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Topic 09 - Topic 9 Multiple Comparisons Multiple...

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1 Topic 9 – Multiple Comparisons Multiple Comparisons of Treatment Means Reading: 17.7-17.8
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2 Overview Brief Review of One-Way ANOVA Pairwise Comparisons of Treatment Means Multiplicity of Testing Linear Combinations & Contrasts of Treatment Means
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3 Review: One-Way ANOVA An alysis o f Va riance (ANOVA) models provide an efficient way to compare multiple groups. In a single factor ANOVA, The Model F-test will test the equality of all group means at the same time. If this test is significant, then our next goal is to identify specific differences. This is our big topic for this lesson.
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4 Review: Cell Means Model Basic ANOVA Model is: where Notation: i ” subscript indicates the level of the factor j ” subscript indicates observation number within the group ij i ij Y μ ε = + ( 29 2 ~ 0, ij N ε σ 1,2,3,..., i a = 1,2,3,..., i j n =
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5 Review: Factor Effects Model Relationship to Cell Means: 1,2,..., 1,2,..., ij i ij i i k Y j n μ α ε = = + + = ( 29 2 ~ 0, ij N ε σ i i μ μ α = + 0 i α =
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6 Review: Notation DOT indicates “sum” BAR indicates “average” or “divide by cell/sample size” is the mean for all observations is the mean for the observations in Level i of Factor A. Sometimes we omit the “dots” for brevity, but the meaning is the same. Y gg i Y g
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7 Review: Components of Variation Variation between groups gets “explained” by allowing the groups to have different means. This variation contributes to MSR. Variation within groups is unexplained, and contributes to MSE. The ratio F = MSR / MSE forms the basis for testing the hypothesis that all group means are the same.
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8 Review: Components of Variation Of course the individual components would sum to zero, so we must square them. It turns out that all cross-product terms cancel, and we have: BETWEEN WITHIN GROUPS GROUPS ( ) ( ) ( ) - = - + - å å å gg g gg g 1444442 444443 1444442 444443 1444442 444443 2 2 2 , , , SST SSE SSA ij i ij i i j i j i j Y Y Y Y Y Y
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9 Review: ANOVA Table Source SS DF MS F Factor A SSA a – 1 MSA MSA MSE Error SSE N a MSE Total SST N – 1
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10 Review: Model F Test Null Hypothesis (Cell Means) Alternative Hypothesis If we conclude the alternative, then it makes sense to try to determine specific differences. For Factor Effects model: 0 1 2 : a H μ μ μ = = = L : There exists some pair of population means not equal. a H 0 : 0 for all i H i α =
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11 Further Comparisons The F-test is Significant... ...What Next?
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12 Pairwise Comparisons Generally our next step is that we want to find out more specifics about the actual differences between treatment groups. We can compare two groups by looking at the difference between means. 0 : : i j a i j H H μ μ μ μ =
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13 Pairwise Comparisons (2) Can rewrite null hypothesis as and so proceed to look at the difference between means.
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