Check the contraints of KMC

Description

To derive the empirical likelihood with constraints, we need to make sure there are solutions. Dines' method is used here to check whether the linear constraintsare proper or not.

Usage

check_G_mat(gmat)

Arguments

Value

Author(s)

Yifan Yang(yfyang.86@hotmail.com)

References

Dines, L. L. (1926). On positive solutions of a system of linear equations Annals of Mathematics pages 386–392

Zhou, M. and Yang, Y. (2015). A recursive formula for the Kaplan-Meier estimator with mean constraints and its application to empirical likelihood Computational Statistics. Online ISSN 1613-9658.

Examples

#### A Proper Example ####
x <- c( 1, 1.5, 2, 3, 4.2, 5.0, 6.1, 5.3, 4.5, 0.9, 2.1, 4.3) 
d <- c( 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1) 
f1 <-function(x) { x - 3.7} 
f2 <- function(x) {x^2 - 16.5 } 
g <- list(f1, f2)
re = kmc.clean(x, d)
p = length(g)
n = length(re$kmc.time)
gmat<-matrix(0, p, n);
for(i in 1:p){
  gmat[i,] = g[[i]](re$kmc.time)
  }
# You may want to require(Rcpp) on some platforms (such Mac OSX-ARM)
# library(Rcpp)
# check_G_mat(gmat)

Calculate the NPMLE with constriants for accelerated failure time model with given coefficients.

Description

Use the empirical likelihood ratio and Wilks theorem to test if the regression coefficient equals beta.

El(F)=\prod_{i=1}^{n}(\Delta F(T_i))^{\delta_i}(1-F(T_i))^{1-\delta_i}

with constraints

\sum_i g(T_i)\Delta F(T_i)=0,\quad,i=1,2,\ldots

Instead of EM algorithm, this function calculates the Kaplan-Meier estimator with mean constraints recursively to test H_0:~\beta=\beta_0 in the accelerated failure time model:

\log(T_i) = y_i = x_i\beta^\top + \epsilon_i,

where \epsilon is distribution free.

Usage

kmc.bjtest(y, d, x, beta,init.st="naive")

Arguments

Details

The empirical likelihood is the likelihood of the error term when the coefficients are specified. Model assumptions are the same as requirements of a standard Buckley-James estimator.

Value

a list with the following components:

Author(s)

Mai Zhou(mai@ms.uky.edu), Yifan Yang(yfyang.86@hotmail.com)

References

Buckley, J. and James, I. (1979). Linear regression with censored data. Biometrika, 66 429-36

Zhou, M., & Li, G. (2008). Empirical likelihood analysis of the Buckley-James estimator. Journal of multivariate analysis, 99(4), 649-664.

Zhou, M. and Yang, Y. (2015). A recursive formula for the Kaplan-Meier estimator with mean constraints and its application to empirical likelihood Computational Statistics. Online ISSN 1613-9658.

See Also

plotkmc2D, bjtest.

Examples

library(survival)
stanford5 <- stanford2[!is.na(stanford2$t5), ]
y <- log10(stanford5$time)
d <- stanford5$status
oy <- order(y, -d)
d <- d[oy]
y <- y[oy]
x <- cbind(1, stanford5$age)[oy,]
beta0  <-  c(3.2, -0.015)
ss  <-  kmc.bjtest(y, d, x=x, beta=beta0, init.st="naive")

Perform Data Clean for the kmc Algorithm

Description

The kmc.clean function clean the (kmc.time, delta) for the randomized censored data:

- Reorder the data according to the observed time and status;

- Clean the (right) censored data point(s) if they happen before the first uncesored data.

- If there are ties in the data. For the time points contain ties, e.g.

(T_{i_s}, d_{i_s}), i_s \in S \forall j \in S, T_{j} \equiv T

, we re-arranged the data in a manner that those with d=1 are ordered ahead of those with d=0. As d=0 indicates the data point is right censored, such procedure is trivial.

Usage

kmc.clean(kmc.time, delta)

Arguments

Value

A list with the following components:

Author(s)

Yifan Yang(yfyang.86@hotmail.com)

References

Zhou, M. and Yang, Y. (2015). A recursive formula for the Kaplan-Meier estimator with mean constraints and its application to empirical likelihood Computational Statistics Online ISSN 1613-9658.

Examples

x <- c( 1, 1.5, 2, 3, 4.2, 5.0, 6.1, 5.3, 4.5, 0.9, 2.1, 4.3) 
d <- c( 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1) 
kmc.clean(x, d)

Calculate NPMLE with constriants for right censored data

Description

This function calculate the Kaplan-Meier estimator with mean constraints recursively.

El(F)=\prod_{i=1}^{n}(\Delta F(T_i))^{\delta_i}(1-F(T_i))^{1-\delta_i}

with constraints

\sum_i g(T_i)\Delta F(T_i)=0,\quad,i=1,2,...

It uses Lagrange multiplier directly.

Usage

kmc.solve(x, d, g, em.boost = T, using.num = T, using.Fortran =
                 T, using.C = F, tmp.tag = T, rtol = 1e-09, control =
                 list(nr.it = 20, nr.c = 1, em.it = 3),...)

Arguments

Details

The function check_G_mat checks whether the solution space is null or not under the constraint. But due to the computational complexity, it will detect at most two conditions.

Value

A list with the following components:

Author(s)

Mai Zhou(mai@ms.uky.edu), Yifan Yang(yfyang.86@hotmail.com)

References

Zhou, M. and Yang, Y. (2015). A recursive formula for the Kaplan-Meier estimator with mean constraints and its application to empirical likelihood Computational Statistics Online ISSN 1613-9658.

See Also

plotkmc2D.

Examples

# positive time
x <- c( 1, 1.5, 2, 3, 4.2, 5.0, 6.1, 5.3, 4.5, 0.9, 2.1, 4.3) 
# status censored/uncensored
d <- c( 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1)               

#################
# dim =1
#################

f <- function(x) {x-3.7}                     # \sum f(ti) wi ~ 0 
g <- list(f=f) ;                             #define constraint as a list

kmc.solve(x, d, g) ;                         #using default
kmc.solve(x, d, g, using.C=TRUE) ;           #using Rcpp

#################
# dim =2
#################

myfun5 <- function( x)  { 
 x^2-16.5
} 

g <- list( f1=f,f2=myfun5) ;                 #define constraint as a list

re0 <- kmc.solve(x,d,g);

###################################################
# Print Estimation and other information 
# with option: digits = 5
###################################################

#' Print kmc object
#' 
#' @param x kmc object
#' @param digits minimal number of significant digits, see print.default.
f_print <- function(x, digits = 5){
  cat("\n----------------------------------------------------------------\n")
  cat("A Recursive Formula for the Kaplan-Meier Estimator with Constraint\n")
  cat("Information:\n")
  cat("Number of Constraints:\t", length(x$g), "\n")
  cat("lamda(s):\t", x$lambda,'\n');
  cat("\n----------------------------------------------------------------\n")
  names <- c("Log-likelihood(Ha)", "Log-likelihood(H0)",
  "-2LLR", paste("p-Value(df=", length(x$g), ")",sep = ""))
  re <- matrix(c(x[[1]], x[[2]], x[[3]], 1 - pchisq(x[[3]],
  length(x$g))), nrow = 1)
  colnames(re) <- names
  rownames(re) <- "Est"
  print.default(format(re, digits = digits), print.gap = 2,
                quote = FALSE, df = length(x$g))
  cat("------------------------------------------------------------------\n")
}

f_print(re0)

Plot the contour plot of log-likelihood around the H0 (dim=2).

Description

Given a kmc object, this function will produce contour plot if there were two constraints.

Usage

plotkmc2D(resultkmc, flist=list(f1=function(x){x}, f2=function(x){x^2}), 
          range0=c(0.2, 3,20))

Arguments

Value

Author(s)

Yifan Yang(yfyang.86@hotmail.com)

Examples

x <- c( 1, 1.5, 2, 3, 4.2, 5.0, 6.1, 5.3, 4.5, 0.9, 2.1, 4.3) 
d <- c( 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1) 

f<-function( x) { x-3.7} 

myfun5 <- function( x)  { 
 x^2-16.5
} 
# construnct g as a LIST!

g=list( f1=f,f2=myfun5) ; 
kmc.solve( x,d,g) ->re0;

#plotkmc2D(re0) ->ZZ; # run this to generate contour plot
#Advanced PLOT option using ggplot2: not run
#library(reshape2)
#volcano3d <- melt(ZZ$Z)
#names(volcano3d) <- c("x", "y", "z")

#volcano3d$x <- ZZ$X[volcano3d$x];
#volcano3d$y <- ZZ$Y[volcano3d$y];

#### Plot: use ggplot2 ####
#library(ggplot2)
# v <- ggplot(volcano3d, aes(x, y, z=z));
# v + geom_tile(aes(fill = z)) + 
#   stat_contour()+
#   scale_fill_gradientn("Custom Colours",colours=grey.colors(10));

#### Plot: use qplot ####
#qplot(x, y, z = z, data = volcano3d, stat = "contour", geom = "path")