Methods of Multivariate Analysis (Wiley Series in Probability and Statistics)

740 Pages·2002·3.28 MB·English

by  Alvin C. Rencher

Methods of Multivariate Analysis
Second Edition
ALVINC.RENCHER
BrighamYoungUniversity
AJOHNWILEY&SONS,INC.PUBLICATION

Methods of Multivariate Analysis
Second Edition
Methods of Multivariate Analysis
Second Edition
ALVINC.RENCHER
BrighamYoungUniversity
AJOHNWILEY&SONS,INC.PUBLICATION
∞
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LibraryofCongressCataloging-in-PublicationData
Rencher,AlvinC.,1934–
Methodsofmultivariateanalysis/AlvinC.Rencher.—2nded.
p. cm.—(Wileyseriesinprobabilityandmathematicalstatistics)
“AWiley-Intersciencepublication.”
Includesbibliographicalreferencesandindex.
ISBN0-471-41889-7(cloth)
1. Multivariateanalysis. I. Title. II. Series.
QA278.R452001
519.535—dc21 2001046735
′
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1

Contents
1. Introduction 1
1.1 WhyMultivariateAnalysis?, 1
1.2 Prerequisites, 3
1.3 Objectives, 3
1.4 BasicTypesofDataandAnalysis, 3
2. MatrixAlgebra 5
2.1 Introduction, 5
2.2 NotationandBasicDefinitions, 5
2.2.1 Matrices,Vectors,andScalars, 5
2.2.2 EqualityofVectorsandMatrices, 7
2.2.3 TransposeandSymmetricMatrices, 7
2.2.4 SpecialMatrices, 8
2.3 Operations, 9
2.3.1 SummationandProductNotation, 9
2.3.2 AdditionofMatricesandVectors, 10
2.3.3 MultiplicationofMatricesandVectors, 11
2.4 PartitionedMatrices, 20
2.5 Rank, 22
2.6 Inverse, 23
2.7 PositiveDefiniteMatrices, 25
2.8 Determinants, 26
2.9 Trace, 30
2.10 OrthogonalVectorsandMatrices, 31
2.11 EigenvaluesandEigenvectors, 32
2.11.1 Definition, 32
2.11.2 I AandI A, 33
+ −
2.11.3 tr(A)and A, 34
| |
2.11.4 PositiveDefiniteandSemidefiniteMatrices, 34
2.11.5 TheProductAB, 35
2.11.6 SymmetricMatrix, 35
v
vi CONTENTS
2.11.7 SpectralDecomposition, 35
2.11.8 SquareRootMatrix, 36
2.11.9 SquareMatricesandInverseMatrices, 36
2.11.10 SingularValueDecomposition, 36
3. CharacterizingandDisplayingMultivariateData 43
3.1 MeanandVarianceofaUnivariateRandomVariable, 43
3.2 CovarianceandCorrelationofBivariateRandomVariables, 45
3.2.1 Covariance, 45
3.2.2 Correlation, 49
3.3 ScatterPlotsofBivariateSamples, 50
3.4 GraphicalDisplaysforMultivariateSamples, 52
3.5 MeanVectors, 53
3.6 CovarianceMatrices, 57
3.7 CorrelationMatrices, 60
3.8 MeanVectorsandCovarianceMatricesforSubsetsof
Variables, 62
3.8.1 TwoSubsets, 62
3.8.2 ThreeorMoreSubsets, 64
3.9 LinearCombinationsofVariables, 66
3.9.1 SampleProperties, 66
3.9.2 PopulationProperties, 72
3.10 MeasuresofOverallVariability, 73
3.11 EstimationofMissingValues, 74
3.12 DistancebetweenVectors, 76
4. TheMultivariateNormalDistribution 82
4.1 MultivariateNormalDensityFunction, 82
4.1.1 UnivariateNormalDensity, 82
4.1.2 MultivariateNormalDensity, 83
4.1.3 GeneralizedPopulationVariance, 83
4.1.4 DiversityofApplicationsoftheMultivariateNormal, 85
4.2 PropertiesofMultivariateNormalRandomVariables, 85
4.3 EstimationintheMultivariateNormal, 90
4.3.1 MaximumLikelihoodEstimation, 90
4.3.2 DistributionofyandS, 91
4.4 AssessingMultivariateNormality, 92
4.4.1 InvestigatingUnivariateNormality, 92
4.4.2 InvestigatingMultivariateNormality, 96
CONTENTS vii
4.5 Outliers, 99
4.5.1 OutliersinUnivariateSamples, 100
4.5.2 OutliersinMultivariateSamples, 101
5. TestsonOneorTwoMeanVectors 112
5.1 MultivariateversusUnivariateTests, 112
5.2 Testson(cid:1)with(cid:2)Known, 113
5.2.1 ReviewofUnivariateTestfor H µ µ
0 0
: =
withσ Known, 113
5.2.2 MultivariateTestfor H (cid:1) (cid:1) with(cid:2)Known, 114
0 0
: =
5.3 Testson(cid:1)When(cid:2)IsUnknown, 117
5.3.1 ReviewofUnivariatet-Testfor H µ µ withσ
0 0
: =
Unknown, 117
5.3.2 Hotelling’sT2-Testfor H (cid:1) (cid:1) with(cid:2)Unknown, 117
0 0
: =
5.4 ComparingTwoMeanVectors, 121
5.4.1 ReviewofUnivariateTwo-Samplet-Test, 121
5.4.2 MultivariateTwo-SampleT2-Test, 122
5.4.3 LikelihoodRatioTests, 126
5.5 TestsonIndividualVariablesConditionalonRejectionof H by
0
theT2-Test, 126
5.6 ComputationofT2, 130
5.6.1 ObtainingT2fromaMANOVAProgram, 130
5.6.2 ObtainingT2fromMultipleRegression, 130
5.7 PairedObservationsTest, 132
5.7.1 UnivariateCase, 132
5.7.2 MultivariateCase, 134
5.8 TestforAdditionalInformation, 136
5.9 ProfileAnalysis, 139
5.9.1 One-SampleProfileAnalysis, 139
5.9.2 Two-SampleProfileAnalysis, 141
6. MultivariateAnalysisofVariance 156
6.1 One-WayModels, 156
6.1.1 UnivariateOne-WayAnalysisofVariance(ANOVA), 156
6.1.2 MultivariateOne-WayAnalysisofVarianceModel
(MANOVA), 158
6.1.3 Wilks’TestStatistic, 161
6.1.4 Roy’sTest, 164
6.1.5 PillaiandLawley–HotellingTests, 166