Correspondence Analysis

562 Pages·2014·2.054 MB·english

by  Eric J. Beh, Rosaria Lombardo

Correspondence Analysis
Theory, Practice and New Strategies
Eric J. Beh
SchoolofMathematics&PhysicalSciences
UniversityofNewcastle,Australia
Rosaria Lombardo
DepartmentofEconomics,SecondUniversityofNaples, Italy
Thiseditionfirstpublished2014
©2014JohnWiley&Sons,Ltd
LibraryofCongressCataloging-in-PublicationData
Beh,EricJ.,author.
Correspondenceanalysis:theory,practiceandnewstrategies/EricBeh,RosariaLombardo.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-1-119-95324-1(hardback)
1.Correspondenceanalysis(Statistics)I.Lombardo,Rosaria,author.II.Title.
QA278.5.B442014
519.5’37--dc23
2014017301
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN:978-1-119-95324-1
Setin10/12ptTimesLTStd-RomanbyThomsonDigital,Noida,India.
12014
Contents
Foreword xv
Preface xvii
PartOne Introduction 1
1 DataVisualisation 3
1.1 AVeryBriefIntroductiontoDataVisualisation 3
1.1.1 AVeryBriefHistory 3
1.1.2 IntroductiontoVisualisationToolsforNumericalData 4
1.1.3 IntroductiontoVisualisationToolsforUnivariateCategoricalData 6
1.2 DataVisualisationforContingencyTables 10
1.2.1 FourfoldDisplays 11
1.3 OtherPlots 12
1.4 StudyingExposuretoAsbestos 13
1.4.1 AsbestosandIrvingJ.Selikoff 13
1.4.2 Selikoff’sData 17
1.4.3 NumericalAnalysisofSelikoff’sData 17
1.4.4 AGraphicalAnalysisofSelikoff’sData 18
1.4.5 ClassicalCorrespondenceAnalysisofSelikoff’sData 20
1.4.6 OtherMethodsofGraphicalAnalysis 22
1.5 HappinessData 25
1.6 CorrespondenceAnalysisNow 29
1.6.1 ABibliographicTaste 29
1.6.2 TheIncreasingPopularityofCorrespondenceAnalysis 29
1.6.3 TheGrowthoftheCorrespondenceAnalysisFamilyTree 32
1.7 OverviewoftheBook 34
1.8 RCode 35
References 36
2 Pearson’sChi-SquaredStatistic 44
2.1 Introduction 44
2.2 Pearson’sChi-SquaredStatistic 44
2.2.1 Notation 44
2.2.2 MeasuringtheDeparturefromIndependence 45
2.2.3 Pearson’sChi-SquaredStatistic 47
2.2.4 Other𝜒2MeasuresofAssociation 48
2.2.5 ThePowerDivergenceStatistic 49
2.2.6 DealingwiththeSampleSize 50
2.3 TheGoodman--KruskalTauIndex 51
2.3.1 OtherMeasuresandIssues 52
2.4 The2×2ContingencyTable 52
2.4.1 Yates’ContinuityCorrection 53
2.5 EarlyContingencyTables 54
2.5.1 TheImpactofAdolphQuetelet 55
2.5.2 Gavarret’s(1840)LegitimateChildrenData 58
2.5.3 Finley’s(1884)TornadoData 58
2.5.4 Galton’s(1892)FingerprintData 59
2.5.5 FinalComments 61
2.6 RCode 61
2.6.1 ExpectationandVarianceofthePearsonChi-SquaredStatistic 61
2.6.2 Pearson’sChi-SquaredTestofIndependence 62
2.6.3 TheCressie--ReadStatistic 64
References 67
PartTwo CorrespondenceAnalysisofTwo-WayContingencyTables 71
3 MethodsofDecomposition 73
3.1 Introduction 73
3.2 ReducingMultidimensionalSpace 73
3.3 ProfilesandCloudofPoints 74
3.4 PropertyofDistributionalEquivalence 79
3.5 TheTripletandClassicalReciprocalAveraging 79
3.5.1 One-DimensionalReciprocalAveraging 80
3.5.2 MatrixFormofOne-DimensionalReciprocalAveraging 81
3.5.3 𝑀-DimensionalReciprocalAveraging 83
3.5.4 SomeHistoricalComments 83
3.6 SolvingtheTripletUsingEigen-Decomposition 84
3.6.1 TheDecomposition 84
3.6.2 Example 85
3.7 SolvingtheTripletUsingSingularValueDecomposition 86
3.7.1 TheStandardDecomposition 86
3.7.2 TheGeneralisedDecomposition 88
3.8 TheGeneralisedTripletandReciprocalAveraging 89
3.9 SolvingtheGeneralisedTripletUsingGram--SchmidtProcess 91
3.9.1 OrderedCategoricalVariablesandaprioriScores 91
3.9.2 OnFindingOrthogonalisedVectors 92
3.9.3 ARecurrenceFormulaeApproach 94
3.9.4 ChangingtheBasisVector 96
3.9.5 GeneralisedCorrelations 97
3.10 BivariateMomentDecomposition 100
3.11 HybridDecomposition 100
3.11.1 AnAlternativeSinglyOrderedApproach 102
3.12 RCode 103
3.12.1 Eigen-DecompositioninR 103
3.12.2 SingularValueDecompositioninR 103
3.12.3 SingularValueDecompositionforMatrixApproximation 104
3.12.4 GeneratingEmerson’sPolynomials 106
3.13 APreliminaryGraphicalSummary 109
3.14 AnalysisofAnalgesicDrugs 112
References 115
4 SimpleCorrespondenceAnalysis 120
4.1 Introduction 120
4.2 Notation 121
4.3 MeasuringDeparturesfromCompleteIndependence 122
4.3.1 The‘DuplicationConstant’ 123
4.3.2 PearsonRatios 123
4.4 DecomposingthePearsonRatio 124
4.5 CoordinateSystems 126
4.5.1 StandardCoordinates 126
4.5.2 PrincipalCoordinates 127
4.5.3 BiplotCoordinates 132
4.6 Distances 136
4.6.1 DistancefromtheOrigin 136
4.6.2 Intra-VariableDistancesandthe𝐿𝑝 Metric 137
4.6.3 Inter-VariableDistances 138
4.7 TransitionFormulae 140
4.8 MomentsofthePrincipalCoordinates 141
4.8.1 TheMeanof𝑓𝑖𝑚 142
4.8.2 TheVarianceof𝑓𝑖𝑚 142
4.8.3 TheSkewnessof𝑓𝑖𝑚 143
4.8.4 TheKurtosisof𝑓𝑖𝑚 143
4.8.5 MomentsoftheAsbestosData 144
4.9 HowManyDimensionstoUse? 145
4.10 RCode 147
4.11 OtherTheoreticalIssues 154
4.12 SomeApplicationsofCorrespondenceAnalysis 156
4.13 AnalysisofaMother’sAttachmenttoHerChild 158
References 165
5 Non-SymmetricalCorrespondenceAnalysis 177
5.1 Introduction 177
5.2 TheGoodman--KruskalTauIndex 180
5.2.1 TheTauIndexasaMeasureoftheIncreaseinPredictability 180
5.2.2 TheTauIndexintheContextofANOVA 182
5.2.3 TheSensitivityof𝜏 182
5.2.4 ADemonstration:RevisitingSelikoff’sAsbestosData 185
5.3 Non-SymmetricalCorrespondenceAnalysis 186
5.3.1 TheCentredColumnProfileMatrix 186
5.3.2 Decompositionof𝜏 187
5.4 TheCoordinateSystems 188
5.4.1 StandardCoordinates 188
5.4.2 PrincipalCoordinates 189
5.4.3 BiplotCoordinates 193
5.5 TransitionFormulae 197
5.5.1 SupplementaryPoints 198
5.5.2 ReconstructionFormulae 198
5.6 MomentsofthePrincipalCoordinates 199
5.6.1 TheMeanof𝑓𝑖𝑚 199
5.6.2 TheVarianceof𝑓𝑖𝑚 200
5.6.3 TheSkewnessof𝑓𝑖𝑚 201
5.6.4 TheKurtosisof𝑓𝑖𝑚 201
5.7 TheDistances 201
5.7.1 ColumnDistances 201
5.7.2 RowDistances 203
5.8 ComparisonwithSimpleCorrespondenceAnalysis 204
5.9 RCode 204
5.10 AnalysisofaMother’sAttachmenttoHerChild 209
References 212
6 OrderedCorrespondenceAnalysis 216
6.1 Introduction 216
6.2 Pearson’sRatioandBivariateMomentDecomposition 221
6.3 CoordinateSystems 222
6.3.1 StandardCoordinates 222
6.3.2 TheGeneralisedCorrelations 223
6.3.3 PrincipalCoordinates 225
6.3.4 Location,DispersionandHigherOrderComponents 229
6.3.5 TheCorrespondencePlotandGeneralisedCorrelations 230
6.3.6 ImpactontheChoiceofScores 232
6.4 ArtificialDataRevisited 233
6.4.1 OntheStructureoftheAssociation 233
6.4.2 AGraphicalSummaryoftheAssociation 233
6.4.3 AnInterpretationoftheAxesandComponents 234
6.4.4 TheImpactoftheChoiceofScores 235
6.5 TransitionFormulae 236
6.6 DistanceMeasures 238
6.6.1 DistancefromtheOrigin 238
6.6.2 Intra-VariableDistances 239
6.7 SinglyOrderedAnalysis 239
6.8 RCode 241
6.8.1 GeneralisedCorrelationsandPrincipalInertias 241
6.8.2 DoublyOrderedCorrespondenceAnalysis 245
References 248
7 OrderedNon-SymmetricalCorrespondenceAnalysis 251
7.1 Introduction 251
7.2 GeneralConsiderations 252
7.2.1 OrthogonalPolynomialsInsteadofSingularVectors 253
7.3 DoublyOrderedNon-SymmetricalCorrespondenceAnalysis 254
7.3.1 BivariateMomentDecomposition 254
7.3.2 GeneralisedCorrelationsinBivariateMomentDecomposition 255
7.4 SinglyOrderedNon-SymmetricalCorrespondenceAnalysis 257
7.4.1 HybridDecompositionforanOrderedPredictorVariable 257
7.4.2 HybridDecompositionintheCaseofOrderedResponseVariables 258
7.4.3 GeneralisedCorrelationsinHybridDecomposition 258
7.5 CoordinateSystemsforOrderedNon-SymmetricalCorrespondence
Analysis 259
7.5.1 PolynomialPlotsforDoublyOrderedNon-Symmetrical
CorrespondenceAnalysis 260
7.5.2 PolynomialBiplotforDoublyOrderedNon-Symmetrical
CorrespondenceAnalysis 262
7.5.3 PolynomialPlotforSinglyOrderedNon-Symmetrical
CorrespondenceAnalysiswithanOrderedPredictorVariable 262
7.5.4 PolynomialBiplotforSinglyOrderedNon-Symmetrical
CorrespondenceAnalysiswithanOrderedPredictorVariable 263
7.5.5 PolynomialPlotforSinglyOrderedNon-Symmetrical
CorrespondenceAnalysiswithanOrderedResponseVariable 264
7.5.6 PolynomialBiplotforSinglyOrderedNon-Symmetrical
CorrespondenceAnalysiswithanOrderedResponseVariable 265
7.6 TestsofAsymmetricAssociation 265
7.7 DistancesinOrderedNon-SymmetricalCorrespondenceAnalysis 266
7.7.1 DistancesinDoublyOrderedNon-Symmetrical
CorrespondenceAnalysis 267
7.7.2 DistancesinSinglyOrderedNon-Symmetrical
CorrespondenceAnalysis 269
7.8 DoublyOrderedNon-SymmetricalCorrespondenceofAsbestosData 269
7.8.1 Trends 270
7.9 SinglyOrderedNon-SymmetricalCorrespondenceAnalysisofDrugData 277
7.9.1 PredictabilityofOrderedRowsGivenColumns 278
7.10 RCodeforOrderedNon-SymmetricalCorrespondenceAnalysis 283
References 300
8 ExternalStabilityandConfidenceRegions 302
8.1 Introduction 302
8.2 OntheStatisticalSignificanceofaPoint 303
8.3 CircularConfidenceRegionsforClassicalCorrespondenceAnalysis 304
8.4 EllipticalConfidenceRegionsforClassicalCorrespondenceAnalysis 306
8.4.1 TheInformationintheOptimalCorrespondencePlot 306
8.4.2 TheInformationintheFirstTwoDimensions 308
8.4.3 EccentricityofEllipticalRegions 309
8.4.4 ComparisonofConfidenceRegions 309
8.5 ConfidenceRegionsforNon-SymmetricalCorrespondenceAnalysis 311
8.5.1 CircularRegionsinNon-SymmetricalCorrespondenceAnalysis 312
8.5.2 EllipticalRegionsinNon-SymmetricalCorrespondenceAnalysis 312
8.6 Approximate𝑝-valuesandClassicalCorrespondenceAnalysis 313
8.6.1 Approximate𝑝-valuesBasedonConfidenceCircles 313
8.6.2 Approximate𝑝-valuesBasedonConfidenceEllipses 314
8.7 Approximate𝑝-valuesandNon-SymmetricalCorrespondenceAnalysis 315
8.8 BootstrapEllipticalConfidenceRegions 315
8.9 Ringrose’sBootstrapConfidenceRegions 316
8.9.1 ConfidenceEllipsesandCovarianceMatrix 317
8.10 ConfidenceRegionsandSelikoff’sAsbestosData 318
8.11 ConfidenceRegionsandMother--ChildAttachmentData 322
8.12 RCode 325
8.12.1 CalculatingthePathofaConfidenceEllipse 326
8.12.2 ConstructingEllipticalRegionsinaCorrespondencePlot 327
References 335
9 VariantsofCorrespondenceAnalysis 337
9.1 Introduction 337
9.2 CorrespondenceAnalysisUsingAdjustedStandardisedResiduals 337
9.3 CorrespondenceAnalysisUsingtheFreeman--TukeyStatistic 340
9.4 CorrespondenceAnalysisofRankedData 342
9.5 RCode 343
9.5.1 AdjustedStandardisedResiduals 343
9.5.2 Freeman--TukeyStatistic 349
9.6 TheCorrespondenceAnalysisFamily 353
9.6.1 DetrendedCorrespondenceAnalysis 353
9.6.2 CanonicalCorrespondenceAnalysis 354
9.6.3 InverseCorrespondenceAnalysis 355
9.6.4 OrderedCorrespondenceAnalysis 355
9.6.5 GradeCorrespondenceAnalysis 355
9.6.6 SymbolicCorrespondenceAnalysis 356
9.6.7 CorrespondenceAnalysisofProximityData 356
9.6.8 Residual(Scaling)CorrespondenceAnalysis 360
9.6.9 Log-RatioCorrespondenceAnalysis 362
9.6.10 ParametricCorrespondenceAnalysis 364
9.6.11 SubsetCorrespondenceAnalysis 364
9.6.12 Foucart’sCorrespondenceAnalysis 365
9.7 OtherTechniques 365
References 366
PartThree CorrespondenceAnalysisofMulti-WayContingencyTables 373
10 CodingandMultipleCorrespondenceAnalysis 375
10.1 IntroductiontoCoding 375
10.2 CodingData 377
10.2.1 B-Splines 377
10.2.2 CrispCoding 380
10.2.3 FuzzyCoding 382
10.3 CodingOrderedCategoricalVariablesbyOrthogonalPolynomials 382
10.4 BurtMatrix 384
10.5 AnIntroductiontoMultipleCorrespondenceAnalysis 386
10.6 MultipleCorrespondenceAnalysis 388
10.6.1 Notation 388
10.6.2 DecompositionMethods 389
10.6.3 Coordinates,TransitionFormulaeandAdjustedInertia 393
10.7 VariantsofMultipleCorrespondenceAnalysis 395
10.7.1 JointCorrespondenceAnalysis 396
10.7.2 StackingandConcatenation 397
10.8 OrderedMultipleCorrespondenceAnalysis 398
10.8.1 OrthogonalPolynomialsinMultipleCorrespondenceAnalysis 398
10.8.2 HybridDecompositionofMultipleIndicatorTables 399
10.8.3 TwoOrderedVariablesandTheirContingencyTable 400
10.8.4 TestofStatisticalSignificance 401
10.8.5 PropertiesofOrderedMultipleCorrespondenceAnalysis 403
10.8.6 GraphicalDisplaysinOrderedMultipleCorrespondence
Analysis 404
10.9 Applications 405
10.9.1 CustomerSatisfactioninHealthCareServices 406
10.9.2 TwoQualityAspects 411
10.10 RCode 417
10.10.1 B-SplineFunction 417
10.10.2 CrispandFuzzyCodingUsingB-SplinesinR 421
10.10.3 CrispCodingandtheBurtTablebyIndicatorFunctionsinR 425
10.10.4 ClassicalandMultipleCorrespondenceAnalysisinR 428
References 444
11 SymmetricalandNon-SymmetricalThree-WayCorrespondenceAnalysis 451
11.1 Introduction 451
11.2 Notation 453
11.3 SymmetricandAsymmetricAssociationinThree-WayContingency
Tables 454
11.4 PartitioningThree-WayMeasuresofAssociation 455
11.4.1 PartitioningPearson’sThree-WayStatistic 457
11.4.2 PartitioningMarcotorchino’sandGray--William’s
Three-WayIndices 458
11.4.3 Marcotorchino’sIndex 460
11.4.4 PartitioningtheThree-WayDeltaIndex 461
11.4.5 Three-WayDeltaIndex 463
11.5 FormalTestsofPredictability 463
11.5.1 TestingPearson’sStatistic 464
11.5.2 TestingtheMarcotorchino’sIndex 464
11.5.3 TestingtheDeltaIndex 465
11.5.4 Discussion 465
11.6 Tucker3DecompositionforThree-WayTables 466
11.7 CorrespondenceAnalysisofThree-WayContingencyTables 467
11.7.1 SymmetricallyAssociatedVariables 467
11.7.2 AsymmetricallyAssociatedVariables 468
11.7.3 AdditionalProperty 469
11.8 ModellingofPartialandMarginalDependence 470
11.9 GraphicalRepresentation 471
11.9.1 InteractivePlot 471
11.9.2 InteractiveBiplot 472
11.9.3 CategoryContribution 474
11.10 OntheApplicationofPartitions 474
11.10.1 OliveData:PartitioningtheAsymmetricAssociation 474
11.10.2 JobSatisfactionData:PartitioningtheAsymmetricAssociation 476
11.11 OntheApplicationofThree-WayCorrespondenceAnalysis 477
11.11.1 JobSatisfactionandThree-WaySymmetricalCorrespondence
Analysis 477
11.11.2 JobSatisfactionandThree-WayNon-Symmetrical
CorrespondenceAnalysis 483
11.12 RCode 490
References 511
PartFour TheComputationofCorrespondenceAnalysis 517
12 ComputingandCorrespondenceAnalysis 519
12.1 Introduction 519
12.2 ALookThroughTime 519
12.2.1 Pre-1990 519
12.2.2 From1990to2000 520
12.2.3 TheEarly2000s 522
12.3 TheImpactofR 523
12.3.1 OverviewofCorrespondenceAnalysisinR 523
12.3.2 MASS 524
12.3.3 Nenadic´ andGreenacre’s(2007)ca 525
12.3.4 Murtagh(2005) 527
12.3.5 ade4 530
12.4 SomeStand-AlonePrograms 533
12.4.1 JMP 533
12.4.2 SPSS 533
12.4.3 PAST 534
12.4.4 DtmVic5.6+ 535
References 540
Index 545