Empirical likelihood for hazard with right censored, left truncated data

Description

Use empirical likelihood ratio and Wilks theorem to test the null hypothesis that

\int f(t) dH(t) = θ

with right censored, left truncated data. Where H(t) is the unknown cumulative hazard function; f(t) can be any given function and θ a given constant. In fact, f(t) can even be data dependent, just have to be ‘predictable’.

Usage

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emplikH1.test(x, d, y= -Inf, theta, fun, tola=.Machine$double.eps^.5)

Arguments

x

a vector of the censored survival times.

d

a vector of the censoring indicators, 1-uncensor; 0-censor.

y

a vector of the observed left truncation times.

theta

a real number used in the H_0 to set the hazard to this value.

fun

a left continuous (weight) function used to calculate the weighted hazard in H_0. fun must be able to take a vector input. See example below.

tola

an optional positive real number specifying the tolerance of iteration error in solve the non-linear equation needed in constrained maximization.

Details

This function is designed for the case where the true distributions are all continuous. So there should be no tie in the data.

The log empirical likelihood used here is the ‘Poisson’ version empirical likelihood:

∑_{i=1}^n δ_i \log (dH(x_i)) - [ H(x_i) - H(y_i) ] ~.

If there are ties in the data that are resulted from rounding, you may break the tie by adding a different tiny number to the tied observation(s). If those are true ties (thus the true distribution is discrete) we recommend use emplikdisc.test().

The constant theta must be inside the so called feasible region for the computation to continue. This is similar to the requirement that in testing the value of the mean, the value must be inside the convex hull of the observations. It is always true that the NPMLE values are feasible. So when the computation complains that there is no hazard function satisfy the constraint, you should try to move the theta value closer to the NPMLE. When the computation stops prematurely, the -2LLR should have value infinite.

Value

A list with the following components:

times

the location of the hazard jumps.

wts

the jump size of hazard function at those locations.

lambda

the Lagrange multiplier.

"-2LLR"

the -2Log Likelihood ratio.

Pval

P-value

niters

number of iterations used

Author(s)

Mai Zhou

References

Pan, X. and Zhou, M. (2002), “Empirical likelihood in terms of hazard for censored data”. Journal of Multivariate Analysis 80, 166-188.

Examples

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fun <- function(x) { as.numeric(x <= 6.5) }
emplikH1.test( x=c(1,2,3,4,5), d=c(1,1,0,1,1), theta=2, fun=fun) 
fun2 <- function(x) {exp(-x)}  
emplikH1.test( x=c(1,2,3,4,5), d=c(1,1,0,1,1), theta=0.2, fun=fun2) 

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