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

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

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• 1. Georgia State University
EMPIRICAL LIKELIHOOD INFERENCE FOR THE ACCELERATED FAILURE TIME MODEL USING KENDALL ESTIMATING EQUATUION
By Yinghua Lu
June 29th 2009
• 2. Contents
• Introduction
• 3. Main Procedure
• 4. Simulation Study
• 5. Real Application
• 6. Conclusion
• Introduction – AFT Model
Accelerated Failure Time (AFT) Model:
• Very popular.
• 7. Similar to the classic linear regression:
where Y=ln(T).
Different methods are developed
• OLS
• 8. Non-monotone estimating equations
• 9. Monotone estimating equations with normal approximation.
• Introduction – Kendall’s Tau
Let {X1,Y1} and {X2, Y2} be two observations of two variables.
Kendall’s tau coefficient is defined as:
where nc is the number of [sign(X1-X2) = sign(Y1-Y2)], nd is the number of [sign(X1-X2) = -sign(Y1-Y2)].
Sen(1968) proposed
ε(b)=Y-bX
U(b) is non-increasing in b.
• 10. Introduction – Empirical Likelihood
• A nonparametric method
• 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
• 14. Joins the reliability of the nonparametric methods and the efficiency of the likelihood methods.
• Introduction – Empirical Likelihood
For X1,X2,…,Xn, the likelihood function is defined by
Let X1,X2,…,Xnbe n independent samples, the empirical cumulative distribution (ECDF) at x is
The nonparametric likelihood of the CDF can be defined as
• 15. Introduction – Empirical Likelihood
Likelihood ratio:
Owen (2001) proved
• 16. Introduction – Brief History
• Traced back to Thomas and Grunkemeier (1975)
• 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.
• 19. Zhou (2005) and Zhou and Li (2008): Logrank and Gehan estimators, and Buckley-James estimator.
• Main Procedure – Preliminaries
Let T1,…,Tn be a sequence of random variables and Ti &gt; 0. Let Z1,…,Zn be their corresponding covariates sequence.
Z and β are px1 vectors.
We observe and
Define
We employee the estimating equation as follow:
• 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
• 27. Simulation Study – EL vs. NA
Consider the AFT model:
Model 1: (skewed error distribution)
• Z ~ Uniform distribution in [-1, 1].
• 28. The censoring time C ~ Uniform distribution in [0, c], where c controls the censoring rate.
• 29. The error term has the standard extreme value distribution, which is skewed to the right.
• Simulation Study – EL vs. NA
Model 2: (symmetric error distribution ).
• Z ~ Uniform distribution in [0.5, 1.5].
• 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:
• 40. Simulation Study – EL vs. NA
Summary:
• As the sample size increase, the coverage probabilities (CP) for both methods increase.
• 41. As the censoring rate increase, the coverage probabilities (CP) for both methods decrease.
• 42. When the sample size is small, the CP for EL is better than NA, for very heavy censoring rate, both are not good enough though.
• Simulation Study – EL vs. NA
Summary:
• Average length for the EL is a little longer than the NA in all cases.
• 43. A little over-coverage problem with the EL.
• 44. Under-coverage problem with the NA.
• Simulation Study – Kendall vs. others
Consider the following AFT model:
We observe and
Model 3:
• Z ~ Normal distribution as N(1, 0.52).
• 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
• 48. Repetition: 5000
• Simulation Study – Kendall
Results for model 3:
• 49. Simulation Study – Kendall
Results for model 3:
• When the sample size is small (n=50) and the censoring rate is heavy, Kendall’s rank regression estimator is better an all the other estimators.
• 50. In other cases, Kendall’s rank regression estimator is also comparative.
• Real Application
Bone marrow transplants are a standard treatment for acute leukemia.
Total of 137 patients were treated.
For simplicity, the model contains only one covariate at a time, which is where Ti is Time to Death.
The response variable Time to Death takes values from 1 day to 2640 days with mean equal to 839.16 days.
• 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.
• 54. Conclusion & Discussion
• Average length of the CI by the EL are slightly longer than that by NA.
• 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.
• 56. Kendall’s rank regression estimator is better than the Buckley-James, Logrank and Gehan estimators in terms of coverage probabilities.
• Conclusion & Discussion
• The combination of the Kendall estimating equation and the EL CI has strong advantages over the other considered approaches in the case of small sample size and heavy censoring rate.
• 57. The combination shows a problem of over-coverage.
• 58. A smoothing kernel is suggested to eliminate such a problem in the future work.
• Thank you !