# kmc.bjtest: Calculate NPMLE with constriants for accelerated failure time... In kmc: Kaplan-Meier Estimator with Constraints for Right Censored Data -- a Recursive Computational Algorithm

## Description

Use the empirical likelihood ratio and Wilks theorem to test if the regression coefficient is equal to beta.

El(F)=∏_{i=1}^{n}(Δ F(T_i))^{δ_i}(1-F(T_i))^{1-δ_i}

with constraints

Instead of EM algorithm, this function calculate the Kaplan-Meier estimator with mean constraints recursively to test H_0:~β=β_0 in accelerated failure time model:

\log(T_i) = y_i = x_iβ^\top + ε_i,

where ε is distribution free.

## Usage

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

## 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. Default uses 1/n

## 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.

## 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

## Author(s)

Mai Zhou([email protected]), Yifan Yang([email protected])

## 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.

plotkmc2D, print.kmcS3 bjtest.
 1 2 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