Empirical Likelihood with Arbitrary Censored/Truncated Data by Constrained EM Algorithm

1. Empirical Likelihood with Arbitrary Censored/Truncated Data by Constrained EM Algorithm Min Chen, Jingyu Luan, Mai Zhou Department of Statistics University of Kentucky 817 Patterson Office Tower Lexington, KY 40506 minchen@ms.uky.edu

2. Outline • Introduction • Maximize Empirical Likelihood under parameters of weighted hazard • Maximize Empirical Likelihood under parameters of weighted hazards for data with covariates • Examples and Numerical results • Conclusion we can compute empirical likelihood ratio for arbitrary censored/truncated data with/without covariates under weighted hazard parameter.

3. Introduction (1) • Empirical Likelihood Ratio Method • Empirical Distribution for n iid observations • Empirical Likelihood function • Empirical Likelihood Ratio

4. Introduction (1) • Empirical Likelihood ratio Function by Owen(1988)

5. Introduction (2) • Arbitrary Censored/Truncated Data • According to Turnbull (1976) and Frydman (1994), any observation in an arbitrary censored/Truncated Data can be described by , with , where And we associate each observation with , the covariate, if any.

6. Introduction (2) For example • Any exact observation without covariate could be described as • Any right censored observation with one covariate could be described as

7. Introduction (2) The likelihood for the arbitrary censored and/or truncated data is proportion to

8. Introduction (3) Based on Turnbull (1976) and Alioum (1996), for arbitrary censored/truncated data, we could construct a set such that • Any c.d.f. jumps outside of C could not be MLE of the unknown distribution function • The likelihood is independent of the behavior of the distribution inside each interval .

9. Introduction (3) Now the problem of maximizing emprical likelihood reduces to maximize The MLE of can be obtained by self- Consistency/EM algorithm.

10. Maximum of Empirical Likelihood under parameters of weighted hazard • The empirical log likelihood for , i=1,…,N is proportional to • and could be written in terms of hazards as

11. Maximum Empirical Likelihood under parameters of weighted hazard • Under hazard constraint , we may think of using the Lagrange Multiplier method to find the maximum log likelihood, but it turns out to be intractable.

12. Maximum Empirical Likelihood under parameters of weighted hazard • Modified self-consistency/EM algorithm E-Step: Given current Estimate of H(.), we can compute a weight on each pseudo jump point of H(.). M-Step: For the pseudo jump points associated with weights in E-Step, we could compute a new hazard jump.

13. Maximum Empirical Likelihood under parameters of weighted hazard Theorem 1 Under hazard type constraint , the NPMLE of hazard jumps obtained by the modified EM algorithm for the arbitrary censored and/or truncated data is equivalent to the solution by Lagrange Multiplier Method.

14. Empirical ML under parameters of weighted hazards for data with covariates • For arbitrary censored/truncated data with covariates, by Cox proportional hazards regression model, the log likelihood could be written as

15. Empirical ML under parameters of weighted hazards for data with covariates • Under hazard constraint , and hypothesis about , obtain the maximum log likelihood by using Lagrange Multiplier method is even more complicate.

16. Empirical ML under parameters of weighted hazards for data with covariates Theorem 2 Under hazard type constraint, the NPMLE of hazard jumps obtained by the modified EM algorithm for the arbitrary censored and/or truncated data with covariates are equivalent to the solution by Lagrange Multiplier Method.

17. Examples and Numerical results (1) Left Truncated Right Censored Data without covariates under hazard constraint The maximum log likelihood is achieved at =0.33

18. Examples and Numerical results (1) Left Truncated Right Censored Data without covariates under hazard constraint The maximum log likelihood is achieved at =0.71

19. Examples and Numerical results (2) Right Censored Data with one covariate under hazard constraint The maximum log likelihood is achieved at =-0.01

20. Examples and Numerical results (3) Interval Censored Data with one covariate and no hazard constraint The maximum log likelihood is achieved at =0.1

21. Conclusion we can compute empirical likelihood ratio for arbitrary censored/truncated data with/without covariates under weighted hazard parameter.

22. References • Alioum A. and Commenges D. (1996) A proportional Hazards Model for Arbitrarily Censored and Truncated Data. Biometrics, 52, 512-524. • Cox, D.R. (1972) Regression models and life tables (with discussion). J. of the Royal Statistical Society, Series B, 34, 187-220. • Frydman, H. (1994) A note on nonparametric estimation of the distribution function from interval-censored and truncated observations. Journals of the Royal Statistical Society, Series B, 56, 71-74. • Gentleman, R. and Ihaka, R. (1996) R: A Language for data analysis and graphics. J. of Computational and Graphical Statistics, 5, 299-314. • Luan J.Y., Chen M. and Zhou M. (2003) Empirical Likelihood Ratio with Right Censoring and Left Truncation Data. Technical Report.

23. References • Klein and Moeschberger (1997) Survival Analysis: Techniques for Censored and Truncated Data. Springer, New York. • Owen, A. (2001) Empirical Likelihood. Chapman \& Hall. London. • Pan, X.R. and Zhou, M. (1999). Using one parameter sub-family of distributions in empirical likelihood with censored data. J.Statist. Planning and Infer. 75, 379-392. • Thomas, D. R. and Grunkemeier, G.L. (1975). Confidence Interval estimation of survival probabilities for censored data. Amer. Statist. Assoc. 70, 865-871. • Turnbull B, The empirical distribution function with arbitrary grouped, censored and truncated data. JRSS B, 290-295. • Zhou M. (2003). Empirical likelihood ratio with arbitrary censored/truncated data by EM algorithm. Technical Report.