2 H23 BioS (i) data d1; input group patno t sex censor; cards;
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- しなつ ちづ
- 5 years ago
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1 H BioS (i) data d1; input group patno t sex censor; cards; ; run; 1
2 proc lifetest data d1 plot(lls); time t * censor (0); strata sex group; run; log-log 1 log-log log(t) Cox Kaplan-Meier 4 1: sex 0 group 0 t *
3 Note : sex 0 group 1 t * * * *... 0 Note : sex 1 group 0 t * * *... 0 Note
4 4: sex 1 group 1 t * * *... 0 Note (ii) log-log S ij (t) (i : j : ) log( log(s ij (t))) log(t) t j S i (t j ) log( log(s i (t i ))) t j S 00 (t j ) log(t j ) log( log(s 00 (t j ))) t j S 10 (t j ) log(t j ) log( log(s 10 (t j ))) t j S 01 (t j ) log(t j ) log( log(s 01 (t j ))) t j S 11 (t j ) log(t j ) log( log(s 11 (t j ))) (i) (iii) Proc Phreg Cox proc phreg data d1; model t * censor(0) group sex / ties efron; run; 4
5 PHREG WORK.D1 t censor 0 EFRON (GCONV1E-8) - LOG L AIC SBC : BETA0 Pr > ChiSq Score Wald
6 Pr > ChiSq group sex Wald group 1.87 χ.564 p *1 (iv) Cox (a) Cox λ(t x i, y j ) λ 0 (t) exp(αx i + βy j ) (i 0, 1, j 0, 1) x 0 0, x 1 1, y 0 0, y 1 1 ( ) λ 00 (t) λ(t 0, 0) λ 0 (t) (1) ( ) λ 10 (t) λ(t 1, 0) λ 0 (t) exp(α) () ( ) λ 01 (t) λ(t 0, 1) λ 0 (t) exp(β) () ( ) λ 11 (t) λ(t 1, 1) λ 0 (t) exp(α + β) (4) * 0 t λ ij (t) λ(t x i, y j ) Λ(t x i, y j ) t 0 t 0 ( t 0 λ(u x i, y j )du λ 0 (u) exp(αx i + βy j )du ) λ 0 (u)du exp(αx i + βy j ) t 0 λ 0(u)du Λ 0 (t) ( ) Λ 00 (t) Λ(t 0, 0) Λ 0 (t) ( ) Λ 10 (t) Λ(t 1, 0) Λ 0 (t) exp(α) ( ) Λ 01 (t) Λ(t 0, 1) Λ 0 (t) exp(β) ( ) Λ 11 (t) Λ(t 1, 1) Λ 0 (t) exp(α + β) (b) (a) H 0 : β 0 H 1 : β 0 *1 Wald Wald * λ ij (t) λ(t x i, y j ) i j λ ij (t) x i, y j λ(t x i, y j ) 6
7 (c) λ(t 0, 0) λ 10(t) λ 00 (t) λ 0(t) exp(α) exp(α) λ 0 (t) λ 01(t) λ 00 (t) λ 0(t) exp(β) exp(β) λ 0 (t) λ 11(t) λ 00 (t) λ 0(t) exp(α + β) exp(α + β) λ 0 (t) t log(λ 00 (t)) log(λ 0 (t)) log(λ 10 (t)) log(λ 0 (t)) + α log(λ 01 (t)) log(λ 0 (t)) + β log(λ 11 (t)) log(λ 0 (t)) + α + β t β 1 α, β t log( log) α, β, α + β t 4 * Cox + 1 λ(t x i, y j ) λ 0 (t) t t exp(αx i + βy j ) α, β 1 (d) λ ij (t) T λ ij (t) λ(t x i, y j ) lim dt 0 P r(t < T t + dt x i, y j, t T ) dt F ij (t) i j *4 F ij (t) 1 S ij (t) 1 P r(t > t x i, y j ) P r(t t x i, y j ) λ(t x i, y j ) lim dt 0 P r(t < T t + dt x i, y j, t T ) dt P r(t < T t + dt x i, y j ) 1 lim dt 0 dt P r(t T x i, y j ) F ij (t + dt) F ij (t) 1 lim dt 0 dt 1 F ij (t) * y t + 1 y t + 5 y t y t + 10 *4 F ij (t) F (t x i, y j ) 7
8 F ij (t) S ij (t) (1 S ij(t)) S ij (t) S ij (t) S ij (t) t 0 d log(s ij (t)) dt d log(s ij (t)) dt dt λ(t x i, y j ) t 0 λ(t x i, y j )dt log(s ij (t)) log(s ij (0)) Λ(t x i, y j ) (a) S ij (t) exp (Λ(t x i, y j )) ( S ij (0) 1) S 00 (t) exp (Λ(t 0, 0)) exp (λ 0 (t)) S 10 (t) exp (Λ(t 1, 0)) exp (λ 0 (t)) exp(α) S 01 (t) exp (Λ(t 0, 1)) exp (λ 0 (t)) exp(β) S 11 (t) exp (Λ(t 1, 1)) exp (λ 0 (t)) exp(α + β) (e) log( log(s ij (t))) S 00 (t) exp (λ 0 (t)) log(s 00 (t)) λ 0 (t) log( log(s 00 (t))) log(λ 0 (t)) (5) S 10 (t) exp (λ 0 (t) exp(α)) log(s 10 (t)) λ 0 (t) exp(α) log( log(s 10 (t))) log(λ 0 (t)) + α (6) S 01 (t) exp (λ 0 (t) exp(β)) log(s 01 (t)) λ 0 (t) exp(β) log( log(s 01 (t))) log(λ 0 (t)) + β (7) S 11 (t) exp (λ 0 (t) exp(α + β)) log(s 11 (t)) λ 0 (t) exp(α + β) log( log(s (t))) log(λ 0 (t)) + α + β (8) (f) (5) (8) log( log(s ij (t))) y log(t) y log( log(s ij (t))) y log( log(s 00 (t)) log(λ 0 (t)) 8
9 y log( log(s 10 (t)) log(λ 0 (t)) + α y log( log(s 01 (t)) log(λ 0 (t)) + β y log( log(s 11 (t)) log(λ 0 (t)) + α + β log(λ 0 (t)) log(λ 0 (t)) α, β, α + β *5 log( log(s ij (t))) 1 (g) at risk number *5 a, b log( log(λ 0 (t))) a + bt log( log(λ 0 (t))) exp(a + bt + bt ) Cox 9
10 (1) (0 1 ) at risk number ( ) (0 1 ) (0) t L t 4 at risk number L 4 λ(4 0, 0) λ(4 0, 0) + 5 λ(4 1, 0) + 6 λ(4 0, 1) + 5 λ(4 1, 1) λ 0 (4) λ 0 (4) + 5 λ 0 (4) exp(α) + 6 λ 0 (4) exp(β) + 5 λ 0 (4) exp(α + β) exp(α) + 6 exp(β) + 5 exp(α + β) 10
11 1 1 7 at risk number 6 4 L 7 λ(7 0, 1) λ(7 0, 0) + λ(7 1, 0) + 6 λ(7 0, 1) + 4 λ(4 1, 1) λ 0 (7) exp(β) λ 0 (7) + λ 0 (7) exp(α) + 6 λ 0 (7) exp(β) + 4 λ 0 (7) exp(α + β) exp(β) + exp(α) + 6 exp(β) + 4 exp(α + β) 8 Kaplan-Meier at risk number at risk number 5 4 L 8 λ(8 1, 1) λ(8 0, 0) + λ(8 1, 0) + 5 λ(8 0, 1) + 4 λ(4 1, 1) λ 0 (8) exp(α + β) λ 0 (8) + λ 0 (8) exp(α) + 5 λ 0 (8) exp(β) + 4 λ 0 (8) exp(α + β) exp(α + β) + exp(α) + 5 exp(β) + 4 exp(α + β) 0 at risk number 0 L 0 λ(0 1, 1) λ(0 1, 0) + λ(8 0, 1) + λ(4 1, 1) λ 0 (0) exp(α + β) λ 0 (0) exp(α) + λ 0 (0) exp(β) + λ 0 (0) exp(α + β) exp(α + β) exp(α) + exp(β) + exp(α + β) Efron at risk number L λ( 0, 0) 4 λ( 0, 0) + 6 λ( 1, 0) + 6 λ( 0, 1) + 5 λ( 1, 1) ( 4 1 ) λ( 0, 0) + ( 6 1 λ( 1, 0) ) (10) λ( 1, 0) + 6 λ( 0, 1) + 5 λ( 1, 1) λ 0 () 4 λ 0 () + 6 λ 0 () exp(α) + 6 λ 0 () exp(β) + 5 λ 0 () exp(α + β) λ 0 () exp(α) 7 λ 0() + 11 λ 0() exp(α) + 6 λ() exp(β) + 5 λ 0 () exp(α + β) exp(α) + 6 exp(β) + 5 exp(α + β) exp(α) exp(α) + 6 exp(β) + 5 exp(α + β) 1 (9) 1 1 (9) 11
12 1/ (10) 1/ 1 5 at risk number / 1 L 5 λ(5 1, 0) λ(5 0, 0) + 5 λ(5 1, 0) + 6 λ(5 0, 1) + 5 λ(5 1, 1) λ(5 1, 0) λ(5 0, 0) + 4 λ(5 1, 0) + 6 λ(5 0, 1) + 5 λ(5 1, 1) λ 0 (5) exp(α) λ 0 (5) + 5 λ 0 (5) exp(α) + 6 λ 0 (5) exp(β) + 5 λ 0 (5) exp(α + β) λ 0 (5) exp(α) λ 0 (5) + 4 λ 0 (5) exp(α) + 6 λ(5) exp(β) + 5 λ 0 (5) exp(α + β) exp(α) + 5 exp(α) + 6 exp(β) + 5 exp(α + β) exp(α) + 4 exp(α) + 6 exp(β) + 5 exp(α + β) 15 at risk number / / 1/ L 15 λ(15 0, 0) λ(15 0, 0) + λ(15 1, 0) + 5 λ(15 0, 1) + λ(15 1, 1) λ(15 0, 0) ( ) ( λ(15 0, 0) + λ(15 1, 0) ) λ(15 0, 1) + λ(15 1, 1) λ(15 0, 1) ( ) ( λ(15 0, 0) + λ(15 1, 0) ) λ(15 0, 1) + λ(15 1, 1) λ 0 (15) λ 0 (15) + λ 0 (15) exp(α) + 5 λ 0 (15) exp(β) + λ 0 (15) exp(α + β) λ 0 (15) 4 λ 0(15) + λ 0 (15) exp(α) + 14 λ(15) exp(β) + λ 0(15) exp(α + β) λ 0 (15) exp(β) λ 0(15) + λ 0 (15) exp(α) + 1 λ(15) exp(β) + λ 0(15) exp(α + β) 1 + exp(α) + 5 exp(β) + exp(α + β) exp(α) + exp(β) + exp(α + β) + exp(α) + 1 exp(β) exp(β) + exp(α + β) α, β L(α, β) L t L(α, β) L L 4 L 5 L 7 L 8 L 15 L 0 (11) 1
13 l(α, β) (11) l(α, β) log L + log L 4 + log L 5 + log L 7 + log L 8 + log L 15 + log L 0 (1) log L α log (4 + 6 exp(α) + 6 exp(β) + 5 exp(α + β)) log log L 4 log ( + 5 exp(α) + 6 exp(β) + 5 exp(α + β)) ( ) exp(α) + 6 exp(β) + 5 exp(α + β) log L 5 α log ( + 5 exp(α) + 6 exp(β) + 5 exp(α + β)) + α log ( + 4 exp(α) + 6 exp(β) + 5 exp(α + β)) α log ( + 5 exp(α) + 6 exp(β) + 5 exp(α + β)) log ( + 4 exp(α) + 6 exp(β) + 5 exp(α + β)) log L 7 β log ( + exp(α) + 6 exp(β) + 4 exp(α + β)) log L 8 α + β log ( + exp(α) + 5 exp(β) + 4 exp(α + β)) ( 4 log L 15 log ( + exp(α) + 5 exp(β) + exp(α + β)) log ( ) 1 +β log + exp(α) + exp(β) + exp(α + β) β log ( + exp(α) + 5 exp(β) + exp(α + β)) log ( ) 1 log + exp(α) + exp(β) + exp(α + β) log L 0 (α + β) log ( exp(α) + exp(β) + exp(α + β)) (1) l(α, β) 5α + 4β log (4 + 6 exp(α) + 6 exp(β) + 5 exp(α + β)) log + exp(α) + 14 ) exp(β) + exp(α + β) ( ) exp(α) + exp(β) + exp(α + β) ( ) exp(α) + 6 exp(β) + 5 exp(α + β) log ( + 5 exp(α) + 6 exp(β) + 5 exp(α + β)) log ( + 5 exp(α) + 6 exp(β) + 5 exp(α + β)) log ( + 4 exp(α) + 6 exp(β) + 5 exp(α + β)) log ( + exp(α) + 6 exp(β) + 4 exp(α + β)) log ( + exp(α) + 5 exp(β) + 4 exp(α + β)) log ( + exp(α) + 5 exp(β) + exp(α + β)) ( ) ( ) log + exp(α) + exp(β) + exp(α + β) log + exp(α) + exp(β) + exp(α + β) log ( exp(α) + exp(β) + 4 exp(α + β)) (h) Newton-Raphson α, β α, β 11 l α (α, β) 5 6 exp(α) + 5 exp(α + β) exp(α) + 6 exp(β) + 5 exp(α + β) exp(α) + 5 exp(α + β) exp(α) + 6 exp(β) + 5 exp(α + β) exp(α) + 5 exp(α + β) + 5 exp(α) + 6 exp(β) + 5 exp(α + β) 5 exp(α) + 5 exp(α + β) + 5 exp(α) + 6 exp(β) + 5 exp(α + β) 4 exp(α) + 5 exp(α + β) + 4 exp(α) + 6 exp(β) + 5 exp(α + β) exp(α) + 4 exp(α + β) + exp(α) + 6 exp(β) + 4 exp(α + β) exp(α) + 4 exp(α + β) + exp(α) + 5 exp(β) + 4 exp(α + β) exp(α) + exp(α + β) + exp(α) + 5 exp(β) + exp(α + β) exp(α) + exp(α + β) 4 + exp(α) + 14 exp(β) + exp(α + β) exp(α) + exp(α + β) 1 + exp(α) + exp(β) + exp(α + β) exp(α) + 4 exp(α + β) exp(α) + exp(β) + exp(α + β) 1
14 l β (α, β) 4 6 exp(β) + 5 exp(α + β) exp(α) + 6 exp(β) + 5 exp(α + β) 6 exp(β) + 5 exp(α + β) 7 exp(α) + 6 exp(β) + 5 exp(α + β) exp(β) + 5 exp(α + β) + 5 exp(α) + 6 exp(β) + 5 exp(α + β) 6 exp(β) + 5 exp(α + β) + 5 exp(α) + 6 exp(β) + 5 exp(α + β) 6 exp(β) + 5 exp(α + β) + 4 exp(α) + 6 exp(β) + 5 exp(α + β) 6 exp(β) + 4 exp(α + β) + exp(α) + 6 exp(β) + 4 exp(α + β) 5 exp(β) + 4 exp(α + β) + exp(α) + 5 exp(β) + 4 exp(α + β) 5 exp(β) + exp(α + β) + exp(α) + 5 exp(β) + exp(α + β) exp(α) + 14 exp(β) + exp(α + β) exp(β) + exp(α + β) exp(β) + 4 exp(α + β) exp(α) + exp(β) + exp(α + β) F(θ) 1 + exp(α) + 1 exp(β) + exp(α + β) exp(β) + exp(α + β) θ (α, β) l(α, β) 1 F(θ) l α (α, β) l β (α, β) Newton-Raphson F(θ) α, β l exp(α) + 56 exp(α + β) + 0 exp(α + β) (α) 4 α (4 + 6 exp(α) + 6 exp(β) + 5 exp(α + β)) exp(α) + exp(α + β) + 0 exp(α + β) exp(α) + 6 exp(β) + 5 exp(α + β)) ( exp(α) + 45 exp(α + β) + 0 exp(α + β) 10 exp(α) + 40 exp(α + β) + 0 exp(α + β) ( + 5 exp(α) + 6 exp(β) + 5 exp(α + β)) ( + 5 exp(α) + 6 exp(β) + 5 exp(α + β)) 8 exp(α) + 4 exp(α + β) + 0 exp(α + β) 6 exp(α) + 6 exp(α + β) + 4 exp(α + β) ( + 4 exp(α) + 6 exp(β) + 5 exp(α + β)) ( + exp(α) + 6 exp(β) + 4 exp(α + β)) 6 exp(α) + exp(α + β) + 0 exp(α + β) 4 exp(α) + 14 exp(α + β) + 10 exp(α + β) ( + exp(α) + 5 exp(β) + 4 exp(α + β)) ( + exp(α) + 5 exp(β) + exp(α + β)) exp(α) + 1 exp(α + β) + exp(α + β) exp(α) + 10 exp(α + β) + exp(α + β) ( exp(α) + exp(β) + exp(α + β)) ( + exp(α) + 1 exp(β) + exp(α + β)) 6 exp(α + β) + 1 exp(α + β) ( exp(α) + exp(β) + exp(α + β)) l α β 16 exp(α + β) (4 + 6 exp(α) + 6 exp(β) + 5 exp(α + β)) + 1 exp(α + β) ( exp(α) + 6 exp(β) + 5 exp(α + β)) 15 exp(α + β) ( + 5 exp(α) + 6 exp(β) + 5 exp(α + β)) 0 exp(α + β) ( + 5 exp(α) + 6 exp(β) + 5 exp(α + β)) 14 exp(α + β) ( + 4 exp(α) + 6 exp(β) + 5 exp(α + β)) 10 exp(α + β) ( + exp(α) + 6 exp(β) + 4 exp(α + β)) 7 exp(α + β) ( + exp(α) + 5 exp(β) + 4 exp(α + β)) 6 exp(α + β) ( + exp(α) + 5 exp(β) + exp(α + β)) ( 4 0 exp(α + β) 14 + exp(α) + exp(β) + exp(α + exp(α + β) β)) ( + exp(α) + 1 exp(β) + exp(α + β)) 6 exp(α + β) ( exp(α) + exp(β) + exp(α + β)) 101 l exp(β) + 56 exp(α + β) + 0 exp(α + β) 1 exp(β) + 4 β (4 + 6 exp(α) + 6 exp(β) + 5 exp(α + β)) + exp(α + β) + 55 exp(α + β) ( exp(α) + 6 exp(β) + 5 exp(α + β)) 14
15 18 exp(β) + 45 exp(α + 0β) + exp(α + β) 1 exp(β) + 40 exp(α + β) + 5 exp(α + β) ( + 5 exp(α) + 6 exp(β) + 5 exp(α + β)) ( + 5 exp(α) + 6 exp(β) + 5 exp(α + β)) 1 exp(β) + 4 exp(α + β) + 0 exp(α + β) 1 exp(β) + 0 exp(α + β) + 1 exp(α + β) ( + 4 exp(α) + 6 exp(β) + 5 exp(α + β)) ( + exp(α) + 6 exp(β) + 4 exp(α + β)) 10 exp(β) + exp(α + β) + 1 exp(α + β) 10 exp(β) + 14 exp(α + β) + 4 exp(α + β) ( + exp(α) + 5 exp(β) + 4 exp(α + β)) ( + exp(α) + 5 exp(β) + exp(α + β)) 56 9 ( 4 exp(β) + 1 exp(α + β) + 4 exp(α + β) exp(β) + 10 exp(α + β) + 4 exp(α + β) 14 + exp(α) + exp(β) + exp(α + β)) ( + exp(α) + 1 exp(β) + exp(α + β)) 6 exp(α + β) + 8 exp(α + β) ( exp(α) + exp(β) + exp(α + β)) Newton-Raphson θ (α, β) m θ m (α m, β m ) 6 9 ( ) 1 F θ m θ m1 θ (θ m1) F(θ) * * ; %macro ln(alpha, beta, a, b, c, d); log(&a + &b * exp(&alpha) + &c * exp(&beta) + &d * exp( &alpha + &beta)) %mend ln; * ; %macro f(alpha, beta); 5*&alpha + 4*&beta - %ln(&alpha, &beta, 4, 6, 6, 5) - %ln(&alpha, &beta, 7/, 11/, 6, 5) - %ln(&alpha, &beta,, 5, 6, 5) - %ln(&alpha, &beta,, 5, 6, 5) - %ln(&alpha, &beta,, 4, 6, 5) - %ln(&alpha, &beta,,, 6, 4) - %ln(&alpha, &beta,,, 5, 4) - %ln(&alpha, &beta,,, 5, ) - %ln(&alpha, &beta, 4/,, 14/, ) - %ln(&alpha, &beta, /,, 1/, ) - %ln(&alpha, &beta, 0,,, ) %mend f; * 1 ; %macro frac1(alpha, beta, a, b, c, d); (&b * exp(&alpha) + &d * exp(&alpha + &beta)) / (&a + &b*exp(&alpha) + &c*exp(&beta) + &d*exp(&alpha + &beta)) %mend frac1; %macro frac(alpha, beta, a, b, c, d); (&c * exp(&beta) + &d * exp(&alpha + &beta)) / (&a + &b*exp(&alpha) + &c*exp(&beta) + &d*exp(&alpha + &beta)) %mend frac; 15
16 * ; %macro frac11(alpha, beta, a, b, c, d); (&a * &b * exp(&alpha) + (&a * &d + &b * &c)*exp(&alpha + &beta) + &c * &d * exp(&alpha + *&beta)) / (&a + &b*exp(&alpha) + &c*exp(&beta) + &d*exp(&alpha + &beta))** %mend frac11; %macro frac1(alpha, beta, a, b, c, d); ( (&a * &d - &b * &c)*exp(&alpha + &beta) ) / (&a + &b*exp(&alpha) + &c*exp(&beta) + &d*exp(&alpha + &beta))** %mend frac1; %macro frac(alpha, beta, a, b, c, d); (&a * &c * exp(&beta) + (&a * &d + &b * &c)*exp(&alpha + &beta) + &b * &d * exp(*&alpha + &beta)) / (&a + &b*exp(&alpha) + &c*exp(&beta) + &d*exp(&alpha + &beta))** %mend frac; * 1 l alpha; %macro f1(alpha, beta); 5 - %frac1(&alpha, &beta, 4, 6, 6, 5) - %frac1(&alpha, &beta, 7/, 11/, 6, 5) - %frac1(&alpha, &beta,, 5, 6, 5) - %frac1(&alpha, &beta,, 5, 6, 5) - %frac1(&alpha, &beta,, 4, 6, 5) - %frac1(&alpha, &beta,,, 6, 4) - %frac1(&alpha, &beta,,, 5, 4) - %frac1(&alpha, &beta,,, 5, ) - %frac1(&alpha, &beta, 4/,, 14/, ) - %frac1(&alpha, &beta, /,, 1/, )- %frac1(&alpha, &beta, 0,,, ) %mend f1; * 1 l beta; %macro f(alpha, beta); 4 - %frac(&alpha, &beta, 4, 6, 6, 5) - %frac(&alpha, &beta, 7/, 11/, 6, 5) - %frac(&alpha, &beta,, 5, 6, 5) - %frac(&alpha, &beta,, 5, 6, 5) - %frac(&alpha, &beta,, 4, 6, 5) - %frac(&alpha, &beta,,, 6, 4) - %frac(&alpha, &beta,,, 5, 4) - %frac(&alpha, &beta,,, 5, ) - %frac(&alpha, &beta, 4/,, 14/, ) - %frac(&alpha, &beta, /,, 1/, ) - %frac(&alpha, &beta, 0,,, ) %mend f; * l alpha alpha; %macro f11(alpha, beta); - %frac11(&alpha, &beta, 4, 6, 6, 5) - %frac11(&alpha, &beta, 7/, 11/, 6, 5) - %frac11(&alpha, &beta,, 5, 6, 5) - %frac11(&alpha, &beta,, 5, 6, 5) - %frac11(&alpha, &beta,, 4, 6, 5) - %frac11(&alpha, &beta,,, 6, 4) - %frac11(&alpha, &beta,,, 5, 4) - %frac11(&alpha, &beta,,, 5, ) - %frac11(&alpha, &beta, 4/,, 14/, ) - %frac11(&alpha, &beta, /,, 1/, ) - %frac11(&alpha, &beta, 0,,, ) %mend f11; 16
17 * l alpha beta; %macro f1(alpha, beta); - %frac1(&alpha, &beta, 4, 6, 6, 5) - %frac1(&alpha, &beta, 7/, 11/, 6, 5) - %frac1(&alpha, &beta,, 5, 6, 5) - %frac1(&alpha, &beta,, 5, 6, 5) - %frac1(&alpha, &beta,, 4, 6, 5) - %frac1(&alpha, &beta,,, 6, 4) - %frac1(&alpha, &beta,,, 5, 4) - %frac1(&alpha, &beta,,, 5, ) - %frac1(&alpha, &beta, 4/,, 14/, ) - %frac1(&alpha, &beta, /,, 1/, ) - %frac1(&alpha, &beta, 0,,, ) %mend f1; * l beta beta; %macro f(alpha, beta); - %frac(&alpha, &beta, 4, 6, 6, 5) - %frac(&alpha, &beta, 7/, 11/, 6, 5) - %frac(&alpha, &beta,, 5, 6, 5) - %frac(&alpha, &beta,, 5, 6, 5) - %frac(&alpha, &beta,, 4, 6, 5) - %frac(&alpha, &beta,,, 6, 4) - %frac(&alpha, &beta,,, 5, 4) - %frac(&alpha, &beta,,, 5, ) - %frac(&alpha, &beta, 4/,, 14/, ) - %frac(&alpha, &beta, /,, 1/, ) - %frac(&alpha, &beta, 0,,, ) %mend f; *Newton-Raphson ; * Newton-Raphson Fisher s scoring ; %macro seq n(x1, x); x1 &x1; x &x; f1 %f1(x1, x); f %f(x1, x); f11 %f11(x1, x); f1 %f1(x1, x); f %f(x1, x); det f11*f - f1**; y1 x1 - (f * f1 - f1*f) / det; y x - (- f1*f1 + f11*f ) / det; %mend seq n; 17
18 * ; %macro exe(init1, init, method, no); data d&no; length i.0 diff x1 x y1 y f1 f %if &method n %then %do; f11 f1 f %end; %else %if &method f %then %do; I11 I1 I %end; det sol1 sol check1 check 8.0; i 1; diff 10; x1 &init1; x &init; do until(diff < 10**(-8) or det 0 or i > 1000); %seq &method(x1, x) output; diff sqrt((y1-x1)** + (y-x)**); x1 y1; x y; i i +1; end; sol1 y1;*newton-raphson ; sol y; check1 %f1(sol1, sol);* 0 ; check %f(sol1, sol);* 0 ; output; run; %mend exe; %exe(0.1, 0.1, n, 1) proc print proc print datad1; run; 18
19 i diff x1 x y1 y f1 f f11 f1 f det sol1 sol check1 check E E-16 α sol β sol SAS 19
20 (j) α, β (i) *6 λ(t x i, 1) λ(t x i, 0) λ 0(t) exp( αx i + β) exp( β) exp(1.86) 0.90 λ 0 (t) exp( αx i ) λ(t 1, y j) λ(t 0, y j ) λ 0(t) exp( α + βy j ) exp( α) exp(0.4684) 0.66 λ 0 (t) exp( βy j ) (k) Wald Wald H 0 H 1 H 0 : β 0 H 1 : β 0 L L ( 0 1 ) ( L θ 0 1 ) ( α β ) β H 0 H 1 H 0 : L θ 0 H 1 : L θ 0 θ (α, β) θ ( α, β) Fisher I(θ) I 11(θ) I 1 (θ) (I(θ)) 1 I11 (θ) I 1 (θ) Wald χ I 1 (θ) I (θ) I 1 (θ) I (θ) χ W ald (L θ) (L I( θ) 1 L) 1 (L θ) ( ( 0 1 ) ( α β )) ( ( 0 1 ) ( I 11 ( θ) I 1 ( θ) I 1 ( θ) I ( θ) ) ( 0 1 )) 1 ( ( 0 1 ) ( α β )) ( β) I ( θ) χ W ald 5% χ W ald > χ (1, 0.95) *6 x i 0
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More informationy = x 4 y = x 8 3 y = x 4 y = x 3. 4 f(x) = x y = f(x) 4 x =,, 3, 4, 5 5 f(x) f() = f() = 3 f(3) = 3 4 f(4) = 4 *3 S S = f() + f() + f(3) + f(4) () *4
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More information5 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 )
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