censboot                package: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$strata[s]
               out <- c(out,md)
          }
          out
     }
     aml.case <- censboot(aml,aml.fun,R=499,strata=aml$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(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$ulcer))
          ind <- !duplicated(d$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!
     system.time(mel.mod <- censboot(melanoma,mel.fun,R=999,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)$point
     plot(seq(0.25,10,by=0.25),mel.env[1,], ylim=c(-2,2),
          xlab="thickness (mm)", ylab="linear predictor",type="n")
     lines(seq(0.25,10,by=0.25),mel.env[1,],lty=2)
     lines(seq(0.25,10,by=0.25),mel.env[2,],lty=2)
     lines(seq(0.25,10,by=0.25),mel.mod$t0,lty=1)

