emplik2 (version 1.00)

el2.cen.EMm: Computes p-value for multiple mean-type hypotheses, based on two independent samples that may contain censored data.

Description

This function uses the EM algorithm to calculate a maximized empirical likelihood ratio for a set of $p$ 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(p).

Usage

el2.cen.EMm(x, dx, y, dy, p, H, xc=1:length(x), yc=1:length(y),
  mean, maxit=10)

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

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

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

yc

a vector containing the indices of the y datapoints

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 10

Value

• el2.cen.EMm returns a list of values as follows:
• xd1
  a vector of the unique, uncensored $x$-values in ascending order
• yd1
  a vector of the 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
  the 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, can be used to assess convergence of the EM algorithm)
• 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

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 $mean_k$.)

References

Barton, W. (2009). PhD dissertation at University of Kentucky, estimated completion Dec. 2009. 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. Dr. Zhou is my PhD advisor at the University of Kentucky. My el2.cen.EMm function extends Dr. Zhou's el.cen.EM2 function 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)
maxit<-10
mean=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
# Ho2: mean of X equals mean of Y

el2.cen.EMm(x, dx, y, dy, p, H, xc=1:length(x), yc=1:length(y),
  mean, maxit=10)

# Result: Pval is 0.6310234, so we cannot with 95 percent confidence reject the two
# simultaneous hypotheses Ho1 and Ho2