Empirical likelihood ratio test for a vector of means with right, left or doubly censored data, by EM algorithm

Description

This function is similar to el.cen.EM(), but for multiple constraints. In the input there is a vector of observations x = (x_1, \cdots , x_n) and a function fun. The function fun should return the (n by k) matrix

( f_1(x), f_2(x), \cdots, f_k (x) ) .

Also, the ordering of the observations, when consider censoring or redistributing-to-the-right, is according to the value of x, not fun(x). So the probability distribution is for values x. This program uses EM algorithm to maximize (wrt p_i) empirical log likelihood function for right, left or doubly censored data with the MEAN constraint:

j = 1,2, \cdots ,k ~~~~ ∑_{d_i=1} p_i f_j(x_i) = \int f_j(t) dF(t) = μ_j ~.

Where p_i = Δ F(x_i) is a probability, d_i is the censoring indicator, 1(uncensored), 0(right censored), 2(left censored). It also returns those p_i. The log likelihood function is defined as

∑_{d_i=1} \log Δ F(x_i) + ∑_{d_i=2} \log F(x_i) + ∑_{d_i=0} \log [ 1-F(x_i)] ~.

Usage

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el.cen.EM2(x,d,xc=1:length(x),fun,mu,maxit=25,error=1e-9,...)

Arguments

x

a vector containing the observed survival times.

d

a vector containing the censoring indicators, 1-uncensored; 0-right censored; 2-left censored.

xc

an optional vector of collapsing control values. If xc[i] xc[j] have different values then (x[i], d[i]), (x[j], d[j]) will not merge into one observation with weight two, even if they are identical. Default is not to merge.

fun

a left continuous (weight) function that returns a matrix. The columns (=k) of the matrix is used to calculate the means and will be tested in H_0. fun(t) must be able to take a vector input t.

mu

a vector of length k. Used in the constraint, as the mean of f(X).

maxit

an optional integer, used to control maximum number of iterations.

error

an optional positive real number specifying the tolerance of iteration error. This is the bound of the L_1 norm of the difference of two successive weights.

...

additional inputs to pass to fun().

Details

This implementation is all in R and have several for-loops in it. A faster version would use C to do the for-loop part. (but this version is easier to port to Splus, and seems faster enough).

We return the log likelihood all the time. Sometimes, (for right censored and no censor case) we also return the -2 log likelihood ratio. In other cases, you have to plot a curve with many values of the parameter, mu, to find out where the log likelihood becomes maximum. And from there you can get -2 log likelihood ratio between the maximum location and your current parameter in Ho.

In order to get a proper distribution as NPMLE, we automatically change the d for the largest observation to 1 (even if it is right censored), similar for the left censored, smallest observation. μ is a given constant vector. When the given constants μ is too far away from the NPMLE, there will be no distribution satisfy the constraint. In this case the computation will stop. The -2 Log empirical likelihood ratio should be infinite.

The constant vector mu must be inside ( \min f(x_i) , \max f(x_i) ) for the computation to continue. It is always true that the NPMLE values are feasible. So when the computation stops, try move the mu closer to the NPMLE —

\hat μ _j = ∑_{d_i=1} p_i^0 f_j(x_i)

where p_i^0 taken to be the jumps of the NPMLE of CDF. Or use a different fun.

Difference to the function el.cen.EM: due to the introduction of input xc here in this function, the output loglik may be different compared to the function el.cen.EM due to not collapsing of duplicated input survival values. The -2LLR should be the same from both functions.

Value

A list with the following components:

loglik

the maximized empirical log likelihood under the constraints.

times

locations of CDF that have positive mass.

prob

the jump size of CDF at those locations.

"-2LLR"

If available, it is Minus two times the Empirical Log Likelihood Ratio. Should be approx. chi-square distributed under Ho.

Pval

If available, the P-value of the test, using chi-square approximation.

lam

the Lagrange multiplier in the final EM step. (the M-step)

Author(s)

Mai Zhou

References

Zhou, M. (2005). Empirical likelihood ratio with arbitrary censored/truncated data by EM algorithm. Journal of Computational and Graphical Statistics, 643-656.

Zhou, M. (2002). Computing censored empirical likelihood ratio by EM algorithm. Tech Report, Univ. of Kentucky, Dept of Statistics

Examples

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## censored regression with one right censored observation.
## we check the estimation equation, with the MLE inside myfun7. 
y <- c(3, 5.3, 6.4, 9.1, 14.1, 15.4, 18.1, 15.3, 14, 5.8, 7.3, 14.4)
x <- c(1, 1.5, 2,   3,   4,    5,    6,    5,    4,  1,   2,   4.5)
d <- c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0)
### first we estimate beta, the MLE
lm.wfit(x=cbind(rep(1,12),x), y=y, w=WKM(x=y, d=d)$jump[rank(y)])$coef
## you should get 1.392885 and 2.845658
## then define myfun7 with the MLE value
myfun7 <- function(y, xmat) {
temp1 <- y - ( 1.392885 +  2.845658 * xmat)
return( cbind( temp1, xmat*temp1) )
}
## now test 
el.cen.EM2(y,d, fun=myfun7, mu=c(0,0), xmat=x)
## we should get, Pval = 1 , as the MLE should.
## for other values of (a, b) inside myfun7, you get other Pval
##
rqfun1 <- function(y, xmat, beta, tau = 0.5) {
temp1 <- tau - (1-myfun55(y-beta*xmat))
return(xmat * temp1)
}
myfun55 <- function(x, eps=0.001){
u <- x*sqrt(5)/eps
INDE <- (u < sqrt(5)) & (u > -sqrt(5))
u[u >= sqrt(5)] <- 0
u[u <= -sqrt(5)] <- 1
y <- 0.5 - (u - (u)^3/15)*3/(4*sqrt(5))
u[ INDE ] <- y[ INDE ]
return(u)
}
## myfun55 is a smoothed indicator fn. 
## eps should be between (1/sqrt(n), 1/n^0.75) [Chen and Hall]
el.cen.EM2(x=y,d=d,xc=1:12,fun=rqfun1,mu=0,xmat=x,beta=3.08,tau=0.44769875)
## default tau=0.5 
el.cen.EM2(x=y,d=d,xc=1:12,fun=rqfun1,mu=0,xmat=x,beta=3.0799107404)
###################################################
### next 2 examples are testing the mean/median residual time
###################################################
mygfun <- function(s, age, muage) {as.numeric(s >= age)*(s-(age+muage))}
mygfun2 <- function(s, age, Mdage) 
          {as.numeric(s <= (age+Mdage)) - 0.5*as.numeric(s <= age)}
## Not run: 
library(survival) 
time <- cancer$time
status <- cancer$status-1
###for mean residual time 
el.cen.EM2(x=time, d=status, fun=mygfun, mu=0, age=365.25, muage=234)$Pval
el.cen.EM2(x=time, d=status, fun=mygfun, mu=0, age=365.25, muage=323)$Pval
### for median resudual time
el.cen.EM2(x=time, d=status, fun=mygfun2, mu=0.5, age=365.25, Mdage=184)$Pval
el.cen.EM2(x=time, d=status, fun=mygfun2, mu=0.5, age=365.25, Mdage=321)$Pval

## End(Not run)

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