Question

The following table contains data from a study on the survival times T (in days) of

eighteen patients with lung diseases. Right censored observations are denoted by “+”.

Days to death
8 8 26+ 29 69 76+ 77 78+ 79 108 115 122+ 153 154 157+ 180+ 190+ 211+

1. (a) Estimate the survival function S(t) using the Kaplan-Meier estimator. Compute the standard error of the estimate at each of the uncensored time points.

2. (b) Estimate the median and the 25th-percentile of the survival time based on the Kaplan-Meier estimates.

3. (c) Construct a 95% confidence interval for S(t) at t = 100 days.

4. (d) Based on the Epanechnikov kernel function given by K(u) = 0.75(1 − u2) for−1 ≤ u ≤ 1 with a bandwidth of h = 30, estimate λ(100) with a pointwise 95% confidence interval.

5. (e) Assume that T can be modeled by an exponential distribution with density functionf(t) = λexp(−λt), for t,λ > 0.

   i. Find the maximum likelihood estimate for λ.
   ii. Estimate the mean of T and P(T > 100). Give a point estimate and a 95% confidence interval for each case.

Answer 1

time=c(8,8,26,29,69,76,77,78,79,108,115,122,153,154,157,180,190,211);time
length(time)
satus=c(1,1,0,1,1,0,1,0,1,1,1,0,1,1,0,0,0,0);satus
length(satus)
ctc=data.frame(time,satus);ctc
plot(surv.bygr,conf.int=F,xlab="time",ylab="survival function",cex=0.6)