el2.cen.EMs: 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)

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.

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

Author

William H. Barton <bbarton@lexmark.com> ; modified by Mai Zhou.

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.)

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

Run this code

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

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