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1 Armitage t 1.2 SAS Proc GLM

2 t sex N y y y Diff (1-2) t t Pr > t y Pooled Equal < t p t 2

3 sex N x x x Diff (1-2) t t Pr > t x Pooled Equal < kg p * *2 *1 *2 3

4 F Pr > F Model <.0001 Error Corrected Total Type III F Pr > F sex x <.0001 sex p x 3 model / solution clparm *3 t Pr > t 95% Intercept B sex B sex B..... x < Note X X B *4 x *5 100 *3 solution clparm 95 *4 Note 0 B *5 (better ) PPK 4

5 2.1.4 t t t t *6 Error 37.3 = 6.1 * y t 4 y *6 *7 Error 5

6 t dose N y y y Diff (1-2) t t Pr > t y Pooled Equal *8 t p x y x x 5 x x t *

7 dose N x x x Diff (1-2) t t Pr > t x Pooled Equal <.0001 x y x 6 x y 4 x F Pr > F Model <.0001 Error Corrected Total

8 Type III F Pr > F dose <.0001 x <.0001 dose x 1 model /solution clparm t Pr > t 95% Intercept B < dose B < dose B..... x < Note X X B y x y = x (= x) y = x (= x) *9 x 6.45 x 1 y 2.91 x x x 1 * 10 2 x Proc MEANS : x x N x = ŷ = x = ŷ = x = Proc GLM model lsmeans lsmeans dose; * 11 2 dose y *9 2 *10 *11 8

9 2.2.4 t t t t 2.5 x x Error 7.7 = y x x y x BioS (2) * 12 (1) (2) (3) (1), (2) 1 2 (3) * 13 (1) (3) *12 Greenland, S. and Robins, J. M. Identifiability, exchangeability, and epidemiological confounding. International Journal of Epidemiology 1986;15: Rothman, K. J., Greenland, S. Modern Epidemiology, 2nd ed. Philadelphia: Lippincott and Raven, *13 9

10 3 3 * 14 * 15 (2) x y 60 y 7 y y t dose N y y y Diff (1-2) t t Pr > t y Pooled Equal p y x *14 *15 10

11 8 x 1 2 t dose N x x x Diff (1-2) t t Pr > t x Pooled Equal y x 11

12 9 x y * 16 2 * 17 F Pr > F Model <.0001 Error Corrected Total Type III F Pr > F dose x <.0001 dose p t Pr > t Intercept B <.0001 dose B dose B... x <.0001 Note X X B *16 y *

13 y = x (= x) y = x (= x) x 6.06 x Proc MEANS : x x N x = 5.07 * 18 2 ŷ = x = ŷ = x = SAS model lsmeans dose; 2 dose y t t t t Error = y y x x 3.3 *18 13

14 = * x * 20 x 4 * 21 *19 1 t *20 t x 6.1 =0 *21 14

15 1 SAS SAS y x dose proc glm data=d1; class dose; model y= dose x / solution clparm ; lsmeans dose; run; quit; model solution clparm dose 2 lsmeans 2 t t t y 11, y 12,, y 1n N(µ 1, σ 2 ) y 21, y 22,, y 2n N(µ 2, σ 2 ) y ij = µ i + ɛ ij ( ɛij N(0, σ 2 ) ) i = 1, 2, j = 1,, n i j x ij y ij = µ i + βx ij + ɛ ij ( ɛij N(0, σ 2 ) ) i = 1, 2, j = 1,, n ( ) x ij x y x 15

16 3 1 2 x 10 x * 22 2 x *22 y x = 10 x y 2 x x x x y 16

H22 BioS (i) I treat1 II treat2 data d1; input group patno treat1 treat2; cards; ; run; I

H22 BioS (i) I treat1 II treat2 data d1; input group patno treat1 treat2; cards; ; run; I H BioS (i) I treat II treat data d; input group patno treat treat; cards; 8 7 4 8 8 5 5 6 ; run; I II sum data d; set d; sum treat + treat; run; sum proc gplot data d; plot sum * group ; symbol c black

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H22 BioS t (i) treat1 treat2 data d1; input patno treat1 treat2; cards; ; run; 1 (i) treat = 1 treat =

H22 BioS t (i) treat1 treat2 data d1; input patno treat1 treat2; cards; ; run; 1 (i) treat = 1 treat = H BioS t (i) treat treat data d; input patno treat treat; cards; 3 8 7 4 8 8 5 5 6 3 ; run; (i) treat treat data d; input group patno period treat y; label group patno period ; cards; 3 8 3 7 4 8 4 8 5

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5 Armitage x 1,, x n y i = 10x i + 3 y i = log x i {x i } {y i } 1.2 n i i x ij i j y ij, z ij i j 2 1 y = a x + b ( cm) x ij (i j )

5 Armitage x 1,, x n y i = 10x i + 3 y i = log x i {x i } {y i } 1.2 n i i x ij i j y ij, z ij i j 2 1 y = a x + b ( cm) x ij (i j ) 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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α 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 +

α 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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(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

(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,

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,, Poisson 3 3. t t y,, y n Nµ, σ 2 y i µ + ɛ i ɛ i N0, σ 2 E[y i ] µ * i y i x i y i α + βx i + ɛ i ɛ i N0, σ 2, α, β *3 y i E[y i ] α + βx i

,, Poisson 3 3. t t y,, y n Nµ, σ 2 y i µ + ɛ i ɛ i N0, σ 2 E[y i ] µ * i y i x i y i α + βx i + ɛ i ɛ i N0, σ 2, α, β *3 y i E[y i ] α + βx i Armitage.? SAS.2 µ, µ 2, µ 3 a, a 2, a 3 a µ + a 2 µ 2 + a 3 µ 3 µ, µ 2, µ 3 µ, µ 2, µ 3 log a, a 2, a 3 a µ + a 2 µ 2 + a 3 µ 3 µ, µ 2, µ 3 * 2 2. y t y y y Poisson y * ,, Poisson 3 3. t t y,, y n Nµ,

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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 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 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

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