Empirical Likelihood for Right Empirical Likelihood for Right Censored and Left Truncated dataCensored and Left Truncated data

Jingyu (Julia) LuanJingyu (Julia) Luan

University of Kentucky, Johns Hopkins UniversityUniversity of Kentucky, Johns Hopkins UniversityMarch 30, 2004March 30, 2004

Outline of the Presentation:Outline of the Presentation:

• Part I: Introduction and BackgroundPart I: Introduction and Background

• Part II: Empirical Likelihood Theorem for Part II: Empirical Likelihood Theorem for Right-Censored and Left-Truncated DataRight-Censored and Left-Truncated Data

• Part III: Future ResearchPart III: Future Research

Part I: Introduction and Background 1.1 Empirical Likelihood Ratio Test

1.2 Censoring and Truncation

1.3 Literature Review

1.4 Counting Process and Survival Analysis

1.1 Empirical Likelihood Ratio Test

HH Under Likelihood of Maximum

H Under Likelihood of Maximum

a0

0

R

Two cases:

A. Parametric

B. Non-parametric

Likelihood Ratio Test:

1.1 Empirical Likelihood Ratio Test

Parametric situation: Wilks (1938): -2logR has an

asymptotic p2 distribution under

null hypothesis..

1.1 Empirical Likelihood Ratio Test

Nonparametric situation: Empirical Likelihood is defined as (Owen 1988):

Where X1, X2, …, Xn are independent random variables from an unknown distribution F. And it is well known that L(F) is maximized by the empirical distribution function Fn over all possible CDFs, where

1.1 Empirical Likelihood Ratio Test

Owen focused on studying the properties of the likelihood ratio function when F0 satisfies certain constraint, (the null hypothesis)

for a given g(.).

1.1 Empirical Likelihood Ratio Test

Owen defined the empirical likelihood ratio function as

where L(F|θ) is the maximized empirical likelihood function among all CDFs satisfying the above constraint (null hypothesis).

Owen (1988) demonstrated when θ=θ0,

-2logR(F|θ) has an asymptotic χ2 distribution with df=1.

1.2 Censoring and Truncation

Survival analysis is the analysis of time-to-event data.

Two important features of time-to-event data:

A. Censoring

B. Truncation

1.2 Censoring and Truncation

Censoring occurs when an individual’s life length is known to happen only in a certain period of time.

A. Right Censoring

B. Left Censoring

C. Interval Censoring

1.2 Censoring and Truncation

Truncation A. Left Truncation: it occurs when subjects enter a

study at a particular time and are followed from this delayed entry time until the event happens or until the subject is censored;

B. Right Truncation: it occurs when only individuals who have experienced the event of interest are included in the sample.

1.2 Censoring and Truncation

Example of Right-Censored and Left-Truncated data: [Klein, Moeschberger, p65]

In a survival study of the Channing House retirement center located in California, ages at which individuals entered the retirement community (truncation event) and ages when members of the community died or still alive at the end of the study (censoring event) were recorded.

1.3 Literature Review

Product limit estimator (based on right censored and left truncated data) of survival function: an analogue of the Kaplan-Meier estimator of survival function under censoring;

For solely truncation data, Wang and Jewell (1985) and Woodroofe (1985) independently proved consistency results for the product limit estimator and showed weak convergence to a Brownian Motion process;

1.3 Literature Review

Wang, Jewell, and Tsai (1987): described a description of the asymptotic behavior of the product limit estimator for right censoring and left truncation data

For pure truncated data, Li (1995) studied the empirical likelihood theorem.

1.3 Literature Review

For solely censoring data: Pan and Zhou (1999) showed that the empirical

likelihood ratio with continuous mean and hazard constraint also have a chi-square limit.

Fang (2000) proved the empirical likelihood ratio with discrete hazard constraint follows a chi-square distribution under one sample case and two sample case.

1.4 Counting Process and Survival Analysis

Counting process provides an elegant martingale based approach to study time-to-event data.

Martingale methods can be used to obtain simple expressions for the moments of complicated statistics and to calculate and verify asymptotic distributions for test statistics and estimators.

Part II: Empirical Likelihood Theorem for Part II: Empirical Likelihood Theorem for Right-Censored and Left-Truncated DataRight-Censored and Left-Truncated Data

2.1 One Sample Case2.1 One Sample Case

2.2 Two Sample Case2.2 Two Sample Case

Part III: One Sample CasePart III: One Sample Case

First, introduce some notations:First, introduce some notations:

Part III: One Sample CasePart III: One Sample Case

Then, the likelihood function is: Then, the likelihood function is:

Part III: One Sample CasePart III: One Sample Case

Part III: One Sample CasePart III: One Sample Case

Part III: One Sample CasePart III: One Sample Case

Part III: Two Sample CasePart III: Two Sample Case

Part III: Future researchPart III: Future research

The same setting as one sample case. Under k constraints:The same setting as one sample case. Under k constraints:

Part III: Future researchPart III: Future research

LetLet

be k observed samples.

Part III: Future researchPart III: Future research

Part III: Future ResearchPart III: Future Research

Owen (1991) has demonstrated that empirical Owen (1991) has demonstrated that empirical likelihood ratio can be applied to regression models.likelihood ratio can be applied to regression models.

However, for censored/truncated data the empirical However, for censored/truncated data the empirical likelihood results are rare. Exception: Li (2002).likelihood results are rare. Exception: Li (2002).

Notice we are talking about ordinary regression Notice we are talking about ordinary regression model, not the Cox proportional hazards regression model, not the Cox proportional hazards regression model.model.

Part III: Regression modelsPart III: Regression models

We propose a “redistribution” algorithm of estimating We propose a “redistribution” algorithm of estimating the parameters in the regression models with the parameters in the regression models with censored/truncated data.censored/truncated data.

We think the Empirical likelihood method can be We think the Empirical likelihood method can be used there to do inference.used there to do inference.

Reference:

Reference:

Acknowledgement

Dr. Mai Zhou

Dr. Arne Bathke

Dr. William Griffith

Dr. William Rayens

Dr. Rencang Li