EMPIRICAL LIKELIHOOD INFERENCE FOR THE ACCELERATED FAILURE TIME MODEL USING KENDALL ESTIMATING EQUATUION

1. Georgia State University EMPIRICAL LIKELIHOOD INFERENCE FOR THE ACCELERATED FAILURE TIME MODEL USING KENDALL ESTIMATING EQUATUION By Yinghua Lu June 29th 2009

3. Main Procedure

4. Simulation Study

5. Real Application

8. Non-monotone estimating equations

11. Based on a data-driven likelihood ratio function

12. Without specifying a parametric family of distributions for the data.

13. The shape of confidence regions

15. Introduction – Empirical Likelihood Likelihood ratio: Owen (2001) proved

17. Summarized and discussed in Owen (1988, 1990, 1991, 2001)

18. Qin and Jing (2001) and Li and Wang (2003): the limiting distribution EL ratio is a weighted chi-square distribution.

20. Main Procedure – Preliminaries We can rewrite it as a U-statistic with symmetric kernel, Similar to Fygenson and Ritov (1994), where R and J are defined similarly in Fygenson and Ritov (1994).

21. Main Procedure – Preliminaries The asymptotic variance of generalized estimate of β is The numerator can be estimated by The denominator can be estimated by Then we can construct the confidence interval as

22. Main Procedure – Empirical Likelihood Let and Apply the idea of Sen (1960), we define where W’s are independently distributed.

23. Main Procedure – Empirical Likelihood Let be a probability vector. Then the empirical likelihood function at the value β is given by For this function, reaches its maximum when Thus, the empirical likelihood ratio at β is defined by

24. Main Procedure – Empirical Likelihood By Lagrange Multiplier method for logarithm transformation of above equation, we write Setting the partial derivative of G with respect to p to 0, we have then

25. Main Procedure – Empirical Likelihood Plug into the previous equation, we obtain So, for all the p’s We have

26. Main Procedure – Empirical Likelihood Theorem 1 Under the above conditions, converges in distribution to , where is a chi-square random variable with p degrees of freedom. Confidence region for β is given by EL confidence region for the q sub-vector Of Theorem 2 Under the above conditions, converges in distribution to , where is a chi-square random variable with q degrees of freedom. confidence region for is given by

28. The censoring time C ~ Uniform distribution in [0, c], where c controls the censoring rate.

30. The censoring time C is defined as 2exp(1)+c.

31. The error term has the standard Normal distribution N(0,1), which is symmetric.Setting: Repetition: 10000

32. Simulation Study – EL vs. NA Results for model 1:

33. Simulation Study – EL vs. NA Results for model 1:

34. Simulation Study – EL vs. NA Results for model 1:

35. Simulation Study – EL vs. NA Results for model 1:

36. Simulation Study – EL vs. NA Results for model 2:

37. Simulation Study – EL vs. NA Results for model 2:

38. Simulation Study – EL vs. NA Results for model 2:

39. Simulation Study – EL vs. NA Results for model 2:

41. As the censoring rate increase, the coverage probabilities (CP) for both methods decrease.

43. A little over-coverage problem with the EL.

45. The censoring time C ~ Normal distribution as N(µ, 42), where µ produce samples with censoring rate equal to 10%, 30%, 50%, 75%.

46. The error term has Normal distribution as N(0, 0.52).

47. Sample Size: 50, 100 and 200

51. Real Application We consider the following four variables: Disease Group (3 groups) Waiting Time to Transplant in Days (from 24 to 2616 days, mean=275 days) Recipient and Donor Age (from 7 to 52 and from 2 to 56) French-American-British (FAB): classification based on standard morphological criteria.

52. Real Application

53. Real Application Results: Two methods show similar results. Two exceptions may due to asymmetric CI of the EL. Average lengths of the EL are a little longer than that of the NA. Same results with the simulation study.

55. The coverage probabilities of the EL are closer to the nominal levels than NA, especially when the sample size is very small and censoring rate is heavy.

57. The combination shows a problem of over-coverage.