kmc (version 0.2-2)

kmc.solve: Calculate NPMLE with constriants for right censored data

Description

This function calculate the Kaplan-Meier estimator with mean constraints recursively. $$El(F)=\prod_{i=1}^{n}(\Delta F(T_i))^{\delta_i}(1-F(T_i))^{1-\delta_i}$$ with constraints $$\sum_i g(T_i)\Delta F(T_i)=0,\quad,i=1,2,...$$ It uses Lagrange multiplier directly.

Usage

kmc.solve(x, d, g, em.boost = T, using.num = T, using.Fortran =
                 T, using.C = F, tmp.tag = T, rtol = 1e-09, control =
                 list(nr.it = 20, nr.c = 1, em.it = 3),...)

Arguments

x

Possitive time

d

Status, 0: right censored; 1 uncensored

g

list of contraint functions. It should be a list of functions list(f1,f2,...)

em.boost

logical asking whether to use EM to get the initial value, default=TRUE. See 'Details' for EM control.

using.num

logical asking whether to use numeric derivative in iterations, default=TRUE.

using.Fortran

logical asking whether to use Fortran in root solving, default=F.

using.C

logical asking whether to use Rcpp in iteraruib, default=T. This option will promote the performance of KMC algorithm. Development version works on one constraint only, otherwise it will generate a Error information. It won't work on using.num=F.

tmp.tag

Development version needs it, keep it as TRUE.

rtol

Tolerance used in rootSolve(multiroot) package, see 'rootSolve::multiroot'.

control

nr.it controls max iterations allowed in N-R algorithm default=20, nr.c is the scaler used in N-R algorithm default=1,em.it is max iteration if use EM algorithm (em.boost) to get the initial value of lambda, default=3.

...

Unspecified yet.

Value

• a list with the following components:
• loglik.ha
  log empirical likelihood without constraints
• loglik.h0
  log empirical likelihood with constraints
• "-2llr"
  -2 Log empirical likelihood
• phat
  $$\Delta F(T_i)$$
• pvalue
  p-value of the test
• df
  Degree(s) of freedom. It equals the number of constraints.
• lambda
  The lambda is the Lagrangian multiplier described in reference.

Details

The development version won't check the constraint is proper or not. Besides, an extrordinary large number will be generated when numeric method/constraint(s) fails.

References

Zhou, M. and Yang, Y. (2015). A recursive formula for the Kaplan-Meier estimator with mean constraints and its application to empirical likelihood Computational Statistics Online ISSN 1613-9658.

See Also

plotkmc2D, print.kmcS3.

Examples

x <- c( 1, 1.5, 2, 3, 4.2, 5.0, 6.1, 5.3, 4.5, 0.9, 2.1, 4.3) # positive time
d <- c( 1,   1, 0, 1, 0, 1, 1, 1, 1, 0, 0,   1)               # status censored/uncensored

#################
# dim =1
#################

f<-function(x) { x-3.7}                                       # \sum f(ti) wi ~ 0
g=list( f=f) ;                                                #define constraint as a list

kmc.solve( x,d,g) ;                                           #using default
kmc.solve( x,d,g,using.C=TRUE) ;                              #using Rcpp

#################
# dim =2
#################

myfun5 <- function( x)  {
 x^2-16.5
}
g=list( f1=f,f2=myfun5) ;                                     #define constraint as a list
kmc.solve( x,d,g) ->re0;
print(re0);                                                   #print method, plain ASCII
print(re0,5);                                                 #print method, digits=5,plain ASCII
print(re0,type='md');                                         #print method, output GIT Markdown
#summary(re0) is under developing.