Theory

The Kaplan-Meier [KM] estimator of the survival function is $$\widehat{S}(t)=\prod_{\{i: t_i\leq t\}}\biggl[1-\frac{\Delta\overline{N}(t_i)}{\overline{Y}(t_i)}\biggr].$$ A uniformly consistent estimator of the variance of $\sqrt{n}[\widehat{S}(t)-S(t)]$ is $$\widehat{V}(t)=\widehat{S}^2(t)\widehat{\sigma}(t)= \widehat{S}^2(t)\int_0^t \frac{nd\overline{N}(u)}{\overline{Y}(u)[\overline{Y}(u)-\Delta\overline{N}(u)]}$$ (Greenwood formula).

An asymptotic pointwise $100(1-\alpha)$% confidence interval for $S(t)$ at a fixed $t$ is $$\biggl( \widehat{S}(t)\Bigl[1-u_{1-\alpha/2}\sqrt{\widehat{\sigma}(t)/n}\Bigr],\, \widehat{S}(t)\Bigl[1+u_{1-\alpha/2}\sqrt{\widehat{\sigma}(t)/n}\Bigr] \biggr).$$ Asymptotic simultaneous $100(1-\alpha)$% Hall-Wellner confidence bands for $S(t)$ over $t\in\langle 0,\tau\rangle$ (for some pre-specified $\tau$) are $$\biggl( \widehat{S}(t)\Bigl\{1-k_{1-\alpha}(\widehat{K}(\tau))\bigl[1+\widehat{\sigma}(t)\bigr]/\sqrt{n}\Bigr\},\, \widehat{S}(t)\Bigl\{1+k_{1-\alpha}(\widehat{K}(\tau))\bigl[1+\widehat{\sigma}(t)\bigr]/\sqrt{n}\Bigr\},\, \biggr),$$ where $\widehat{K}(t)=\widehat{\sigma}(t)/[1+\widehat{\sigma}(t)]$ and $k_{1-\alpha}(t)$, $t\in(0,1\rangle$, satisfies the equation $$\text{P}\bigl[\sup_{u\in\langle 0,t\rangle}|B(u)|>k_{1-\alpha}(t)\bigr]=\alpha,$$ where $B$ is the Brownian bridge.

There are variations of confidence intervals and confidence bounds for $S(t)$ based on various transformations ($\log$, $\log(-\log)$, $\arcsin$, …). Formulae for these intervals can be derived by the delta method.

Implementation in R

library(survival)
fit <- survfit(Surv(x,delta)~1,data=dn,conf.type="plain",conf.int=0.9)
fit2 <- survfit(Surv(x,delta)~grp,data=dataname,conf.type="plain",conf.int=0.9)
cbind(fit$time,fit$lower,fit$upper)
summary(fit)
plot(fit2[1],conf.int=TRUE)
lines(fit2[2],conf.int=TRUE,col="red")

library(OIsurv)
out <- confBands(Surv(dn$x,dn$delta),confType="plain",confLevel=0.95,type="hall",tU=240)
lines(out,lty=3,col="blue")

library(km.ci)
fit <- survfit(Surv(x,delta)~1,data=dn,conf.type="plain",conf.int=0.95)
out <- km.ci(fit,conf.level=0.95,tl=0.03,tu=240,method="hall-wellner") 
summary(out)
lines(out,lty=3,col="blue")
plot(out)

Task 1

Download the dataset km_all.RData.

The dataframe inside is called all. It includes 101 observations and three variables. The observations are acute lymphatic leukemia [ALL] patients who had undergone bone marrow transplant. The variable time contains time (in months) since transplantation to either death/relapse or end of follow up, whichever occured first. The outcome of interest is time to death or relapse of ALL (relapse-free survival). The variable delta includes the event indicator (1 = death or relapse, 0 = censoring). The variable type distinguishes two different types of transplant (1 = allogeneic, 2 = autologous).

Calculate and plot the Kaplan-Meier estimate, 95% pointwise confidence intervals and 95% Hall-Wellner confidence bounds for all patients together, for patients with allogeneic transplants, and for patients with autologous transplants.

Task 2

Generate $n=50$ censored observations as follows: the survival distribution is Weibull with shape parameter $\alpha=0.7$ and scale parameter $1/\lambda=2$. Its expectation is $\Gamma(1+1/\alpha)/\lambda=2\Gamma(17/7)\doteq 2.53$. The censoring distribution is exponential with rate $\lambda=0.2$ (the expectation is $1/\lambda=5$), independent of survival.

Calculate and plot the Kaplan-Meier estimator of $S(t)$ together with 95% pointwise confidence intervals and 95% Hall-Wellner confidence bounds. Include the true survival function in the plot (use a different color). Include a legend explaining which curve is which.

Task 3 (voluntary)

Conduct a simulation study with data created according to Task 2 assignment. Generate 500 such datasets and estimate the probability that the true survival curve is wholly covered by the 95% pointwise confidence intervals and 95% Hall-Wellner confidence bounds (restrict the task to a reasonable finite interval).