d011ch: Compute NPMLE of CDF from doubly censored data, with and without a constraint, plus an empirical likelihood ratio

Description

d011ch computes the NPMLE of CDF, with and without a constraint, from doubly censored data. It also computes the -2 log empirical likelihood ratio for testing the given constraint via empirical likelihood theorem, i.e. under Ho it should be distributed as chi-square with df=1.

It uses EM algorithm starting from an initial CDF estimator that have jumps at uncensored points as well as the mid-point of those censoring times that have a pattern of (0,2), (see below for definition and example.)

The constraint on the CDF are given in the form F(K) = konst. where you specify the time K and probability `konst'.

When there are ties among censored and uncensored observations, the left (right) censored points are treated as happened slightly before (after), to break tie. Also the last right censored observation and first left censored observation are changed to uncensored, in order to obtain a proper distribution as estimator (though this can be modified easily as they are written in R language).

Usage

d011ch(z, d, K, konst, 
     identical = rep(0, length(z)), maxiter = 49, error = 0.00001)

Value

a list contain the NPMLE of CDF with and without the constraint, -2loglik ratio and other informations.

time: survival times. Those corresponding to d=2 are removed. Those corresponding to (0,2) censoring pattern are added, at mid-point.
status: Censoring status of the above times. Since left censored times are removed, there is no status =2. There may be -1, indicating that this is an added time for (0,2) censoring pattern.
surv: The survival function at the above times.
jump: Jumps of NPMLE at the above times.
exttime: Similar to time but now include the left censored times.
extstatus: Censoring status of exttime. -1 has same meaning as status before.
extjump: Jumps of the unconstrained NPMLE on extended times.
extsurv.Sx: Survival probability at exttime.
konstdist: The constrained NPMLE of distribution.
konstjump: Jumps of the constrained NPMLE of CDF.
konsttime: Location of the constraint, same as K in the input.
theta: is the same value konst in the input.
"-2loglikR": the Wilks statistics. Distributed approximately chi-square df=1 under Ho
maxiter: the actual number of iterations for the unconstrained NPNLE. The constrained NPMLE usually took less iterations to converge.

Arguments

z: a vector of length n denoting observed times, (ties permitted)
d: a vector of length n that contains censoring indicator: d= 2 or 1 or 0, (according to z being left, not, right censored)
K: the constraint time.
konst: the constraint value, i.e. F(K)=konst.
identical: optional. a vector of length n that has values either 0 or 1. identical[i]=1 means even if (z[i],d[i]) is identical with (z[j],d[j]), for some $j \not= i$, they still stay as 2 observations, not 1 observation with weight 2, which only happen if identical[i]=0 and identical[j] =0. One reason to do this is because they may have different covariates not shown here. This flexibility may be useful for regression applications. Default value is identical = 0.
maxiter: optional integer value. Default to 49
error: optional. Default to 0.00001. maxiter and error are used to control the time of computing.

Author

Kun Chen, Mai Zhou

Details

A true NPMLE may have probability mass inside an interval when two consecutive time having censoring pattern d= (0, 2). See the help file of d011() for more details.

The definition of the double censoring is same as the Example 6.4 of Owen's 2001 book (except he uses d=0, 1, -1. we use d=0, 1 and 2). Our empirical likelihood function for the doubly censored data is also the same with the definition L(F) given on the middle of p. 148 of the book. In our natation it is $$EL_c(F) = \prod_{i=1}^n [\Delta F (z_i) ]^{d_i=1} [1-F(z_i)]^{d_i=0}[F(z_i)]^{d_i=2}.$$

References

Chang, M. N. and Yang, G. L. (1987). Strong consistency of a nonparametric estimator of the survival function with doubly censored data. Ann. Statist. 15, 1536-1547.

Murphy, S. and Van der Varrt. (1997). Semiparametric Likelihood Ratio Inference. Ann. Statist. 25, 1471-1509.

Chen, K. and Zhou, M. (2003). Nonparametric Hypothesis Testing and Confidence Intervals with Doubly Censored Data. Lifetime Data Analysis. 9, 71-91.

Owen, A. (2001). Empirical Likelihood. Chapman and Hall CRC press, Boca Raton.

Examples

Run this code

d011ch(z=c(1,2,3,4,5), d=c(1,0,2,2,1), K=3.5, konst=0.6)
#
# Here we are testing Ho: F(3.5) = 0.6 with a two-sided alternative
# you should get something like
#
#       $time:
#       [1] 1.0 2.0 2.5 5.0    (notice the times, (3,4), corresponding
#                                   to d=2 are removed, and time 2.5 added
#       $status:               since there is a (0,2) pattern at
#       [1]  1  0 -1  1        times 2, 3. The status indicator of -1
#                                   show that it is an added time )
#       $surv
#       [1] 0.5000351 0.5000351 0.3333177 0.0000000
#
#       $jump
#       [1] 0.4999649 0.0000000 0.1667174 0.3333177
#
#       $exttime
#       [1] 1.0 2.0 2.5 3.0 4.0 5.0       (exttime include all the times,
#                                         censor or not, plus the added time)
#       $extstatus
#       [1]  1  0 -1  2  2  1
#
#       $extjump
#       [1] 0.4999649 0.0000000 0.1667174 0.0000000 0.0000000 0.3333177
#
#       $extsurv.Sx
#       [1] 0.5000351 0.5000351 0.3333177 0.3333177 0.3333177 0.0000000
#
#       $konstdist
#       [1] 0.4999365 0.4999365 0.6000000 0.6000000 0.6000000 1.0000000
#
#       $konstjump
#       [1] 0.4999365 0.0000000 0.1000635 0.0000000 0.0000000 0.4000000
#
#       $konsttime
#       [1] 3.5
#
#       $theta
#       [1] 0.6
#
#       $"-2loglikR"                  (the Wilks statistics to test Ho:
#       [1] 0.05679897                  F(K)=konst)
#
#       $maxiter
#       [1] 33
#
#  The Wilks statistic is 0.05679897, there is no evidence against Ho: F(3.5)=0.6

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