Empirical likelihood for weighted 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) = K

with right censored, left truncated data, where H(t) is the (unknown) cumulative hazard function; f(t, ... ) can be any given left continuous function in t; (of course the integral must be finite).

Usage

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emplikH2.test(x, d, y= -Inf, K, fun, tola=.Machine$double.eps^.5,...)

Arguments

x

a vector containing the censored survival times.

d

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

y

a vector containing the left truncation times. If left as default value, -Inf, it means no truncation.

K

a real number used in the constraint, i.e. to set the weighted integral of hazard to this value.

fun

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

tola

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

...

additional parameter(s), if any, passing along to fun. This allows an implicit function of fun.

Details

This version works for implicit function f(t, ...).

This function is designed for continuous distributions. Thus we do not expect tie in the observation x. If you believe the true underlying distribution is continuous but the sample observations have tie due to rounding, then you might want to add a small number to the observations to break tie.

The likelihood used here is the ‘Poisson’ version of the empirical likelihood

∏_{i=1}^n ( dH(x_i) )^{δ_i} \exp [-H(x_i)+H(y_i)] .

For discrete distributions we recommend use emplikdisc.test().

Please note here the largest observed time is NOT automatically defined to be uncensored. In the el.cen.EM( ), it is (to make F a proper distribution always).

The constant K must be inside the so called feasible region for the computation to continue. This is similar to the requirement that when testing the value of the mean, the value must be inside the convex hull of the observations for the computation to continue. It is always true that the NPMLE value is feasible. So when the computation cannot continue, that means there is no hazard function dominated by the Nelson-Aalen estimator satisfy the constraint. You may try to move the theta and K closer to the NPMLE. When the computation cannot continue, 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, XR and Zhou, M. (2002), “Empirical likelihood in terms of cumulative hazard for censored data”. Journal of Multivariate Analysis 80, 166-188.

See Also

emplikHs.test2

Examples

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z1<-c(1,2,3,4,5)
d1<-c(1,1,0,1,1)
fun4 <- function(x, theta) { as.numeric(x <= theta) }
emplikH2.test(x=z1,d=d1, K=0.5, fun=fun4, theta=3.5)
#Next, test if H(3.5) = log(2) .
emplikH2.test(x=z1,d=d1, K=log(2), fun=fun4, theta=3.5)
#Next, try one sample log rank test
indi <- function(x,y){ as.numeric(x >= y) }
fun3 <- function(t,z){rowsum(outer(z,t,FUN="indi"),group=rep(1,length(z)))} 
emplikH2.test(x=z1, d=d1, K=sum(0.25*z1), fun=fun3, z=z1) 
##this is testing if the data is from an exp(0.25) population.

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