kmc (version 0.4-2)

kmc.bjtest: Calculate the NPMLE with constriants for accelerated failure time model with given coefficients.

Description

Use the empirical likelihood ratio and Wilks theorem to test if the regression coefficient equals beta. $$El(F)=\prod_{i=1}^{n}(\Delta F(T_i))^{\delta_i}(1-F(T_i))^{1-\delta_i}$$ with constraints $$ \sum_i g(T_i)\Delta F(T_i)=0,\quad,i=1,2,\ldots $$ Instead of EM algorithm, this function calculates the Kaplan-Meier estimator with mean constraints recursively to test \(H_0:~\beta=\beta_0\) in the accelerated failure time model: $$ \log(T_i) = y_i = x_i\beta^\top + \epsilon_i, $$ where \(\epsilon\) is distribution free.

Usage

kmc.bjtest(y, d, x, beta,init.st="naive")

Value

a list with the following components:

prob

the probabilities that max the empirical likelihood under estimating equation.

logel1

the log empirical likelihood without constraints, i.e. under Kaplan-Merier of residuals'

logel2

the log empirical likelihood with constraints, i.e. under null hypotheses or estimation equations.

"-2LLR"

the -2 loglikelihood ratio; have approximate chisq distribution under null hypotheses

convergence

an indicator:

0: fails to converge

1: converged

Arguments

y

Response variable vector (length n).

d

Status vector (length n), 0: right censored; 1 uncensored.

x

n by p explanatory variable matrix.

beta

The value of the regression coeffiecnt vector (length p) to be tested.

init.st

Type of methods to initialize the algorithm. By default, init.st is set to 1/n

Author

Mai Zhou([e-mail]), Yifan Yang([e-mail])

Details

The empirical likelihood is the likelihood of the error term when the coefficients are specified. Model assumptions are the same as requirements of a standard Buckley-James estimator.

References

Buckley, J. and James, I. (1979). Linear regression with censored data. Biometrika, 66 429-36

Zhou, M., & Li, G. (2008). Empirical likelihood analysis of the Buckley-James estimator. Journal of multivariate analysis, 99(4), 649-664.

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, bjtest.

Examples

library(survival)
stanford5 <- stanford2[!is.na(stanford2$t5), ]
y <- log10(stanford5$time)
d <- stanford5$status
oy <- order(y, -d)
d <- d[oy]
y <- y[oy]
x <- cbind(1, stanford5$age)[oy,]
beta0  <-  c(3.2, -0.015)
ss  <-  kmc.bjtest(y, d, x=x, beta=beta0, init.st="naive")