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, σ 2 y i µ + ɛ i ɛ i N0, σ 2 E[y i ] µ *2 3.2 2 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 *2 ɛ i *3 x i 2
3.3 3 i p i x i x i p i p i + exp α βx i α, β y +exp α βx y +exp x *4 0.8 0.6 0.4 0.2-0 -5 5 0 y +exp x x y +exp x 0 *5 p i 0 *6 4 2 x i α, β, µ, µ i 4. y *4 α 0, β *5 y +exp α βx y x y +exp x 0 *6 expα + βx expα + βx + exp α βx + expα + βx 3
4.. t t t fixed effect 2 3 i j y ij Nµ i, σ 2 y ij µ i + ɛ ij ɛ ij N0, σ 2 y ij E[y ij ] µ i µ i *7 y ij *8 fixed effect y ij µ + α i + β j + γ ij + ɛ ij ɛ ij N0, σ 2 E[y ij ] µ + α i + β j + γ ij µ, α i, β j, γ ij y ij 4..2 y i α + βx i + ɛ i ɛ i N0, σ 2 *9 E[y i ] α + βx i α, β y i 2 E[y i ] α + β x i + β 2 x 2i + + β k x ki E[y i ] α + β x i + β 2 x 2 i + + β k x k i α, β,, β k α, β,, β k *7 E[y ij ] µ i *8 µ i µ + α i E[y ij ] µ + α i µ, α i E[y ij ] µ + α i *9 y i Nα + βx i, σ 2 4
4..3 2 i j x ij y ij µ + α i + βx ij + ɛ ij ɛ ij N0, σ 2 α i x ij β E[y ij ] µ + α i + βx ij µ, α i, β y i 4.2 4.2. 3 y i i 0, y i p i y i Bernoulli Bep i * 0 hx +exp x y i Bep i p i hα + βx i hx α+βx i hx +exp x h x log * h pi p i log α + βx i p i h x log log x x x x x x * 2 *0 Bernoulli * y +exp x x y + exp x y + exp x y exp x log y «x y log y x y y log x x *2 logitx log x x logitx log «y x log y y ««y x x log y +exp x x x 5
y i Bernoulli * 3 logitp i α + βx i 4.2.2 Φx 2π exp t2 2 dt x y i Bep i p i Φα + βx i Φx Φ x * 4 Φ p i α + βx i α, β Φ x y i Bernoulli Φ x 4.2.3 Poisson Poisson 2 0 i y i λ i Poisson P oλ i λ i x i λ i e α+βx i log log λ i α + βx i α, β y i P oλ i log λ i α + βx i y i Poisson * 5 log x 5 * 6 5. 5 y i x i i,, 5 *3 Bernoulli *4 *5 Poisson *6 3 6
y i, 0 x i mmhg 0 20 30 40 0 50 60 y i 0 5 * 7 p i Bernoulli p i Bernoulli Bep i y Bep, y 2 Bep 2, y 3 Bep 3, y 4 Bep 4, y 5 Bep 5 y i fy i p i p y i i p i yi i,, 5 5 fy p p y p y, fy 2 p 2 p y2 2 p 2 y 2, fy 3 p 3 p y3 3 p 3 y 3, fy 4 p 4 p y 4 4 p 4 y 4, fy 5 p 5 p y 5 5 p 5 y 5 p i p i + exp α βx i expα + βx i + expα + βx i 5 x i p expα + βx expα + 20β + expα + βx + expα + 20β, p 2 expα + βx 2 expα + 30β + expα + βx 2 + expα + 30β p 3 expα + βx 3 expα + 40β + expα + βx 3 + expα + 40β, p 4 expα + βx 4 expα + 50β + expα + βx 4 + expα + 50β p 5 expα + βx 5 expα + 60β + expα + βx 5 + expα + 60β y i Bep i p i expα + βx i + expα + βx i pi log α + βx i logitp i α + βx i p i *7 n x Bernoulli 7
p i p i α, β * 8 α, β 5.2 α, β 5.2. p i p i Lα, β y,, y 5 5 fy i p i i fy p fy 2 p 2 fy 3 p 3 fy 4 p 4 fy 5 p 5 p p 2 p 3 p 4 p 5 y 0, y 2, y 3, y 4 0, y 5 expα + 20β + expα + 20β expα + 40β + expα + 40β + expα + 20β expα + 30β + expα + 30β expα + 50β + expα + 50β expα + 30β + expα + 30β expα + 60β + expα + 60β expα + 40β + expα + 40β + expα + 50β expα + 60β + expα + 60β *8 β H 0 : β 0 8
lα, β y,, y 5 log Lα, β y,, y 5 log log + expα + 20β + expα + 20β + log + expα + 50β expα + 30β + expα + 30β expα + 30β + log + expα + 30β + log expα + 60β + expα + 60β expα + 40β + expα + 40β + expα + 50β expα + 40β + log + expα + 40β expα + 60β + expα + 60β log + expα + 20β + {log expα + 30β log + expα + 30β} + {log expα + 40β log + expα + 40β} log + expα + 50β + {log expα + 60β log + expα + 60β} log + expα + 20β + α + 30β log + expα + 30β + α + 40β log + expα + 40β log + expα + 50β + α + 60β log + expα + 60β log + expα + 20β log + expα + 30β log + expα + 40β log + expα + 50β log + expα + 60β + 3α + 430β α, β α l α α, β y,, y 5 expα + 20β + expα + 20β β l β α, β y,, y 5 expα + 30β expα + 40β expα + 50β expα + 60β + expα + 30β + expα + 40β + expα + 50β + expα + 60β + 3 20 expα + 20β 30 expα + 30β expα + 40β 50 expα + 50β 60 expα + 60β + expα + 20β + expα + 30β + expα + 40β + expα + 50β + expα + 60β + 430 l α α, β y,, y 5 0 l β α, β y,, y 5 0 exp α + 20 β + exp α + 20 β exp α + 30 β exp α + 40 β exp α + 50 β exp α + 60 β + exp α + 30 β + exp α + 40 β + exp α + 50 β + exp α + 60 β + 3 0 20 exp α + 20 β 30 exp α + 30 β exp α + 40 β 50 exp α + 50 β 60 exp α + 60 β + exp α + 20 β + exp α + 30 β + exp α + 40 β + exp α + 50 β + exp α + 60 β + 430 0 9
α, β * 9 SAS 5.2.2 SAS proc logistic SAS data d; input y x; cards; 0 20 30 40 0 50 60 run; proc logistic proc genmod 2 proc logistic proc genmod proc logistic proc logistic datad descending; model yx; run; * 20 WORK.D y 2 binary logit Fisher s scoring binary logit y i Bernoulli Fisher s scoring α, β * 2 *9 5 0 *20 descending y 0, y y 0 proc genmod *2 Fisher s scoring 0
y y y 0 0 descending Probability modeled is y y GCONVE-8 AIC 8.730 0.302 SC 8.340 9.52-2 Log L 6.730 6.302 * 22 H0: BETA0 2 Pr > ChiSq 0.4277 0.53 Score 0.467 0.586 Wald 0.3947 0.5298 p i +exp α βx i H 0 : β 0 x i * 23 * 24 Wald 2 Pr > ChiSq Intercept -5.7094 9.72 0.3450 0.5570 x 0.0439 0.0700 0.3947 0.5298 *22 *23 β 0 p i +exp α x i *24
p i + exp α βx i logitp i α + βx i Intercept α x β Intercept α x β { α 5.7094 β 0.0439 * 25 x Pr > Chisq H 0 : β 0 H 0 : β 0 Wald p i + exp5.7094 0.0439x i x 20,, x 5 60 p + exp5.7094 0.0439 20 0.39, p 2 + exp5.7094 0.0439 30 0.50 p 3 + exp5.7094 0.0439 40 0.6, p 4 + exp5.7094 0.0439 50 0.7 p 5 + exp5.7094 0.0439 60 0.79 x 0.8 0.6 0.4 0.2 2 y 80 00 20 40 60 80 200 +exp5.7094 0.0439x 95% Wald x.045 0.9.98 *25 x x x β 2
x x x + px p i log pi ôddspx + exp5.7094 0.0439x, px + + exp5.7094 0.0439x + p i α + βx i pi p i expα + βx i px px + exp 5.7094 + 0.0439x, ôddspx + exp 5.7094 + 0.0439x + px px + * 26 OR ÔR ôddspx + ÔR ôddspx exp0.0439.045 exp 5.7094 + 0.0439x + exp 5.7094 + 0.0439x exp 5.7094 + 0.0439x exp0.0439 exp 5.7094 + 0.0439x x ÔR exp β 0 x + x + 0 ÔR exp0 β.552 SAS proc logistic datad descending; model yx; units x0; run; x 0 0.552 5.2.3 SAS 2 proc genmod proc genmod proc genmod datad descending; model yx / linklogit distbin; run; model Bernoulli *26 p oddsp p p p [oddsp bp bp 3
WORK.D Binomial Logit y proc logistic PROC GENMOD y PROC GENMOD is modeling the probability with y. PROC GENMOD y descending y 0 / 3 6.3024 2.008 3 6.3024 2.008 Pearson 2 3 4.9694.6565 Pearson X2 3 4.9694.6565-3.52 proc logistic AIC proc logistic 2Log 2 deviance Wald 95% 2 Pr > ChiSq Intercept -5.7094 9.72-24.7624 3.3435 0.34 0.5570 x 0.0439 0.0700-0.0932 0.8 0.39 0.5298 Scale 0.0000 0.0000.0000.0000 Note: α, β proc logistic Scale Scale overdispersion 4
6 General Linear Model Generalized Linear Model 2 GLM GLM SAS Proc GLM Proc GLM GLM Linear Model LM * 27 Proc GLM LM GLM 7 2 proc logistic proc genmod 2 * 28 7. data d; input y x; cards; 0 20 0 30 0 40 50 60 run; x 45 x < 45 y 0 x 45 y 2 y 0 * 29 7.. proc logistic proc logistic *27 GLM GLIM LM GLM *28 *29 OK 5
WARNING WARNING LOGISTIC 7..2 proc genmod proc genmod WARNING: Hessian.0868635408 0.000 proc logistic 7.2 data d; input y x; cards; 0 20 0 30 30 50 60 run; x 30 y 0 y x 30 x < 30 y 0 x > 30 y x 30 y 0, y 0, * 30 7.2. proc logistic proc logistic *30 SAS 6
WARNING WARNING LOGISTIC * 3 7.2.2 proc genmod proc genmod WARNING: Hessian * 32 *3 *32 7