el2.cen.EMs: Computes empirical likelihood ratio and p-value for a single...

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

Computes empirical likelihood ratio and p-value for a single mean-type hypothesis, 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 the hypothesis

H_o: E(g(x,y)-mean)=0

where E indicates expected value; g(x,y) is a user-defined function of x and y; and mean is the hypothesized value of E(g(x,y)). The default: g(x,y)=I[x \geq y], mean=0.5. The 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 approximately distributed as chisq(df=1).

Usage

el2.cen.EMs(x,dx,y,dy,fun=function(x,y){x>=y},mean=0.5,
                                tol.u=1e-6,tol.v=1e-6,maxit=50)

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
fun	a user-defined, weight-function g(x,y) used to define the mean in the hypothesis H_o. The default is fun=function(x,y){x>=y}.
mean	the hypothesized value of E(g(x,y)); default is 0.5
tol.u	Error tolerance for iteration control. L1 norm of the u-uOLD is used. Default 1e-6
tol.v	Error tolerance for iteration control. L1 norm of the v-vOLD is used. Default 1e-6
maxit	a positive integer used to set the maximum number of iterations of the EM algorithm; default is 50

Details

The empirical likelihood used here is

EL(mean) = \max_{\mu_i, \nu_j} \left\{ \prod \mu_i \prod \nu_j ; s.t. \sum_i \sum_j g(x_i, y_j) \mu_i \nu_j = mean; \sum \mu_i =1; \sum \nu_j =1. \right\}

for uncensored data. If data were censored, approapriate adjustments are used accordingly. See Owen (2001) section 11.4.

The value of mean should be chosen between the maximum and minimum values of g(x_i,y_j); otherwise there may be no distributions for x and y that will satisfy H_o. If mean is inside this interval, but the convergence is still not satisfactory, then the value of mean should be moved closer to the NPMLE for E(g(x,y)). (The NPMLE itself should always be a feasible value for mean. This NPMLE value is in the output.)

Value

el2.cen.EMs 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.wts function (which is called by el2.cen.EMs)
mean	the hypothesized value of E(g(x,y))
funNPMLE	the non-parametric-maximum-likelihood-estimator of E(g(x,y))
logel00	the log of the unconstrained empirical likelihood
logel	the log of the constrained empirical likelihood
"-2LLR"	-2*(logel-logel00)
Pval	the estimated p-value for H_o, computed as 1-pchisq(-2LLR, df = 1)
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
constraint	the realized value of \sum_{i=1}^n \sum_{j=1}^m (g(x_i,y_j) - mean) \mu_i \nu_j, where \mu_i and \nu_j are the probability jumps at x_i and y_j, respectively, that maximize the empirical likelihood ratio. The value of constraint should be close to 0.

Author(s)

William H. Barton <[e-mail]> ; modified by Mai Zhou.

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 el2.cen.EMs function here extends the el.cen.EM function inside emplik package from one sample to two-samples.

Examples

 
x<-c(10,80,209,273,279,324,391,415,566,785,852,881,895,954,1101,
1133,1337,1393,1408,1444,1513,1585,1669,1823,1941)
dx<-c(1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,0,0,1,0,0,0,0,1,1,0)
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(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0,0,0)

# Ho1:  X is stochastically equal to Y (i.e. P(X>Y)=0.5) 
el2.cen.EMs(x, dx, y, dy, fun=function(x,y){x>=y}, mean=0.5)
# Result: Pval = 0.7090658, so we cannot with 95 percent confidence reject Ho1
# Remark: may be we should be more careful for the (x=y) cases, if any.

# Ho2: mean of X equals mean of Y
el2.cen.EMs(x, dx, y, dy, fun=function(x,y){x-y}, mean=0)
# Result: Pval = 0.9716493, so we cannot with 95 percent confidence reject Ho2

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