Lecturer: PauliinaIlmonen Slides:Ilmo- nen/Virtanen/Ailus/Kantala Analysisofvariance Pairwisecomparison Bartlett’stestfor Introduction to Statistical Inference equalityofvariances Kruskal-Wallistest Lecture 10: ANOVA, Kruskal-Wallis Test References Lecturer: PauliinaIlmonen Slides: Ilmonen/Virtanen/Ailus/Kantala Contents Lecturer: PauliinaIlmonen Slides:Ilmo- nen/Virtanen/Ailus/Kantala Analysisofvariance Pairwisecomparison Analysisofvariance Bartlett’stestfor equalityofvariances Kruskal-Wallistest References Pairwisecomparison Bartlett’stestforequalityofvariances Kruskal-Wallistest References Lecturer: PauliinaIlmonen Slides:Ilmo- nen/Virtanen/Ailus/Kantala Analysisofvariance Pairwisecomparison Bartlett’stestfor equalityofvariances Kruskal-Wallistest References Analysisofvariance Analysis of variance Lecturer: PauliinaIlmonen Slides:Ilmo- nen/Virtanen/Ailus/Kantala Analysisofvariance Pairwisecomparison Bartlett’stestfor equalityofvariances Thetwosamplet-testgeneralizesintoAnalysisofVariance. Kruskal-Wallistest References Inanalysisofvariance—ANOVA—thepopulationconsistsof twoormoregroups. Observationsareassumedtofollow normaldistribution. ANOVAteststheequalityoftheexpected valuesofthegroups. Forexample,wecouldtesttheequalityof theexpectedmonthlysalaryofMScs(Tech.) inHelsinki, TampereandOulu. MANOVA Lecturer: PauliinaIlmonen Slides:Ilmo- nen/Virtanen/Ailus/Kantala Analysisofvariance Pairwisecomparison Bartlett’stestfor equalityofvariances Kruskal-Wallistest References Inmultivariateanalysisofvariance,MANOVA,thetested expectedvaluesarevectors. Wecouldtesttheequalityofthe expectedmonthlysalaryandweeklyovertimeofMScs(Tech.) inHelsinki,TampereandOulu. Multifactor ANOVA Lecturer: PauliinaIlmonen Slides:Ilmo- nen/Virtanen/Ailus/Kantala Analysisofvariance Pairwisecomparison Bartlett’stestfor equalityofvariances Kruskal-Wallistest References Thepopulationcouldalsobedividedintogroupsbasedon multiplefactors(MultifactorANOVA),ofwhichsomecanbe continuous(analysisofcovariance,ANCOVA).Forexample, wecoulddivideMScs(Tech.) intogroupsbasedonthecity theyliveinandbasedontheirgender. Multifactor MANOVA Lecturer: PauliinaIlmonen Slides:Ilmo- nen/Virtanen/Ailus/Kantala Analysisofvariance Pairwisecomparison Bartlett’stestfor equalityofvariances Kruskal-Wallistest References InmultifactorMANOVA,thetestedexpectedvaluesarevectors. ANOVA Lecturer: PauliinaIlmonen Slides:Ilmo- nen/Virtanen/Ailus/Kantala Analysisofvariance Pairwisecomparison Bartlett’stestfor equalityofvariances Kruskal-Wallistest References Onthislecturecourse,weonlyconsidercaseswherethe populationisdividedintogroupswithrespecttojustonefactor andtheexpectedvaluethatistestedisunivariate. ANOVA Lecturer: PauliinaIlmonen Slides:Ilmo- nen/Virtanen/Ailus/Kantala Analysisofvariance Pairwisecomparison Bartlett’stestfor Letx ,x ,...,x beobservedvaluesofarandomvariablex, equalityofvariances 1j 2j njj j Kruskal-Wallistest j ∈{1,...,k}. Assumethattheobservationsx ,x ,...,x are 1j 2j njj References i.i.d. andfollownormaldistributionN(µ,σ2),j ∈{1,...,k}. (We j nowhavek randomsamplesfromunivariatenormal distribution. Thevarianceofallthek normaldistributionsisthe same.) Assumethatthek samplesareindependent. ThenullhypothesisH :µ =µ =···=µ . 0 1 2 k ThealternativehypothesisH :µ (cid:54)=µ forsomei,j. 1 j i ANOVA Lecturer: PauliinaIlmonen Slides:Ilmo- nen/Virtanen/Ailus/Kantala Analysisofvariance Pairwisecomparison Bartlett’stestfor equalityofvariances Inanalysisofvariance,thetotalvarianceisdividedintotwo Kruskal-Wallistest parts. Firstpartmeasuresthevariancebetweengroupmeans References andthesecondpartmeasuresvariancewithingroups. Ifthese twovariancesdifferalot,thereisevidenceagainstthenull hypothesis,anditcanberejected. Thetestoftheequalityoftheexpectedvaluesisbasedon comparisonofbetweengroupsvarianceandwithingroups variances. Thisthenametothemethod—analysisof variance.