censboot {boot} | R Documentation

## Bootstrap for Censored Data

### Description

This function applies types of bootstrap resampling which have been suggested to deal with right-censored data. It can also do model-based 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), ...)

### 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 `sim="weird"` when it should only have two columns - the times and censoring indicators. The columns of `data` referenced by the components of `index` are taken to be the times and censoring indicators.

`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 `...{}` argument. In the case of `sim="weird"`, the data passed to `statistic` only contains the times and censoring indicator regardless of the actual number of columns in `data`. In all other cases the data passed to statistic will be of the same form as the original data. When `sim="weird"`, the actual number of observations in the resampled data sets may not be the same as the number in `data`. For this reason, if `sim="weird"` and `strata` is supplied, `statistic` should also take a numeric vector indicating the strata. This allows the statistic to depend on the strata if required.

`R` | The number of bootstrap replicates.
`F.surv` | An object returned from a call to `survfit` giving the survivor function for the data. This is a required argument unless `sim="ordinary"` or `sim="model"` and `cox` is missing.
`G.surv` | Another object returned from a call to `survfit` but with the censoring indicators reversed to give the product-limit estimate of the censoring distribution. Note that for consistency the uncensored times should be reduced by a small amount in the call to `survfit`. This is a required argument whenever `sim="cond"` or when `sim="model"` and `cox` is supplied.

`strata`

The strata used in the calls to `survfit`. It can be a vector or a matrix with 2 columns. If it is a vector then it is assumed to be the strata for the survival distribution, and the censoring distribution is assumed to be the same for all observations. If it is a matrix then the first column is the strata for the survival distribution and the second is the strata for the censoring distribution. When `sim="weird"` only the strata for the survival distribution are used since the censoring times are considered fixed. When `sim="ordinary"`, only one set of strata is used to stratify the observations, this is taken to be the first column of `strata` when it is a matrix.

`sim` | The simulation type. Possible types are `"ordinary"` (case resampling), `"model"` (equivalent to `"ordinary"` if `cox` is missing, otherwise it is model based resampling), `"weird"` (the weird bootstrap - this cannot be used if `cox` is supplied), and `"cond"` (the conditional bootstrap, in which censoring times are resampled from the conditional censoring distribution).
`cox` | An object returned from `coxph`. If it is supplied, then `F.surv` should have been generated by a call of the form `survfit(cox)`.
`index` | A vector of length two giving the positions of the columns in `data` which correspond to the times and censoring indicators respectively.
`...` | Any other arguments which are passed to `statistic`.

### 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.

### Value

An object of class `"boot"` containing the following components:

`t0` | The value of `statistic` when applied to the original data.
`t` | A matrix of bootstrap replicates of the values of `statistic`.
`R` | The number of bootstrap replicates performed.
`sim` | The simulation type used. This will usually be the input value of `sim` unless that was `"model"` but `cox` was not supplied, in which case it will be `"ordinary"`.
`data` | The data used for the bootstrap. This will generally be the input value of `data` unless `sim="weird"`, in which case it will just be the columns containing the times and the censoring indicators.
`seed` | The value of `.Random.seed` when `censboot` was called.
`statistic` | The input value of `statistic`.
`strata` | The strata used in the resampling. When `sim="ordinary"` this will be a vector which stratifies the observations, when `sim="weird"` it is the strata for the survival distribution and in all other cases it is a matrix containing the strata for the survival distribution and the censoring distribution.
`call` | The original call to `censboot`.

### Author(s)

Angelo J. Canty

### References

Andersen, P.K., Borgan, O., Gill, R.D. and Keiding, N. (1993) Statistical Models Based on Counting Processes. Springer-Verlag.

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 NSF-241, Dept. of Statistics, Stanford University.

### See Also

`boot`, `coxph`, `survfit`

### Examples

    data(aml, package="boot")
    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.
    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$time[inds[1-surv$surv[inds]>=0.5]])
              st <- st+surv$strata[s]
              out <- c(out,md)
         }
    }
    aml.case <- censboot(aml,aml.fun,R=499,strata=aml$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(time-0.001*cens,1-cens)~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$time[inds[1-surv$surv[inds]>=0.5]])
              st <- st+surv$strata[s]
              out <- c(out,md)
         }
    }
    aml.wei <- censboot(cbind(aml$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
    # non-ulcerated 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$ulcer))
         eta <- unique(cox$linear.predictors)
         u <- unique(d$thickness)
         sp <- smooth.spline(u,eta,df=20)
         th <- seq(from=0.25,to=10,by=0.25)
         predict(sp,th)$y
    }
    mel.str<-cbind(melanoma$ulcer,agec)
    # this is slow!
    mel.mod <- censboot(melanoma,mel.fun,R=999,F.surv=mel.surv,
         G.surv=mel.cens,cox=mel.cox,strata=mel.str,sim="model")
    proc.time()
    # To plot the original predictor and a 95% pointwise envelope for it
    mel.env <- envelope(mel.mod)$point
    plot(seq(0.25,10,by=0.25),mel.env[1,], ylim=c(-2,2),
         xlab="thickness (mm)", ylab="linear predictor",type="n")
    lines(seq(0.25,10,by=0.25),mel.env[1,],lty=2)
    lines(seq(0.25,10,by=0.25),mel.env[2,],lty=2)
    lines(seq(0.25,10,by=0.25),mel.mod$t0,lty=1)

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