el.cen.EM {emplik} | R Documentation |
Empirical likelihood ratio for mean with right, left or doubly censored data, by EM algorithm
Description
This program uses EM algorithm to compute the maximized
(wrt ) empirical
log likelihood function for right, left or doubly censored data with
the MEAN constraint:
Where is a probability,
is the censoring indicator, 1(uncensored), 0(right censored),
2(left censored).
It also returns those
.
The empirical log likelihood been maximized is
Usage
el.cen.EM(x,d,wt=rep(1,length(d)),fun=function(t){t},mu,maxit=50,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. |
wt |
a weight vector (case weight). positive. same length as d |
fun |
a left continuous (weight) function used to calculate
the mean as in |
mu |
a real number used in the constraint, the mean value of |
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
|
... |
additional arguments, if any, to pass to |
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 seems faster enough and is easier to port to Splus.
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 is the place 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 for the largest observation to 1
(even if it is right censored), similar for the left censored,
smallest observation.
is a given constant.
When the given constants
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 mu
must be inside
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 —
taken to be the jumps of the NPMLE of CDF.
Or use a different
fun
.
Difference to the function el.cen.EM2
: here duplicate (input) observations
are collapsed (with weight 2, 3, ... etc.) but those
will stay separate by default in the el.cen.EM2
. This will lead to
a different loglik
value. But the -2LLR
value should be same
in either version.
Value
A list with the following components:
loglik |
the maximized empirical log likelihood under the constraint. This may be different from the result of el.cen.EM2 because here the tied observations are collapes into 1 with weight. (while el.cen.EM2 do not). However, the -2LLR should be the same. |
times |
locations of CDF that have positive mass. tied obs. are collapesd |
prob |
the jump size of CDF at those locations. |
"-2LLR" |
If available, it is Minus two times the Empirical Log Likelihood Ratio. Should be approximately chi-square distributed under Ho. |
Pval |
The P-value of the test, using chi-square approximation. |
lam |
The Lagrange multiplier. Added 5/2007. |
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.
Murphy, S. and van der Vaart (1997) Semiparametric likelihood ratio inference. Ann. Statist. 25, 1471-1509.
Examples
## example with tied observations
x <- c(1, 1.5, 2, 3, 4, 5, 6, 5, 4, 1, 2, 4.5)
d <- c(1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1)
el.cen.EM(x,d,mu=3.5)
## we should get "-2LLR" = 1.2466....
myfun5 <- function(x, theta, eps) {
u <- (x-theta)*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)
}
el.cen.EM(x, d, fun=myfun5, mu=0.5, theta=3.5, eps=0.1)
## example of using wt in the input. Since the x-vector contain
## two 5 (both d=1), and two 2(both d=0), we can also do
xx <- c(1, 1.5, 2, 3, 4, 5, 6, 4, 1, 4.5)
dd <- c(1, 1, 0, 1, 0, 1, 1, 1, 0, 1)
wt <- c(1, 1, 2, 1, 1, 2, 1, 1, 1, 1)
el.cen.EM(x=xx, d=dd, wt=wt, mu=3.5)
## this should be the same as the first example.