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Chapter 12--Multisample Inference

# If the hypothesis of no interaction is not rejected

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Unformatted text preview: 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Model Cont’d... The mean response of an experimental unit receiving the treatment (i , j ), i.e. the i th level of A combined with the j th level of B is: µij = µ + τi + βj + γij . Notice that the eﬀects of levels of factor A and that of levels of factor B are not additive unless the γ s are zeros. For example, the diﬀerence in mean response for levels 1 and 2 of factor A on level 1 of factor B is, µ11 − µ21 = (τ1 − τ2 ) + (γ11 − γ21 ) and the diﬀerence in mean response for levels 1 and 2 of factor A on level 2 of factor B is, µ12 − µ22 = (τ1 − τ2 ) + (γ12 − γ22 ). Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Model Cont’d... First, the hypothesis of absence of interactions is tested. The hypothesis of no-interaction eﬀects is: H0 : γij = 0 for all (i , j ) vs Ha : Not all γij are zeros. Proﬁle plot is used as a graphical tool to see the form of interaction if there exists one. If the hypothesis of no-interaction is not rejected, then we test the hypothesis of main eﬀects of the two factor, in any order. The hypothesis of no-eﬀects of factor A H0 : τi = 0 for all i vs Ha : Not all τi are zeros. The hypothesis of no-eﬀects of factor B H0 : βj = 0 for all j Chapter 12: Multisample Inference vs Ha : Not all βi are zeros. Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Analysis of Variance The total sum of square, (yijk − y... )2 . ¯ TSS = i j k The sum of square due to A, (¯i .. − y... )2 . y ¯ SSA = bn i The sum of square due to B , (¯.j . − y... )2 . y ¯ SSB = an j Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Analysis of Variance Cont’d... The sum of square due to the interaction AB , (¯ij . − yi .. − y.j . + y... )2 . y ¯ ¯ ¯ SSAB = n i j The error sum of square, (yijk − yij . )2 . ¯ SSE = i j k It is a matter of algebra to show that, TSS = SSA + SSB + SSAB + SSE . Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Analysis of Variance Cont’d... ANOVA table for Two-Way Layout in CRD. Source Mai...
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