emplik2 (version 1.00)

el2.cen.EMs: 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 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

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.

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 $mean$.)

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.EMs function extends Dr. Zhou's 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