i Armitage Q. Bonferroni 1 SAS ver9.1.3 version up 2 *1 *2 FWE *3 2.1 vs vs vs 2.2 5µg 10µg 20µg 5µg 10µg 20µg vs 5µg vs 10µg vs 20µg *1 *2 *3 FWE 1
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1 i Armitage Q Boferroi SAS ver93 versio up * * FWE *3 vs vs vs 5µg 0µg 0µg 5µg 0µg 0µg vs 5µg vs 0µg vs 0µg * * *3 FWE
2 3 A B C D E (i A B C D E (ii A B C D E (iii A B C D E (iv A B C D A < B C D A < B < C < D FWE α *4 (i (ii (i FWE (iii *5 (ii,000,000 (iv (iii (00 α (00 α 3 Z, Z N(0, 005 Z > 96, Z > 96 5 FWE 005 F W E P ( Z > 96 or Z > F W E P ( Z > 4 or Z > *4 *5
3 FWE 005 *6 FWE FWE *7 F W E P ( Z > 4 or Z > F W E P ( max Z i > 4 i, max i, Z i Tukey-Kramer Duett 33 5µg 0µg 0µg Boferroi α α , (i (ii x x x N(µ, x x x N(µ, I x I x I x II N(µ I, *6 P ( Z > F W E P ( Z > 4 or Z > 4 P ( Z 4 ad Z 4 P ( Z 4 P ( Z 4 (* Z, Z *7 3
4 *8 x, x,, x I I I j j (x j x (x j x I (x Ij x I j 4 Tukey-Kramer 4 I * 9 H : µ µ H A : µ µ H 3 : µ µ 3 H3 A : µ µ 3 H I : µ µ I HI A : µ µ I H 3 : µ µ 3 H3 A : µ µ 3 H (I I : µ I µ I H(I I A : µ I µ I I IC I(I 4 t * 0 t i j x i x j t ij ( i + j ij *8 *9 H ij H A ij *0 F t χ F 4
5 * t ij i j i j ij ( ( i i ( i + ( j + ( j j i + j ( i j (x ik x i + (x jk x j k ij I ( ( ( + ( + + ( I + ( + + ( I I I (x j x + (x j x + + (x Ij x I I j j ( I k j I I i (x ij x i i j x i x j T ij ( i + j Tukey-Kramer Tukey * t χ F t T ij ( I t * 3 43 FWE FWE α 3 I(I {T ij } T max i<j T ij max i<j x i x j ( i + j FWE * 4 T T 3 * 005 t > t( +, 0975 (t( +, 0975 ( + t 975 * Tukey Tukey-Kramer *3 t *4 max i<j i < j i, j x i x j x j x i i j 5
6 FWE T * 5 6 FWE FWE FWE FWE I 3 0,, 3 H : µ µ H A : µ µ H 3 : µ µ 3 H A 3 : µ µ 3 H 3 : µ µ 3 H A 3 : µ µ 3 T * 6 µ µ µ 3 0, T (in(0, ( (ii 0 x x x 3 (iii (iv (i (iii T, T 3, T 3 T (v (i (iv,000,000 * 7 T (vi T 5 5 * 8 µ µ µ 3 0, * 9 N(0, 44 ( SAS SAS (iprobmc (iiproc glm * 0 (iprobmc (ca FWE (alpha (df (k *5 *6 6 FWE *7,000,000 *8 *9 µ µ µ 3 *0 proc mixed 6
7 data d; c ProbMC( Rage,, -alpha, df, k; ca c /sqrt(; ru; ca Rage : Tukey-Kramer : -alpha : Studet -( df : ( ( k : c Studet 00(-alpha Studet ca c ProbMC c (iiproc glm d dose bp proc glm datad; class dose; model bp dose; meas dose /tukey; ru; * 45 H ij : µ i µ j alpha T ij > ca (ca ProbMC i, j H ij 46 balaced ubalaced balaced ubalaced balaced Tukey ubalaced Tukey-Kramer proc glm probmc Tukey Tukey-Kramer * * SAS SAS LATEX * Tukey-Kramer 7
8 5 Duett Tukey-Kramer Tukey-Kramer 5 vs (I H : µ µ H A : µ µ H 3 : µ µ 3 H3 A : µ µ 3 H I : µ µ I HI A : µ µ I 5 Tukey-Kramer t Tukey-Kramer x x j T j ( + j 53 FWE Tukey-Kramer T j T max j I T j max j I x x j ( + j ( Tukey-Kramer I 3 0,, 3 H : µ µ H A : µ µ H 3 : µ µ 3 H3 A : µ µ 3 8
9 Tukey-Kramer µ µ µ 3 0, (i N(0, ( (ii 0 x x x 3 (iii (iv (i (iii T, T 3, T (v (i (iv,000,000 T (vi T ( SAS Tukey-Kramer (iprobmc (iiproc glm (iprobmc Tukey-Kramer (abalaced ubalaced (b Rage Duett * 3 (c (d k ( - * 4 (balaced balaced data d; ca ProbMC( Duett,, -alpha, df, k-; ru; Tukey-Kramer Duett : Duett : -alpha : -( df : Tukey-Kramer ( -( k- : ( - Tukey-Kramer Studet *3 Duett Duett Duett *4 9
10 (ubalaced ubalaced data d; 0; 3; 34; k3; df k; lambda sqrt(/(+; lambda sqrt(3/(+3; ca ProbMC( Duett,, -alpha, df, k-, lambda, lambda; ru; (iiproc glm Tukey-Kramer d dose bp (balaced proc glm datad; class dose; model bp dose; meas dose /duett; ru; (ubalaced proc glm datad; class dose; model bp dose; lsmeas dose / pdiff cl adjust duett; ru; 55 Tukey-Kramer H j : µ µ j alpha T j > ca 56 balaced ubalaced balaced ubalaced Duett Tukey-Kramer proc glm probmc balaced ubalaced 0
11 6 FWE T FWE FWE α H : µ µ H 3 : µ µ 3 H 4 : µ µ 4 F W E all P ( H A H A 3 H A 4 µ µ µ 3 µ 4 H, H 3 H 4 H 4 FWE F W E,3 P (H A H A 3 µ µ µ 3 µ 4 F W E,3 F W E all F W E all FWE * 5 T *5 FWE α FWE α
12 7 FWE Tukey-Kramer FWE T max i<j T ij x x T (, T 3 + c x x 3 ( + 3 P (max{t, T 3 } > c P (T > c or T 3 > c P (T c ad T 3 c P (T c P (T 3 c FWE T T 3 T, T 3 t * 6 8 Tukey-Kramer Tukey-Kramer µ µ µ 3, x ij N(0, t N(µ, t 5 N(0, t 5 N(µ, t N(0, t x,, x N(µ, x,, x N(µ, N(0, x,, x, x,, x y,, y N(0, y,, y N(0, x ij y ij + µ (i,, j,, i *6
13 {x ij } x,, x y,, y y,, y x,, x x,, x t {y ij } * 7 x j x j ( y ij + µ j ȳ + µ ȳ y j j x ȳ + µ j j j (x j x ( ( y j + µ ( ȳ + µ (y j ȳ j (y j ȳ ( ( ( + ( + ( ( ( y j ȳ + ( ( + ( j (y j ȳ + (y j ȳ + + j i (y ij ȳ i i j j j (y j ȳ *7 x ij y ij µ, 3
14 t x x t ( + ( + x x i (x ij x i + i j ( ȳ + µ ( ȳ + µ i (y ij ȳ i ( + ( i j ȳ ȳ i (y ij ȳ i i j ȳ ȳ ( + i + (y ij ȳ i ( + i j ȳ ȳ i (y ij ȳ i + i j µ, N(0, {y ij } t N(µ, t N(0, t N(µ, t N(0, t N(0,,000,000 ( + t Tukey-Kramer t pool Tukey-Kramer * 8 N(0, 9 Studet 9 Tukey-Kramer Studet Tukey-Kramer *8 Duett 4
15 Studet u,, u k, χ : u,, u k N(0,, χ χ (ν Q R k,ν max i<j u i u j χ ν Studet Q R k,ν Studet * 9 Tukey-Kramer x i x j T ij ( i + j Studet Tukey Tukey-Kramer i j T ij T ij x i x j ( + x i x j ( x i,, x i N(µ i, x j,, x j N(µ j, x i N (µ i,, x j N (µ j, H ij : µ i µ j ( µ x i, x j N (µ, Q R k,ν x i, x j N(0, ( x i µ, x j µ T ij x i x j N(0, ( x i µ ( x j µ T ij ( x i µ q ( x i µ q ( q x j µ c ( q xj µ c *9 SAS ProbMC ( Rage Q R k,ν 5
16 (i ( q x i µ ( q x j µ N(0, (ii c ν ( ν c χ (ν * 30*3 (iii i, j * 3 ν ν (ii c χ c (ν ν Q R k,ν χ ν max T max i<j T ij T max i<j T ij max i<j max i<j Q R 3,ν ( x i µ q c ( q xj µ c ( x i µ q ( q x j µ c * 33 T 95 Q R 3,ν 95 9 Tukey Tukey-Cramer Tukey Tukey-Cramer FWE *30 *3 ν ( ( ν c χ (ν *3 *33 Q R 3,ν 3 ν ( -( 6
17 0 Duett (balaced Tukey Duett Q D k,ν Duett * 34 u,, u k, χ : u,, u k N(0,, χ χ (ν Duett Q D u u i k,ν max i k χ ν Studet Q D k,ν 95 5 *34 Duett s two-sided rage distributio 7
(iii) x, x N(µ, ) z = x µ () N(0, ) () 0 (y,, y 0 ) (σ = 6) *3 0 y y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 y ( ) *4 H 0 : µ
t 2 Armitage t t t χ 2 F χ 2 F 2 µ, N(µ, ) f(x µ, ) = ( ) exp (x µ)2 2πσ 2 2 0, N(0, ) (00 α) z(α) t * 2. t (i)x N(µ, ) x µ σ N(0, ) 2 (ii)x,, x N(µ, ) x = x + +x ( N µ, σ2 ) (iii) (i),(ii) x,, x N(µ,
More informationα β *2 α α β β α = α 1 β = 1 β 2.2 α 0 β *3 2.3 * *2 *3 *4 (µ A ) (µ P ) (µ A > µ P ) 10 (µ A = µ P + 10) 15 (µ A = µ P +
Armitage 1 1.1 2 t *1 α β 1.2 µ x µ 2 2 2 α β 2.1 1 α β α ( ) β *1 t t 1 α β *2 α α β β α = α 1 β = 1 β 2.2 α 0 β 1 0 0 1 1 5 2.5 *3 2.3 *4 3 3.1 1 1 1 *2 *3 *4 (µ A ) (µ P ) (µ A > µ P ) 10 (µ A = µ P
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Armitage 1 1.1 2 t 1.2 SAS Proc GLM 2 2.1 1 1 2.1.1 50 1 1 t sex N y 50 116.45 119.6 122.75 11.071 1.5657 93.906 154.32 y 50 127.27 130.7 134.13 12.072 1.7073 102.68 163.37 y Diff (1-2) -15.7-11.1-6.504
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5 Armitage. x,, x n y i = 0x i + 3 y i = log x i x i y i.2 n i i x ij i j y ij, z ij i j 2 y = a x + b 2 2. ( cm) x ij (i j ) (i) x, x 2 σ 2 x,, σ 2 x,2 σ x,, σ x,2 t t x * (ii) (i) m y ij = x ij /00 y
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t χ F Q t χ F µ, σ N(µ, σ ) f(x µ, σ ) = ( exp (x ) µ) πσ σ 0, N(0, ) (00 α) z(α) t χ *. t (i)x N(µ, σ ) x µ σ N(0, ) (ii)x,, x N(µ, σ ) x = x+ +x N(µ, σ ) (iii) (i),(ii) z = x µ N(0, ) σ N(0, ) ( 9 97.
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