el2.cen.EMs {emplik2}

Computes 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 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(1).

Usage

el2.cen.EMs(x,dx,y,dy,fun=function(x,y){x>=y}, mean=0.5, maxit=25)

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, continuous-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

maxit

a positive integer used to set the number of iterations of the EM algorithm; default is 25

Details

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.)

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]>

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 extends el.cen.EM function 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
el2.cen.EMs(x, dx, y, dy, fun=function(x,y){x>=y}, mean=0.5, maxit=25)
# Result: Pval = 0.7090658, so we cannot with 95 percent confidence reject Ho1

# Ho2: mean of X equals mean of Y
el2.cen.EMs(x, dx, y, dy, fun=function(x,y){x-y}, mean=0.5, maxit=25)
# Result: Pval = 0.9695593, so we cannot with 95 percent confidence reject Ho2