censboot {boot}  R Documentation 
Bootstrap for Censored Data
Description
This function applies types of bootstrap resampling which have been suggested to deal with rightcensored data. It can also do modelbased resampling using a Cox regression model.
Usage
censboot(data, statistic, R, F.surv, G.surv, strata = matrix(1,n,2),
sim = "ordinary", cox = NULL, index = c(1, 2), ...,
parallel = c("no", "multicore", "snow"),
ncpus = getOption("boot.ncpus", 1L), cl = NULL)
Arguments
data 
The data frame or matrix containing the data. It must have at least two
columns, one of which contains the times and the other the censoring
indicators. It is allowed to have as many other columns as desired
(although efficiency is reduced for large numbers of columns) except for

statistic 
A function which operates on the data frame and returns the required
statistic. Its first argument must be the data. Any other arguments
that it requires can be passed using the 
R 
The number of bootstrap replicates. 
F.surv 
An object returned from a call to 
G.surv 
Another object returned from a call to 
strata 
The strata used in the calls to 
sim 
The simulation type. Possible types are 
cox 
An object returned from 
index 
A vector of length two giving the positions of the columns in

... 
Other named arguments which are passed unchanged to 
parallel , ncpus , cl 
See the help for 
Details
The various types of resampling are described in Davison and Hinkley (1997) in sections 3.5 and 7.3. The simplest is case resampling which simply resamples with replacement from the observations.
The conditional bootstrap simulates failure times from the estimate of
the survival distribution. Then, for each observation its simulated
censoring time is equal to the observed censoring time if the
observation was censored and generated from the estimated censoring
distribution conditional on being greater than the observed failure time
if the observation was uncensored. If the largest value is censored
then it is given a nominal failure time of Inf
and conversely if
it is uncensored it is given a nominal censoring time of Inf
.
This is necessary to allow the largest observation to be in the
resamples.
If a Cox regression model is fitted to the data and supplied, then the
failure times are generated from the survival distribution using that
model. In this case the censoring times can either be simulated from
the estimated censoring distribution (sim = "model"
) or from the
conditional censoring distribution as in the previous paragraph
(sim = "cond"
).
The weird bootstrap holds the censored observations as fixed and also the observed failure times. It then generates the number of events at each failure time using a binomial distribution with mean 1 and denominator the number of failures that could have occurred at that time in the original data set. In our implementation we insist that there is a least one simulated event in each stratum for every bootstrap dataset.
When there are strata involved and sim
is either "model"
or "cond"
the situation becomes more difficult. Since the strata
for the survival and censoring distributions are not the same it is
possible that for some observations both the simulated failure time and
the simulated censoring time are infinite. To see this consider an
observation in stratum 1F for the survival distribution and stratum 1G
for the censoring distribution. Now if the largest value in stratum 1F
is censored it is given a nominal failure time of Inf
, also if
the largest value in stratum 1G is uncensored it is given a nominal
censoring time of Inf
and so both the simulated failure and
censoring times could be infinite. When this happens the simulated
value is considered to be a failure at the time of the largest observed
failure time in the stratum for the survival distribution.
When parallel = "snow"
and cl
is not supplied,
library(survival)
is run in each of the worker processes.
Value
An object of class "boot"
containing the following components:
t0 
The value of 
t 
A matrix of bootstrap replicates of the values of 
R 
The number of bootstrap replicates performed. 
sim 
The simulation type used. This will usually be the input value of

data 
The data used for the bootstrap. This will generally be the input
value of 
seed 
The value of 
statistic 
The input value of 
strata 
The strata used in the resampling. When 
call 
The original call to 
Author(s)
Angelo J. Canty. Parallel extensions by Brian Ripley
References
Andersen, P.K., Borgan, O., Gill, R.D. and Keiding, N. (1993) Statistical Models Based on Counting Processes. SpringerVerlag.
Burr, D. (1994) A comparison of certain bootstrap confidence intervals in the Cox model. Journal of the American Statistical Association, 89, 1290–1302.
Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.
Efron, B. (1981) Censored data and the bootstrap. Journal of the American Statistical Association, 76, 312–319.
Hjort, N.L. (1985) Bootstrapping Cox's regression model. Technical report NSF241, Dept. of Statistics, Stanford University.
See Also
Examples
library(survival)
# Example 3.9 of Davison and Hinkley (1997) does a bootstrap on some
# remission times for patients with a type of leukaemia. The patients
# were divided into those who received maintenance chemotherapy and
# those who did not. Here we are interested in the median remission
# time for the two groups.
data(aml, package = "boot") # not the version in survival.
aml.fun < function(data) {
surv < survfit(Surv(time, cens) ~ group, data = data)
out < NULL
st < 1
for (s in 1:length(surv$strata)) {
inds < st:(st + surv$strata[s]1)
md < min(surv$time[inds[1surv$surv[inds] >= 0.5]])
st < st + surv$strata[s]
out < c(out, md)
}
out
}
aml.case < censboot(aml, aml.fun, R = 499, strata = aml$group)
# Now we will look at the same statistic using the conditional
# bootstrap and the weird bootstrap. For the conditional bootstrap
# the survival distribution is stratified but the censoring
# distribution is not.
aml.s1 < survfit(Surv(time, cens) ~ group, data = aml)
aml.s2 < survfit(Surv(time0.001*cens, 1cens) ~ 1, data = aml)
aml.cond < censboot(aml, aml.fun, R = 499, strata = aml$group,
F.surv = aml.s1, G.surv = aml.s2, sim = "cond")
# For the weird bootstrap we must redefine our function slightly since
# the data will not contain the group number.
aml.fun1 < function(data, str) {
surv < survfit(Surv(data[, 1], data[, 2]) ~ str)
out < NULL
st < 1
for (s in 1:length(surv$strata)) {
inds < st:(st + surv$strata[s]  1)
md < min(surv$time[inds[1surv$surv[inds] >= 0.5]])
st < st + surv$strata[s]
out < c(out, md)
}
out
}
aml.wei < censboot(cbind(aml$time, aml$cens), aml.fun1, R = 499,
strata = aml$group, F.surv = aml.s1, sim = "weird")
# Now for an example where a cox regression model has been fitted
# the data we will look at the melanoma data of Example 7.6 from
# Davison and Hinkley (1997). The fitted model assumes that there
# is a different survival distribution for the ulcerated and
# nonulcerated groups but that the thickness of the tumour has a
# common effect. We will also assume that the censoring distribution
# is different in different age groups. The statistic of interest
# is the linear predictor. This is returned as the values at a
# number of equally spaced points in the range of interest.
data(melanoma, package = "boot")
library(splines)# for ns
mel.cox < coxph(Surv(time, status == 1) ~ ns(thickness, df=4) + strata(ulcer),
data = melanoma)
mel.surv < survfit(mel.cox)
agec < cut(melanoma$age, c(0, 39, 49, 59, 69, 100))
mel.cens < survfit(Surv(time  0.001*(status == 1), status != 1) ~
strata(agec), data = melanoma)
mel.fun < function(d) {
t1 < ns(d$thickness, df=4)
cox < coxph(Surv(d$time, d$status == 1) ~ t1+strata(d$ulcer))
ind < !duplicated(d$thickness)
u < d$thickness[!ind]
eta < cox$linear.predictors[!ind]
sp < smooth.spline(u, eta, df=20)
th < seq(from = 0.25, to = 10, by = 0.25)
predict(sp, th)$y
}
mel.str < cbind(melanoma$ulcer, agec)
# this is slow!
mel.mod < censboot(melanoma, mel.fun, R = 499, F.surv = mel.surv,
G.surv = mel.cens, cox = mel.cox, strata = mel.str, sim = "model")
# To plot the original predictor and a 95% pointwise envelope for it
mel.env < envelope(mel.mod)$point
th < seq(0.25, 10, by = 0.25)
plot(th, mel.env[1, ], ylim = c(2, 2),
xlab = "thickness (mm)", ylab = "linear predictor", type = "n")
lines(th, mel.mod$t0, lty = 1)
matlines(th, t(mel.env), lty = 2)