el2.cen.EMs | R Documentation |
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).
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)
x |
a vector of the data for the first sample |
dx |
a vector of the censoring indicators for |
y |
a vector of the data for the second sample |
dy |
a vector of the censoring indicators for |
fun |
a user-defined, weight-function |
mean |
the hypothesized value of |
tol.u |
Error tolerance for iteration control. L1 norm of the |
tol.v |
Error tolerance for iteration control. L1 norm of the |
maxit |
a positive integer used to set the maximum number of iterations of the EM algorithm; default is 50 |
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.)
el2.cen.EMs
returns a list of values as follows:
xd1 |
a vector of the unique, uncensored |
yd1 |
a vector of the unique, uncensored |
temp3 |
a list of values returned by the |
mean |
the hypothesized value of |
funNPMLE |
the non-parametric-maximum-likelihood-estimator of |
logel00 |
the log of the unconstrained empirical likelihood |
logel |
the log of the constrained empirical likelihood |
"-2LLR" |
|
Pval |
the estimated p-value for |
logvec |
the vector of successive values of |
sum_muvec |
sum of the probability jumps for the uncensored |
sum_nuvec |
sum of the probability jumps for the uncensored |
constraint |
the realized value of |
William H. Barton <bbarton@lexmark.com> ; modified by Mai Zhou.
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.
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
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.