# note4 - STAT5044 Regression and Anova Inyoung Kim Outline 1...

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Unformatted text preview: STAT5044: Regression and Anova Inyoung Kim Outline 1 Testing 2 Confidence interval 3 ANOVA table Testing procedure Decide what question we want to test: ⇒ Null hypothesis and alternative hypothesis Test statistic Decision rule Make conclusion. Hypothesis H : β 1 = β 10 vs H A 1 : β 1 > β 10 H : β 1 = β 10 vs H A 1 : β 1 < β 10 H : β 1 = β 10 vs H A 3 : β 1 6 = β 10 ⇒ What do we need for testing this hypothesis? Test statistics Test statistic? ˆ β 1- β 10 q ˆ Var ( ˆ β 1 ) = ˆ β 1- β 10 p ˆ σ 2 / S xx = ( ˆ β 1- β 10 ) / p σ 2 / S xx p ˆ σ 2 / σ 2 = ∼ t n- 2 under which hypothesis? Test statistic Test statistic? ˆ β 1- β 10 q ˆ Var ( ˆ β 1 ) = ˆ β 1- β 10 p ˆ σ 2 / S xx = ( ˆ β 1- β 10 ) / p σ 2 / S xx p ˆ σ 2 / σ 2 = ∼ t n- 2 under which hypothesis? null hypothesis. Why does the distribution of test statistic follow t n- 2 under H ? Test statistic Numerator: Under H , ˆ β 1- β 10 √ σ 2 / Sxx ∼ N ( , 1 ) Denominator Let ˆ σ 2 = ∑ n i = 1 ( y i- ˆ y i ) 2 ( n- 2 ) r ˆ σ 2 σ 2 ∼ s χ 2 n- 2 ( n- 2 ) ˆ σ 2 σ 2 = SSE / ( n- 2 ) σ 2 ∼ χ 2 n- 2 ( n- 2 ) NOTE: SSE σ 2 = y t ( I- H ) y σ 2 = y t Ay , where A = 1 σ 2 ( I- H ) AV = ( I- H ) 1 σ 2 σ 2 = I- H : idempotent, Rank ( A ) = Rank ( I- H ) = n- 2 Test statistic ˆ β and ˆ σ are independent: ⇒ ( X t X )- 1 X t Y = By and y t ( I- H ) y = y t Ay are independent....
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note4 - STAT5044 Regression and Anova Inyoung Kim Outline 1...

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