chapter8 - Chapter 8 1-Way Analysis of Variance Completely...

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Chapter 81-Way Analysis of Variance - Completely Randomized Design
Comparing t > 2 Groups - Numeric ResponsesExtension of Methods used to Compare 2 GroupsIndependent Samples and Paired Data DesignsNormal and non-normal data distributionsDataDesignNormalNon-normalIndependentSamples(CRD)F-Test1-WayANOVAKruskal-Wallis TestPaired Data(RBD)F-Test2-WayANOVAFriedman’sTest
Completely Randomized Design (CRD)Controlled Experiments - Subjects assigned at random to one of the ttreatments to be comparedObservational Studies - Subjects are sampled from t existing groupsStatistical model yijis measurement from the jthsubject from group i:ijiijiijyεμεαμ+=++=where μis the overall mean, αiis the effect of treatment i, εijis a random error, and μiis the population mean for group i
1-Way ANOVA for Normal Data (CRD)For each group obtain the mean, standard deviation, and sample size:1)(2..--==ijiijiijijinyysnyyObtain the overall mean and sample sizeNyNynynynnNijijttt∑ ∑=++=++=..11........1
Analysis of Variance - Sums of SquaresTotal Variation 1)(112..-=-=∑ ∑==NdfyyTSSTotalkinjijiBetween Group (Sample) Variation∑ ∑===-=-=-=tinjtiTiiiitdfyynyySST1112...2...1)()(Within Group (Sample) VariationETTotalEtiiitinjiijdfdfdfSSESSTTSStNdfsnyySSEi+=+=-=-=-=∑ ∑===12112.)1()(
Analysis of Variance Table and F-TestSource ofVariationSum of SquaresDegrres ofFreedomMean SquareFTreatmentsSSTt-1MST=SST/(t-1)F=MST/MSEErrorSSEN-tMSE=SSE/(N-t)TotalTSSN-1Assumption: All distributions normal with common varianceH0: No differences among Group Means (α1= ⋅ ⋅ ⋅ =αt =0)HA: Group means are not all equal (Not all αiare 0))(:)9(:..:..,1,obstNtobsobsFFPvalPTableFFRRMSEMSTFST-=--α
Expected Mean SquaresModel: yij= μ+αi+ εijwith εij~ N(0,σ2),Σαi= 0:1)()(true),is(otherwise1)()(true,is0:When )1(11)()(1)()(1021221221222====-+=-+=-+=====MSEEMSTEHMSEEMSTEHtntnMSEEMSTEtnMSTEMSEEattiiitiiitiiiαασασασασσ
Expected Mean Squares3 Factors effect magnitude of F-statistic (for fixed t)True group effects (α1,…,αt)Group sample sizes (n1,…,nt

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