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Section 11 Junghoon Moon

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Contents Multiple Regression One-way Anova Two-way Anova

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ANOVA table for multiple regression MSE is estimate of σ 2 SourceDFSum of Squares Mean Square F Model Error P N-p-1 SSM SSE SSM/DFM SSE/DFE F=MSM/ MSE TotalN-1SSTSST/DFT

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Example

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ANOVA I groups Variation within groups Variation between groups Variation within groups < variation between groups Group means are different

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One-way ANOVA Means μ 1,μ 2,μ 3,…..,μ I SD’s σ 1,σ 2,σ 3,….,σ I Hypothesis H 0 : μ 1 =μ 2 =μ 3 =…..=μ I H a : not all μ i are equal. Assumptions I independent samples Populations are normally distributed. Common SD: σ 1 =σ 2 =σ 3 =….=σ I X ij = μ i + ε ij S 2 p =s 2 w (MS within groups = mean square error)

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One-way ANOVA table SourceDFSum of Squares Mean Square F Groups Error I-1 N-I SSG SSE SSM/DFM SSE/DFE F=MSM/ MSE TotalN-1SSTSST/DFT

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Standard Error Bars http://www.uvm.edu/~jleonard/AGRI85/spring2004/Standard_Err or_Bars_in_Excel.html http://www.uvm.edu/~jleonard/AGRI85/spring2004/Standard_Err or_Bars_in_Excel.html

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Contrasts

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Two-way ANOVA table

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Example

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More example Recycling rate: 36 towns in eastern Massachusetts. Region (northeast, northwest, southeast or southwest) “Pay as you throw” program (Pay; Yes or No). a) Complete ANOVA table

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Interaction Qualitative QuantitativeNo Interaction

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Multiple Regression and ANOVA One-way ANOVA Multiple Regression (ANOVA table) Analysis of Variance Source DF SS MS F P Regression 3 207.281 69.094 9.24 0.000 Residual Error 74 553.437 7.479 Total 77 760.718

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