el2.cen.EMm {emplik2} | R Documentation |
Computes empirical likelihood ratio and p-value for multiple mean-type hypotheses, based on two independent samples that may contain censored data.
Description
This function is similar to el2.cen.EMs
but for several mean type restrictions.
This function uses the EM algorithm to calculate a maximized empirical likelihood ratio for a set of p
simultaneous hypotheses
as follows:
H_o: E(g(x,y)-mean)=0
where E
indicates expected value; g(x,y)
is a vector of user-defined functions: g_1(x,y), \ldots,
g_p(x,y)
; and mean
is a vector of p
hypothesized values of E(g(x,y))
. The two samples x
and y
are assumed independent. They may be uncensored, right-censored, left-censored, or left-and-right (“doubly”)
censored. A p-value for H_o
is also calculated, based on the assumption that -2*log(empirical likelihood ratio)
is asymptotically distributed as chisq(df=p).
Usage
el2.cen.EMm(x, dx, wx=rep(1,length(x)), y, dy, wy=rep(1,length(y)),
p, H, xc=1:length(x), yc=1:length(y), mean, maxit=35)
Arguments
x |
a vector of the data for the first sample |
dx |
a vector of the censoring indicators for x: 0=right-censored, 1=uncensored, 2=left-censored |
wx |
a vector of data case weight for x |
y |
a vector of the data for the second sample |
dy |
a vector of the censoring indicators for y: 0=right-censored, 1=uncensored, 2=left-censored |
wy |
a vector of data case weight for y |
p |
the number of hypotheses |
H |
a matrix defined as |
xc |
a vector containing the indices of the |
yc |
a vector containing the indices of the |
mean |
the hypothesized value of |
maxit |
a positive integer used to control the maximum number of iterations of the EM algorithm; default is 35 |
Details
The value of mean_k
should be chosen between the maximum and minimum values of g_k(x_i,y_j)
; otherwise
there may be no distributions for x
and y
that will satisfy H_o
. If mean_k
is inside
this interval, but the convergence is still not satisfactory, then the value of mean_k
should be moved
closer to the NPMLE for E(g_k(x,y))
. (The NPMLE itself should always be a feasible value for mean_k
.)
Value
el2.cen.EMm
returns a list of values as follows:
xd1 |
a vector of unique, uncensored |
yd1 |
a vector of unique, uncensored |
temp3 |
a list of values returned by the |
mean |
the hypothesized value of |
NPMLE |
a non-parametric-maximum-likelihood-estimator vector of |
logel00 |
the log of the unconstrained empirical likelihood |
logel |
the log of the constrained empirical likelihood |
"-2LLR" |
-2*(log-likelihood-ratio) for the |
Pval |
the p-value for the |
logvec |
the vector of successive values of |
sum_muvec |
sum of the probability jumps for the uncensored |
sum_nuvec |
sum of the probability jumps for the uncensored |
Author(s)
William H. Barton <bbarton@lexmark.com>
References
Barton, W. (2010). Comparison of two samples by a nonparametric likelihood-ratio test. PhD dissertation at University of Kentucky.
Chang, M. and Yang, G. (1987). “Strong Consistency of a Nonparametric Estimator of the Survival Function with Doubly Censored Data.” Ann. Stat.
,15, pp. 1536-1547.
Dempster, A., Laird, N., and Rubin, D. (1977). “Maximum Likelihood from Incomplete Data via the EM Algorithm.” J. Roy. Statist. Soc.
, Series B, 39, pp.1-38.
Gomez, G., Julia, O., and Utzet, F. (1992). “Survival Analysis for Left-Censored Data.” In Klein, J. and Goel, P. (ed.),
Survival Analysis: State of the Art.
Kluwer Academic Publishers, Boston, pp. 269-288.
Li, G. (1995). “Nonparametric Likelihood Ratio Estimation of Probabilities for Truncated Data.”
J. Amer. Statist. Assoc.
, 90, pp. 997-1003.
Owen, A.B. (2001). Empirical Likelihood
. Chapman and Hall/CRC, Boca Raton, pp. 223-227.
Turnbull, B. (1976). “The Empirical Distribution Function with Arbitrarily Grouped, Censored and Truncated Data.”
J. Roy. Statist. Soc.
, Series B, 38, pp. 290-295.
Zhou, M. (2005). “Empirical likelihood ratio with arbitrarily censored/truncated data by EM algorithm.”
J. Comput. Graph. Stat.
, 14, pp. 643-656.
Zhou, M. (2009) emplik
package on CRAN website.
The function el2.cen.EMm
here extends el.cen.EM2
inside emplik
package from one-sample to two-samples.
Examples
x<-c(10, 80, 209, 273, 279, 324, 391, 415, 566, 85, 852, 881, 895, 954, 1101, 1133,
1337, 1393, 1408, 1444, 1513, 1585, 1669, 1823, 1941)
dx<-c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0)
y<-c(21, 38, 39, 51, 77, 185, 240, 289, 524, 610, 612, 677, 798, 881, 899, 946, 1010,
1074, 1147, 1154, 1199, 1269, 1329, 1484, 1493, 1559, 1602, 1684, 1900, 1952)
dy<-c(1,1,1,1,1,1,2,2,1,1,1,1,1,2,1,1,1,1,1,1,0,0,1,1,0,0,1,0,0,0)
nx<-length(x)
ny<-length(y)
xc<-1:nx
yc<-1:ny
wx<-rep(1,nx)
wy<-rep(1,ny)
mu=c(0.5,0.5)
p <- 2
H1<-matrix(NA,nrow=nx,ncol=ny)
H2<-matrix(NA,nrow=nx,ncol=ny)
for (i in 1:nx) {
for (j in 1:ny) {
H1[i,j]<-(x[i]>y[j])
H2[i,j]<-(x[i]>1060) } }
H=matrix(c(H1,H2),nrow=nx,ncol=p*ny)
# Ho1: X is stochastically equal to Y (i.e. P(X>Y)=0.5)
# Ho2: P(X>1060)=0.5
el2.cen.EMm(x=x, dx=dx, y=y, dy=dy, p=2, H=H, mean=mu)
# Result: Pval is 0.6310234, so we cannot with 95 percent confidence reject the two
# simultaneous hypotheses Ho1 and Ho2