For the following small data set of survival time: 3, 4, 5+, 6, 6+, 8, 11+, 14+,

Question

For the following small data set of survival time: 3, 4, 5+, 6, 6+, 8, 11+, 14+, 15, 16+, where “+” means a right censored survival time, do the following by hand: (a) Find the Kaplan-Meier estimate of the survival function and its variance at each event time. (b) Use the above Kaplan-Meier estimate to get an estimate and its variance of the cumulative hazard function at each event time. (c) Find the Nelson-Aalen estimate of the cumulative hazard function and its variance at each event time. (d) Find an estimate and its variance of the survival function using the Nelson-Aalen estimate you got in (c) at each event time.

Explanation / Answer

(a)

rm(list=ls(all=TRUE))
library(survival);
time<-c( 3, 4, 5, 6, 6, 8, 11, 14, 15, 16)
length(time);
status<-c(0,0,1,0,1,0,1,1,0,1)
df<-data.frame(time,status)
fit1<-survfit(Surv(time ,status)~1,type="kaplan-meier")
summary(fit1)
par(font=2,font.axis=2,font.lab =2 ,lwd=2)
plot(fit1,xlab="Time to survival",ylab="survival function", main="kaplan-meier")

Call: survfit(formula = Surv(time, status) ~ 1, type = "kaplan-meier")

time n.risk n.event survival std.err lower 95% CI upper 95% CI
    5      8       1    0.875   0.117        0.673            1
    6      7       1    0.750   0.153        0.503            1
   11      4       1    0.562   0.199        0.281            1
   14      3       1    0.375   0.203        0.130            1
   16      1       1    0.000     NaN           NA           NA

(b)

var<-(fit1$surv^2)*cumsum(fit1$n.event/((fit1$n.risk)*(fit1$n.risk-1)))
> var
[1] 0.00000000 0.00000000 0.01367188 0.02343750 0.02343750 0.03955078 0.04101562
[8] 0.04101562        NaN

cbind(fit1$time,-log(fit1$surv),cumsum(fit1$n.event/((fit1$n.risk)*(fit1$n.risk-1))))

    [,1]      [,2]       [,3]
[1,]    3 0.0000000 0.00000000
[2,]    4 0.0000000 0.00000000
[3,]    5 0.1335314 0.01785714
[4,]    6 0.2876821 0.04166667
[5,]    8 0.2876821 0.04166667
[6,]   11 0.5753641 0.12500000
[7,]   14 0.9808293 0.29166667
[8,]   15 0.9808293 0.29166667
[9,]   16       Inf        Inf

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