# Empirical likelihood for weighted hazard with right censored, left truncated data

### Description

Use empirical likelihood ratio and Wilks theorem to test the null hypothesis that

* \int f(t, ... ) dH(t) = K *

with right censored, left truncated data, where * H(t) * is the (unknown)
cumulative hazard function;
* f(t, ... )* can be any given left continuous function in *t*;
(of course the integral must be finite).

### Usage

1 | ```
emplikH2.test(x, d, y= -Inf, K, fun, tola=.Machine$double.eps^.5,...)
``` |

### Arguments

`x` |
a vector containing the censored survival times. |

`d` |
a vector of the censoring indicators, 1-uncensor; 0-censor. |

`y` |
a vector containing the left truncation times. If left as default value, -Inf, it means no truncation. |

`K` |
a real number used in the constraint, i.e. to set the weighted integral of hazard to this value. |

`fun` |
a left continuous (in |

`tola` |
an optional positive real number specifying the tolerance of iteration error in solve the non-linear equation needed in constrained maximization. |

`...` |
additional parameter(s), if any, passing along to |

### Details

This version works for implicit function ` f(t, ...)`

.

This function is designed for continuous distributions.
Thus we do not expect tie in the observation `x`

. If you believe
the true underlying distribution is continuous but the
sample observations have tie due to rounding, then you might want
to add a small number to the observations to break tie.

The likelihood used here is the ‘Poisson’ version of the empirical likelihood

*
∏_{i=1}^n ( dH(x_i) )^{δ_i} \exp [-H(x_i)+H(y_i)] .
*

For discrete distributions we recommend
use `emplikdisc.test()`

.

Please note here the largest observed time is NOT automatically defined to be uncensored. In the el.cen.EM( ), it is (to make F a proper distribution always).

The constant `K`

must be inside the so called
feasible region for the computation to continue. This is similar to the
requirement that when testing the value of the mean, the value must be
inside the convex hull of the observations for the computation to continue.
It is always true that the NPMLE value is feasible. So when the
computation cannot continue, that means there is no hazard function
dominated by the Nelson-Aalen estimator satisfy
the constraint. You may try to move the `theta`

and `K`

closer
to the NPMLE. When the computation cannot continue,
the -2LLR should have value
infinite.

### Value

A list with the following components:

`times` |
the location of the hazard jumps. |

`wts` |
the jump size of hazard function at those locations. |

`lambda` |
the Lagrange multiplier. |

`"-2LLR"` |
the -2Log Likelihood ratio. |

`Pval` |
P-value |

`niters` |
number of iterations used |

### Author(s)

Mai Zhou

### References

Pan, XR and Zhou, M. (2002),
“Empirical likelihood in terms of cumulative
hazard for censored data”.
*Journal of Multivariate Analysis* **80**, 166-188.

### See Also

emplikHs.test2

### Examples

1 2 3 4 5 6 7 8 9 10 11 | ```
z1<-c(1,2,3,4,5)
d1<-c(1,1,0,1,1)
fun4 <- function(x, theta) { as.numeric(x <= theta) }
emplikH2.test(x=z1,d=d1, K=0.5, fun=fun4, theta=3.5)
#Next, test if H(3.5) = log(2) .
emplikH2.test(x=z1,d=d1, K=log(2), fun=fun4, theta=3.5)
#Next, try one sample log rank test
indi <- function(x,y){ as.numeric(x >= y) }
fun3 <- function(t,z){rowsum(outer(z,t,FUN="indi"),group=rep(1,length(z)))}
emplikH2.test(x=z1, d=d1, K=sum(0.25*z1), fun=fun3, z=z1)
##this is testing if the data is from an exp(0.25) population.
``` |

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