Introducing Anova and Ancova: A GLM Approach (Introducing Statistical Methods series)

193 Pages·2001·1.53 MB·English

by  Dr Andrew Rutherford

INTRODUCING ANOVA
AND ANCOVA
ISM Introducing Statistical Methods
Serieseditor:DanielB.Wright,UniversityofSussex
This series provides accessible but in-depth introductions to statistical methods
that are not covered in any detail in standard introductory courses. The books
are aimed both at the beginning researcher who needs to know how to use a
particular technique and the established researcher who wishes to keep up to
datewithrecentdevelopmentsinstatisticaldataanalysis.
Editorialboard
GlynisBreakwell,UniversityofSurrey
JandeLeeuw,UniversityofCalifornia,LosAngeles
ColmO’Muircheartaigh,UniversityofChicago
WillemSaris,UniversiteitvanAmsterdam
HowardSchuman,UniversityofMichigan
KarlvanMeter,CentreNationaldelaRechercheScientifique,Paris
Othertitlesinthisseries
IntroducingMultilevelModeling
ItaKreftandJandeLeeuw
IntroducingSocialNetworks
AlainDegenneandMichelForse´
IntroducingLISREL:AGuidefortheUninitiated
AdamantiosDiamantopoulosandJudySiguaw
INTRODUCING ANOVA
AND ANCOVA
A GLM A
PPROACH
ANDREW RUTHERFORD
SAGE Publications
London • ThousandOaks • NewDelhi
#AndrewRutherford2001
Firstpublished2001
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CONTENTS
1 ANINTRODUCTIONTOGENERALLINEARMODELS:REGRESSION,ANALYSIS
OFVARIANCEANDANALYSISOFCOVARIANCE
1.1 Regression,analysisofvarianceandanalysisofcovariance 1
1.2 Apockethistoryofregression,ANOVAandANCOVA 2
1.3 Anoutlineofgenerallinearmodels(GLMs) 3
1.3.1 Regressionanalysis 4
1.3.2 Analysisofvariance 4
1.3.3 Analysisofcovariance 5
1.4 The‘‘general’’inGLM 5
1.5 The‘‘linear’’inGLM 8
1.6 Leastsquaresestimates 10
1.7 Fixed,randomandmixedeffectsanalyses 10
1.8 ThebenefitsofaGLMapproachtoANOVAandANCOVA 12
1.9 TheGLMpresentation 13
1.10 Statisticalpackagesforcomputers 13
2 TRADITIONALANDGLMAPPROACHESTOINDEPENDENTMEASURESSINGLE
FACTORANOVADESIGNS
2.1 Independentmeasuresdesigns 15
2.1.1 Factorsandindependentvariables 17
2.2 TraditionalANOVAforsinglefactordesigns 18
2.2.1 Variance 18
2.2.2 Example 19
2.3 GLMapproachestosinglefactorANOVA 24
2.3.1 ExperimentaldesignGLMs 24
2.3.2 Estimatingeffectsbycomparingfullandreducedexperimental
designGLMs 29
2.3.3 RegressionGLMs 32
2.3.4 CellmeanGLMs 41
2.3.5 Cellmean,regressionandexperimentaldesignGLMs 42
3 GLMAPPROACHESTOINDEPENDENTMEASURESFACTORIALANOVA
DESIGNS
3.1 Factorialdesigns 43
3.2 Factormaineffectsandfactorinteractions 43
3.2.1 Estimatingeffectsbycomparingfullandreducedexperimental
designGLMs 49
viii CONTENTS
3.3 RegressionGLMsforfactorialANOVA 56
3.3.1 EstimatingmainandinteractioneffectswithregressionGLMs 57
4 GLMAPPROACHESTOREPEATEDMEASURESDESIGNS
4.1 Relatedmeasuresdesigns 60
4.2 Repeatedmeasuresdesigns 61
4.3 Ordereffectcontrols 63
4.3.1 Counterbalancing 63
4.3.2 Randomization 65
4.4 TheGLMapproachtosinglefactorrepeatedmeasuresdesigns 65
4.5 Estimatingeffectsbycomparingfullandreducedsinglefactorrepeated
measuresdesignGLMs 72
4.6 RegressionGLMsforsinglefactorrepeatedmeasuresdesigns 74
5 GLMAPPROACHESTOFACTORIALREPEATEDMEASURESDESIGNS
5.1 Factorialrelatedmeasuresdesigns 78
5.2 ThefullyrelatedfactorialdesignGLM 79
5.3 Estimatingeffectsbycomparingfullandreducedfullyrelatedfactorial
experimentaldesignGLMs 85
5.4 RegressionGLMsforthefullyrelatedfactorialANOVA 88
5.5 MixedfactorialANOVA 94
5.6 Estimatingeffectsbycomparingfullandreducedmixedfactorial
experimentaldesignGLMs 98
5.7 RegressionGLMforthemixedfactorialANOVA 99
6 THEGLMAPPROACHTOANCOVA
6.1 ThenatureofANCOVA 105
6.2 SinglefactorindependentmeasuresANCOVAdesigns 106
6.3 Estimatingeffectsbycomparingfullandreducedsinglefactor
independentmeasuresANCOVAGLMs 111
6.4 RegressionGLMsforthesinglefactorindependentmeasures
ANCOVA 114
6.5 OtherANCOVAdesigns 120
6.5.1 RelatedmeasuresANCOVAdesigns 121
6.5.2 MixedmeasuresfactorialANCOVA 121
7 ASSUMPTIONSUNDERLYINGANOVA,TRADITIONALANCOVAANDGLMS
7.1 ANOVAandGLMassumptions 123
7.1.1 Independentmeasures 123
7.1.2 Relatedmeasures 125
7.1.3 TraditionalANCOVA 126
7.2 AstrategyforcheckingANOVAandtraditionalANCOVAassumptions 127
7.3 Assumptionchecksandsomeassumptionviolationconsequences 128
7.3.1 ANOVAandANCOVA 129
7.3.2 TraditionalANCOVA 136
CONTENTS ix
8 SOMEALTERNATIVESTOTRADITIONALANCOVA
8.1 AlternativestotraditionalANCOVA 145
8.2 Theheterogeneousregressionproblem 145
8.3 TheheterogeneousregressionANCOVAGLM 146
8.4 Singlefactorindependentmeasuresheterogeneousregression
ANCOVA 148
8.5 EstimatingheterogeneousregressionANCOVAeffects 150
8.6 RegressionGLMsforheterogeneousANCOVA 154
8.7 Covariate–experimentalconditionrelations 157
8.7.1 Multicollinearity 158
8.8 Otheralternatives 161
8.8.1 Stratification(blocking) 161
8.8.2 Replacingtheexperimentalconditionswiththecovariate 162
8.9 TheroleofANCOVA 162
9 FURTHERISSUESINANOVAANDANCOVA
9.1 Power 164
9.1.1 Optimalexperimentaldesigns 165
9.1.2 Normalityviolations 166
9.1.3 Maineffectsandinteractions 167
9.2 ErrorrateandtheomnibusF-tests 168
9.3 Errorrateandmultiplecomparisons 169
9.4 TheroleoftheomnibusF-test 171
REFERENCES 173
INDEX 179