emplikH2.test {emplik}R Documentation

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 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

emplikH2.test(x, d, y= -Inf, K, fun, tola=.Machine$double.eps^.25,...)

Arguments

x a vector containing the observed survival times.
d a vector of the censoring indicators, 1-uncensor; 0-censor.
y a vector containing the left truncation times.
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 observations. 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 likelihood

prod_{i=1}^n ( dH(x_i) )^{delta_i} exp [-H(x_i)+H(y_i)] .

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

The constants theta and K 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 stops, that means there is no hazard function satisfy the constraint. You may try to move the theta and K closer to the NPMLE. When the computation stops, 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 and Zhou (2002), ``Empirical likelihood in terms of cumulative hazard for censored data''. Journal of Multivariate Analysis 80, 166-188.

Examples

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) 

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