Empirical Likelihood for Right Censored and Left Truncated data

1. Empirical Likelihood for Right Censored and Left Truncated data Jingyu (Julia) Luan University of Kentucky, Johns Hopkins University March 30, 2004

2. Outline of the Presentation: • Part I: Introduction and Background • Part II: Empirical Likelihood Theorem for Right-Censored and Left-Truncated Data • Part III: Future Research

3. 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

4. 1.1 Empirical Likelihood Ratio Test Likelihood Ratio Test: • Two cases: • Parametric • Non-parametric

5. 1.1 Empirical Likelihood Ratio Test Parametric situation: Wilks (1938): -2logR has an asymptotic p2 distribution under null hypothesis..

6. 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

7. 1.1 Empirical Likelihood Ratio Test • Owen focused on studying the properties ofthe likelihood ratio function when F0 satisfies certain constraint, (the null hypothesis) for a given g(.).

8. 1.1 Empirical Likelihood Ratio Test • Owendefined 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 χ2distribution with df=1.

9. 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

10. 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

11. 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.

12. 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.

13. 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;

14. 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.

15. 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.

16. 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.

17. Part II: Empirical Likelihood Theorem for Right-Censored and Left-Truncated Data • 2.1 One Sample Case • 2.2 Two Sample Case

18. Part III: One Sample Case First, introduce some notations:

19. Part III: One Sample Case • Then, the likelihood function is:

20. Part III: One Sample Case

21. Part III: One Sample Case

22. Part III: One Sample Case

23. Part III: Two Sample Case

24. Part III: Future research The same setting as one sample case. Under k constraints:

25. Part III: Future research Let be k observed samples.

26. Part III: Future research

27. Part III: Future Research Owen (1991) has demonstrated that empirical likelihood ratio can be applied to regression models. However, for censored/truncated data the empirical likelihood results are rare. Exception: Li (2002). Notice we are talking about ordinary regression model, not the Cox proportional hazards regression model.

28. Part III: Regression models We propose a “redistribution” algorithm of estimating the parameters in the regression models with censored/truncated data. We think the Empirical likelihood method can be used there to do inference.

29. Reference:

30. Reference:

31. Acknowledgement • Dr. Mai Zhou • Dr. Arne Bathke • Dr. William Griffith • Dr. William Rayens • Dr. Rencang Li