Chp4 - Simultaneous Inference and Other Topics in...

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Simultaneous Inference and Other Topics in Regression Analysis Simultaneous inference of ( β 0 , β 1 ) and their linear functions Regression through the origin Effects of measurement error Inverse prediction Choice of X levels 1 Simultaneous Inference of β 0 and β 1 We would like to get 1 α confidence regions that the conclusions for both β 0 and β 1 are correct. We call the set of estimates (or tests) of interest the family of estimates (or tests). A statement confidence coefficient is a probability statement about one parameter, which indicates the proportion of correct estimates that are obtained for repeated samples. A family confidence coefficient indicates the proportion of families of estimates that are entirely correct for re- peated samples. 2
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Single confidence intervals Pr | bj β j | s ( b j ) t (1 α, 2 , n 2) = 1 α ; j = 0 , 1 . (1) Generally, Pr | b j β j | s ( b j ) t (1 α, 2 , n 2); j = 0 , 1 (2) = 1 j =0 Pr | b j β j | s ( b j ) t (1 α, 2 , n 2) unless they are independent. 3 Bonferroni Joint Confidence Intervals In theory we can work out the joint probability (2), which involves a complicated two-dimension integral. But we can easily get an lower bound using the following Bonferroni inequality Pr m k =1 A k m k =1 Pr( A k ); Pr m k =1 A c k 1 m k =1 Pr( A k ) Therefore, with A j = | bj β j | s ( b j ) t (1 α, 2 , n 2) , (2) 1 2 α. Generally for m (correlated) tests, we can use α/m significance level for individual test to guarantee an overall Type-I error α . 4
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Simultaneous Estimation of Mean Responses Goal: estimate the mean response at X levels { X h ; h = 1 , · · · , m } . Working-Hotelling Procedure: The Working-Hotelling confidence band covers the entire regression line. So for confidence interval ˆ Y h ± Ws ( ˆ Y h ); W = 2 F (1 α, 2 , n 2) , and the family confidence coefficient for these simultaneous estimates will be at least 1 α . Bonferroni Procedure: With Bonferroni procedure, we just need to use α/m as the individual Type-I error for each X h . ˆ Y h ± Bs ( ˆ Y h ); B = t (1 α 2 m , n 2) . 5 Simultaneous Prediction Intervals for New Observations Goal: predict new observations at X levels { X h ; h = 1 , · · · , m } . Scheff ´ e Procedure: Use the following confidence interval ˆ Y h ± Ws ( pred ); W = mF (1 α, m, n 2) , Bonferroni Procedure With Bonferroni procedure, we just need to use α/m as the individual Type-I error for each X h . ˆ Y h ± Bs ( pred ); B = t (1 α 2 m , n 2) . 6
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Multiple Comparison Coefficients 0.00 0.02 0.04 0.06 0.08 0.10 2.0 2.5 3.0 3.5 4.0 4.5 5.0 α Critical Values (m=3) Sheffe Working-Hotelling Bonferroni 0.00 0.02 0.04 0.06 0.08 0.10 2.0 2.5 3.0 3.5 4.0 4.5 5.0 α Critical Values (m=6) Sheffe Bonferroni Working-Hotelling 7 Toluca Company Example Joint Intervals for β 0 and β 1 : Point estimation b 1 = 25 i =1 ( X i ¯ X ) Y i 25 i =1 ( X i ¯ X ) 2 = 3 .
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