exp( β i z i ) survreg() R survival library(survival) require(survival) 3 survfit() t 1, t 2,... t 1 d 1 t 2 d 2 t 1, t 2,... n 1, n 2,... n i t i n 1
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- しょうすけ たかひ
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1 (nminato@med.gunma-u.ac.jp) web (Survival Analysis Event History Analysis) Kaplan-Meier a x x (T x) x (l x) q x l x (1 q x /2) x x + 1 L x x x q x l x l x a population at risk (1995) 1
2 exp( β i z i ) survreg() R survival library(survival) require(survival) 3 survfit() t 1, t 2,... t 1 d 1 t 2 d 2 t 1, t 2,... n 1, n 2,... n i t i n t 1 1 t 2 3 d 1 = 1 d 2 = 2 n 1 = 5 n 2 = 3 Ŝ(t) Ŝ(t) = (1 d 1 /n 1 )(1 d 2 /n 2 )... = i<t(1 d i /n i ) var(ŝ) = Ŝ2 i<t d i n i (n i d i ) R library(survival) require(survival) Surv( ) survival Recommended Windows R R built-in base, datasets, grdevices, graphics, grid, methods, splines, stats, stats4, tcltk, tools, utils Recommended built-in survival KernSmooth, MASS, boot, class, cluster, foreign, lattice, mgcv, nlme, nnet, rpart, spatial search().packages(all.avail=t) detach(package:survival) 2
3 TRUE FALSE Surv( ) dat time <- c(81,22,29) time2 <- c(92,22,na) event <- c(3,1,0) dat <- Surv(time,time2,event,type="interval") dat res <- survfit(dat) plot(res) summary(res) difftime() ISOdate() x names dob dod difftime() 4 [x$names=="robert"] Robert alivedays as.date() ISOdate(,, ) 4 it r x <- data.frame( names = c("edward","shibasaburo","robert","hideyo"), dob = c(" "," "," "," "), dod = c(" "," "," "," ")) alivedays <- difftime(x$dod,x$dob)[x$names=="robert"] as.numeric(alivedays/365.24) as.numeric(difftime(isodate(2007,1,22),x$dob)/365.24) 4 3
4 1 (1995) p Gehan 42 6-MP R MASS 2 it r require(mass) require(survival) print(res<-survfit(surv(time,cens)~treat,data=gehan)) par(family="sans",las=1) plot(res,lty=c(1,2),main="gehan ") legend(30,0.2,lty=c(1,2),legend=levels(gehan$treat)) summary(res) % Call: survfit(formula = Surv(time, cens) ~ treat, data = gehan) n events median 0.95LCL 0.95UCL treat=6-mp Inf treat=control summary(res) 95% 4
5 Call: survfit(formula = Surv(time, cens) ~ treat, data = gehan) treat=6-mp time n.risk n.event survival std.err lower 95% CI upper 95% CI treat=control time n.risk n.event survival std.err lower 95% CI upper 95% CI NA NA NA 2 survival aml (acute myelogenous leukemia) a time status 0 1 x Maintained Nonmaintained a : Miller RG: Survival Analysis. John Wiley and Sons, Embury SH, Elias L, Heller PH, Hood CE, Greenberg PL, Schrier SL: Remission maintenance therapy in acute myelogenous leukaemia. Western Journal of Medicine, 126, , Gehan it r require(survival) print(res <- survfit(surv(time,status)~x, data=aml)) summary(res) par(family="sans",las=1) plot(res,lty=c(1,2),main=" ") legend(100,0.8,lty=c(1,2),legend=c(" "," ")) 5
6 2 n events median 0.95LCL 0.95UCL x=maintained Inf x=nonmaintained Inf % % 3 1 survdiff() A 2 B 1 4,6,8,9 2 5,7,12, i j e ij i d i i j n ij i n i e ij = d i n ij /n i 5 e 11 = 1 n 11 /n 1 = 4/8 = 0.5 i j d ij w i i j u ij u ij = w i (d ij e ij ) 1 5 6
7 u 1 = i (d i1 e i1 ) u 1 = (1 4/8) + (0 3/7) + (1 3/6) + (0 2/5) + (1 2/4) + (1 1/3) + (0 0/2) + (0 0/1) V = V jj = i (n i n ij )n ij d i (n i d i ) n i2 (n i 1) V = (8 4) 4 (7 3) 3 (6 3) 3 (5 2) 2 (4 2) 2 (3 1) *4/64+4*3/49+3*3/36+3*2/25+2*2/16+2*1/ χ 2 = /1.457 = % % R time event group survdiff(surv(time,event)~group) survdiff(surv(time,event)~group,rho=1) it r require(survival) time <- c(4,6,8,9,5,7,12,14) event <- c(1,1,1,1,1,1,1,1) group <- c(1,1,1,1,2,2,2,2) survdiff(surv(time,event)~group) χ 2 = p = % % survdiff(surv(time,status)~x,data=aml) % R coxph() z i = (z i1, z i2,..., z ip ) i t h(z i, t) 7
8 h(z i, t) = h 0 (t) exp(β 1 z i1 + β 2 z i β p z ip ) h 0 (t) t β 1, β 2,..., β p exp(β x z ix ) Cox z i 1 2 t h 0 (t) exp(β 1 z 11 + β 2 z β p z 1p ) exp(β 1 z 21 + β 2 z β p z 2p ) T S(t) T t S(0) = 1 h(t) t Pr(t T < t + t T t) S(t) S(t + t) h(t) = lim = lim t 0 t t 0 ts(t) = ds(t) dt 1 S(t) = d(log(s(t)) dt H(t) = t h(u)du = log S(t) S(t) = exp( H(t)) 0 z S(z, t) H(z, H(z, t) = t 0 h(z, u)du = t 0 h 0 (u) exp(βz)du = exp(βz)h 0 (t) S(z, t) = exp( H(z, t)) = exp{ exp(βz)h 0 (t)} log( log S(z, t)) = βz + log H 0 (t) βz 8
9 β t i t 1 t i i L L β L Cox 6 R coxph(surv(time,cens)~grp+covar,data=dat) 4 2 it r require(survival) summary(res <- coxph(surv(time,status)~x,data=aml)) loglogplot <- function(x) { S <- X$surv T <- X$time G <- X$ntimes.strata GG <- names(x$strata) GX <- rep(gg,g) xr <- c(0,max(t)*1.5) mas <- ifelse(max(s)==1,0.99,max(s)) mis <- ifelse(min(s)==0,0.01,min(s)) yr <- c(log(-log(mas)),log(-log(mis))) plot(t[gx==gg[1]],log(-log(s[gx==gg[1]])),type="l",lty=1,xlim=xr,ylim=yr, xlab="time",ylab="log(-log(s))",main=" ") for (i in 2:length(GG)) { lines(t[gx==gg[i]],log(-log(s[gx==gg[i]])),lty=i) } legend(max(t),-2,legend=gg,lty=1:length(gg)) } KM <- survfit(surv(time,status)~x,data=aml) par(family="sans",las=1,mfrow=c(1,2)) plot(km,main="aml ") loglogplot(km) 2 6 Exact Breslow Efron Exact Breslow R coxph() Efron Breslow Efron Exact 9
10 Call: coxph(formula = Surv(time, status) ~ x, data = aml) n= 23 coef exp(coef) se(coef) z p xnonmaintained exp(coef) exp(-coef) lower.95 upper.95 xnonmaintained Rsquare= (max possible= ) Likelihood ratio test= 3.38 on 1 df, p= Wald test = 3.2 on 1 df, p= Score (logrank) test = 3.42 on 1 df, p= % 7 exp(coef) % 1 5% Gehan 6-MP it r require(mass) require(survival) res <- coxph(surv(time,cens)~treat,data=gehan) summary(res) plot(survfit(res)) summary(res) plot(survfit(res)) 95% 7 Score (logrank) test Rao Score survdiff() % plot(survfit(coxph(surv(time,cens)~treat+pair,data=gehan))) plot(survfit(surv(time,cens),data=gehan)) 2 coxph() subset=(treat=="6-mp") 2 par(new=t) 10
11 Call: coxph(formula = Surv(time, cens) ~ treat, data = gehan) n= 42 coef exp(coef) se(coef) z p treatcontrol exp(coef) exp(-coef) lower.95 upper.95 treatcontrol Rsquare= (max possible= ) Likelihood ratio test= 16.4 on 1 df, p=5.26e-05 Wald test = 14.5 on 1 df, p= Score (logrank) test = 17.3 on 1 df, p=3.28e-05 5% 6-MP exp(coef) MP % [2.15, 10.8] 6-MP R coxph() strata() time event treat stage coxph(surv(time,event)~treat+strata(stage)) 2 AIC 11
12 6 survival colon 1 Levamisole 5-FU 1 2 id study rx id 1 3 Obs Lev Levamisole Lev+5FU sex 1 0 age obstruct 1 0 perfor 1 0 adhere 1 0 nodes status 1 0 differ extent surg 0 1 node time etype 1 2 Levamisole 5-FU colon2 colon2 <- subset(colon,etype==2) loglogplot() it r colon2$sex <- factor(colon2$sex) KM <- survfit(surv(time,status)~rx,data=colon2) layout(1:2) plot(km) loglogplot(km) res <- coxph(surv(time,status)~rx+age+sex,data=colon2) summary(res) Levamisole 5-FU % [0.545,0.869] R 2 = χ 2 = 12.5 d.f. = 4, p = %
13 attach(colon2) xls <- c(0,max(time)) plot(survfit(coxph(surv(time,status)~age+sex,subset=(rx=="obs"))),col=1,xlim=xls) par(new=t) plot(survfit(coxph(surv(time,status)~age+sex,subset=(rx=="lev"))),col=2,xlim=xls) par(new=t) plot(survfit(coxph(surv(time,status)~age+sex,subset=(rx=="lev+5fu"))),col=3,xlim=xls) detach(colon2) 7 (1995) A R p SAS R coxph() pcancer.r dat <- read.delim(" dat$censor <- 1-dat$CENSOR dat$sex <- factor(dat$sex, labels=c(" "," ")) dat$treat <- factor(dat$treat, labels=c(" "," ")) dat$ch <- ordered(dat$ch,labels=c("ch0","ch1","ch2","ch3")) dat$stage <- ordered(dat$stage,labels=c("iii","iv")) dat$ps <- ordered(dat$ps,labels=c("0,1","2","3","4")) # data from (1995) SAS A.3 # Nishimura et al # # Minato Nakazawa 20/1/2007 R 1 CENSOR # *** *** # CASENO # TIME # CENSOR 1 0 R 1 # AGE # SEX 0 1 # TREAT 0 1 # BUI 0 1 # CH 1 CH0 2:CH1 3:CH2 4:CH3 # P 0 1 # STAGE TNM 3:III 4:IV # PS Performance Status 1:0,1 2:2 3:3 4:4 pcancer.r 10 SAS Breslow method="breslow" Efron Exact 10 source(" 13
14 it r require(survival) summary(coxph(surv(time,censor)~age+sex+treat, data=dat, method="breslow")) summary(coxph(surv(time,censor)~age+sex+treat, data=dat, method="efron")) summary(coxph(surv(time,censor)~age+sex+treat, data=dat, method="exact")) res <- step(coxph(surv(time,censor)~age+sex+treat+bui+ch+p+stage+ps, data=dat, method="breslow")) summary(res) Breslow SAS SAS TREAT BUI STAGE STAGE (1995) SAS. survival ovarian 2 Eastern Cooperative Oncology Group futime fustat age resid.ds 1 2 rx ecog.ps ECOG A4 web 14
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