residual Matrix for mm=2=2

0

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2525

Maximum Likelihood MethodMaximum Likelihood Method

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Example 9.5: Factorization ofExample 9.5: Factorization ofStock Price DataStock Price Data

2929

Example 9.5Example 9.5ML Residual MatrixML Residual Matrix

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3030

Example 9.6Example 9.6Olympic Decathlon DataOlympic Decathlon Data

3131

Example 9.6Example 9.6FactorizationFactorization

3232

Example 9.6Example 9.6PC Residual MatrixPC Residual Matrix

3333

Example 9.6Example 9.6ML Residual MatrixML Residual Matrix

3434

A Large Sample Test for A Large Sample Test for Number of Common FactorsNumber of Common Factors

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3535

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Example 9.7Example 9.7Stock Price Model TestingStock Price Model Testing

model)factor -(two reject tofail

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Example 9.8Example 9.8Examination ScoresExamination Scores

3838

Example 9.8Example 9.8Maximum Likelihood SolutionMaximum Likelihood Solution

3939

Example 9.8Example 9.8Factor Rotation Factor Rotation

4040

Example 9.8Example 9.8Rotated Factor LoadingRotated Factor Loading

4141

Varimax CriterionVarimax Criterion

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Example 9.9: Consumer-Example 9.9: Consumer-Preference Factor AnalysisPreference Factor Analysis

4343

Example 9.9Example 9.9Factor RotationFactor Rotation

4444

Example 9.10 Example 9.10 Stock Price Factor AnalysisStock Price Factor Analysis

4545

Example 9.11Example 9.11Olympic Decathlon Factor AnalysisOlympic Decathlon Factor Analysis

4646

Example 9.11Example 9.11Rotated ML LoadingsRotated ML Loadings

4747

Factor ScoresFactor Scores

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Regression ModelRegression Model

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Factor Scores by RegressionFactor Scores by Regression

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Example 9.12Example 9.12Factor Scores by RegressionFactor Scores by Regression

5555

Example 9.13: Simple Summary Example 9.13: Simple Summary Scores for Stock Price DataScores for Stock Price Data

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5656

A Strategy for Factor AnalysisA Strategy for Factor Analysis

1. Perform a principal component factor 1. Perform a principal component factor analysisanalysis– Look for suspicious observations by plotting Look for suspicious observations by plotting

the factor scoresthe factor scores– Try a varimax rotationTry a varimax rotation2. Perform a maximum likelihood factor 2. Perform a maximum likelihood factor analysis, including a varimax rotationanalysis, including a varimax rotation

5757

A Strategy for Factor AnalysisA Strategy for Factor Analysis3. Compare the solutions obtained from 3. Compare the solutions obtained from the two factor analysesthe two factor analyses– Do the loadings group in the same manner?Do the loadings group in the same manner?– Plot factor scores obtained for PC against Plot factor scores obtained for PC against

scores from ML analysisscores from ML analysis

4. Repeat the first 3 steps for other 4. Repeat the first 3 steps for other numbers of common factorsnumbers of common factors5. For large data sets, split them in half 5. For large data sets, split them in half and perform factor analysis on each part. and perform factor analysis on each part. Compare the two results with each other Compare the two results with each other and with that from the complete data setand with that from the complete data set

5858

Example 9.14Example 9.14Chicken-Bone DataChicken-Bone Data

1

937.01

894.0874.01

878.0877.0926.01

450.0482.0467.0422.01

603.0621.0602.0569.0505.01

length ulna length, humerus :Wing

length tibia length,femur :Leg

breadth skull length, skull :Head

dimensions boneon tsmeasuremen 276

65

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5959

Example 9.14:Principal Component Example 9.14:Principal Component Factor Analysis ResultsFactor Analysis Results

6060

Example 9.14: Maximum Likelihood Example 9.14: Maximum Likelihood Factor Analysis ResultsFactor Analysis Results

6161

Example 9.14Example 9.14Residual Matrix for ML EstimatesResidual Matrix for ML Estimates

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Example 9.14Example 9.14Factor Scores for Factors 1 & 2Factor Scores for Factors 1 & 2

6363

Example 9.14Example 9.14Pairs of Factor Scores: Factor 1Pairs of Factor Scores: Factor 1

6464


6565


6666

Example 9.14Example 9.14Divided Data SetDivided Data Set

1940.0927.0894.0386.0598.0

1911.0909.0420.0587.0

1950.0406.0587.0

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6767

Example 9.14: PC Factor Analysis Example 9.14: PC Factor Analysis for Divided Data Setfor Divided Data Set

6868

WOW CriterionWOW Criterion

In practice the vast majority of In practice the vast majority of attempted factor analyses do not attempted factor analyses do not yield clear-cut resultsyield clear-cut results

If, while scrutinizing the factor If, while scrutinizing the factor analysis, the investigator can shout analysis, the investigator can shout “Wow, I understand these factors,” “Wow, I understand these factors,” the application is deemed successfulthe application is deemed successful

6969

Structural Equation ModelsStructural Equation Models

Sets of linear equations to specify Sets of linear equations to specify phenomena in terms of their phenomena in terms of their presumed cause-and-effect variablespresumed cause-and-effect variables

In its most general form, the models In its most general form, the models allow for variables that can not be allow for variables that can not be measured directlymeasured directly

Particularly helpful in the social and Particularly helpful in the social and behavioral sciencebehavioral science

7070

LISREL (LISREL (LiLinear near SStructural tructural RelRelationships) Modelationships) Model

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ExampleExample: performance of the firm: performance of the firm

: managerial talent: managerial talent

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XX11: years of chief executive experience: years of chief executive experience

XX22: memberships on board of directors: memberships on board of directors

7272

Linear System in Control TheoryLinear System in Control Theory

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7373

Kinds of VariablesKinds of Variables

Exogenous variables: not influenced Exogenous variables: not influenced by other variables in the systemby other variables in the system

Endogenous variables: affected by Endogenous variables: affected by other variablesother variables

Residual: associated with each of the Residual: associated with each of the dependent variablesdependent variables

7474

Construction of a Path DiagramConstruction of a Path DiagramStraight arrowStraight arrow– to each dependent (endogenous) to each dependent (endogenous)

variable from each of its sourcevariable from each of its source

Straight arrowStraight arrow– also to each dependent variables from also to each dependent variables from

its residualits residual

Curved, double-headed arrowCurved, double-headed arrow– between each pair of independent between each pair of independent

(exogenous) variables thought to have (exogenous) variables thought to have nonzero correlationnonzero correlation

7575

Example 9.15Example 9.15Path DiagramPath Diagram

7676

Example 9.15Example 9.15Structural EquationStructural Equation

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Example 9.16Example 9.16Artificial DataArtificial Data

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8080


1.1ˆˆ,6.1ˆˆ

7.3ˆˆˆ,4.6ˆˆˆ,8.12ˆˆˆˆ

2.3ˆˆ,4.6ˆˆˆ,4.55ˆˆˆˆˆ

6.27ˆˆˆˆ,3.14ˆˆˆˆ

1.16.1|4.62.3

6.17.3|8.124.6

4.68.12|4.556.27

2.34.6|6.273.14

42

322121

2222

1

211

2

2221

1211

SS

SS

8181


3.00

05.0ˆ,2.00

05.0ˆ

1

2ˆ,2

1ˆ

1ˆ,4ˆ,8.0ˆ,0.2ˆ2

δε

xy

ΘΘ

ΛΛ

8282

Assessing the Fit of the ModelAssessing the Fit of the Model

The number of observations The number of observations pp for for YY and and qq for for XX must be larger than the must be larger than the total number of unknown parameters total number of unknown parameters tt < < ((pp++qq)()(pp++qq+1)/2+1)/2

Parameter estimates should have Parameter estimates should have appropriate signs and magnitudesappropriate signs and magnitudes

Entries in the residual matrix Entries in the residual matrix SS – – should be uniformly smallshould be uniformly small

8383

Model-Fitting StrategyModel-Fitting StrategyGenerate parameter estimates using Generate parameter estimates using several criteria and compare the several criteria and compare the estimatesestimates– Are signs and magnitudes consistent?Are signs and magnitudes consistent?– Are all variance estimates positive?Are all variance estimates positive?– Are the residual matrices similar?Are the residual matrices similar?

Do the analysis with both Do the analysis with both SS and and RRSplit large data sets in half and Split large data sets in half and perform the analysis on each halfperform the analysis on each half

1 Factor Analysis and Inference for Structured Covariance Matrices Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication

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