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CH6.SeveralMeans.pdf - Chapter 6 Comparisons of Several...

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Chapter 6. Comparisons of Several Multivariate Means 1. Paired comparison and Repeated measurements Paired comparison: Univariate case Let ( X 11 , X 12 ) , . . . , ( X n 1 , X n 2 ) i.i.d N 2 ( µ , Σ ) . We want to do inference on δ = µ 1 µ 2 . Paired t-test statistics Let D i = X i 1 X i 2 Let t = D δ s d / n t n 1 where D = 1 n n X i =1 D i and s 2 d = 1 n 1 n X i =1 ( D i D ) 2 . Paired comparison: Multivariate case Let ( X 11 X 21 ) , . . . , ( X 1 n X 2 n ) i.i.d N 2 p ( µ , Σ ) . We want to do inference on δ = µ 1 µ 2 . Let D i = X 1 i X 2 i Let T 2 = n ( D δ ) S 1 d ( D δ ) ( n 1) p n p F p,n p where D = 1 n n X i =1 D i and S d = 1 n 1 n X i =1 ( D i D )( D i D ) . Reject H 0 : δ = δ 0 in favor of H 1 : δ ̸ = δ 0 iff T 2 0 = n ( ¯ D δ 0 ) t S 1 d ( ¯ D δ 0 ) ( n 1) p n p F p,n p ( α ) 1
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A 100(1 α )% confidence region for δ δ : ( D δ ) S 1 d ( D δ ) ( n 1) p n p F p,n p ( α ) 100(1 α )% T 2 simultaneous confidence intervals for δ i are δ i : D i ± s ( n 1) p n p F p,n p ( α ) s s 2 d i n 100(1 α )% Bonferroni simultaneous confidence intervals for δ i are δ i : D i ± t n 1 α 2 p s s 2 d i n . Inference on contrasts Let X 1 , . . . , X n i.i.d. N q ( µ , Σ ) . contrast : c µ for c 1 = 0 Wish to test H 0 : C µ = 0 versus H 0 : C µ ̸ = 0 where C is a ( q 1) × q matrix s.t. C1 = 0, rank( C ) = q 1. A repeated measure design (p.279) may be represented by C = 1 1 0 · · · 0 1 0 1 · · · 0 . . . . . . . . . . . . . . . 1 0 0 · · · 1 . 2
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Let Y i = CX i N q 1 ( C µ , C Σ C ) . Hence, T 2 test based on Y 1 , . . . , Y n is to reject H 0 if T 2 0 = n ( C ¯ X ) ( CSC ) 1 C ¯ X > ( n 1)( q 1) n q + 1 F q 1 ,n q +1 ( α ) . Note that the d.f.s are q 1 and n q + 1 instead of q and n q. 위의 수식 q ± ± q 1 contrast 을 잡 q 1 × q contrast- C ± q 1 contrast ± r constrat r × q constrat- C . contrast 이유 repeated design 과 같 ± 이 있 C 1 × q (1 , 0 , 0 , . . . , 1) ± . Using T 2 , we can construct a confidence region and simultaneous confidence in- tervals of mean differences ( µ i µ j ). Example 6.2 (p.281) (Anesthetics) repeated measurement design for comparing treatments : CO 2 pressure low high Halothane O trt4 trt3 X trt2 trt1 ( X j 1 , · · · , X j 4 ) ( j = 1 , 2 , · · · , 19) measurement : milliseconds between heartbeats Data (p.282) 3
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Contrasts of interest H contrast ( H ( O ) H ( X )) : ( µ 3 + µ 4 ) ( µ 1 + µ 2 ) CO 2
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