el2.cen.EMm: Computes empirical likelihood ratio and p-value for multiple...

In emplik2: Empirical Likelihood Ratio Test for Two-Sample U-Statistics with Censored Data

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 H = [H_1, H_2, \ldots, H_p], where H_k = [g_k(x_i,y_j)-mu_k], k=1, \ldots, p
xc	a vector containing the indices of the x datapoints, controls if tied x collapse or not
yc	a vector containing the indices of the y datapoints, ditto
mean	the hypothesized value of E(g(x,y))
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 x-values in ascending order
yd1	a vector of unique, uncensored y-values in ascending order
temp3	a list of values returned by the el2.test.wtm function (which is called by el2.cen.EMm)
mean	the hypothesized value of E(g(x,y))
NPMLE	a non-parametric-maximum-likelihood-estimator vector of E(g(x,y))
logel00	the log of the unconstrained empirical likelihood
logel	the log of the constrained empirical likelihood
"-2LLR"	-2*(log-likelihood-ratio) for the p simultaneous hypotheses
Pval	the p-value for the p simultaneous hypotheses, equal to 1 - pchisq(-2LLR, df = p)
logvec	the vector of successive values of logel computed by the EM algorithm (should converge toward a fixed value)
sum_muvec	sum of the probability jumps for the uncensored x-values, should be 1
sum_nuvec	sum of the probability jumps for the uncensored y-values, should be 1

Author(s)

William H. Barton <[e-mail]>

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

emplik2 documentation built on Sept. 12, 2024, 6:52 a.m.