March  2017, 7(1): 107-112. doi: 10.3934/naco.2017007

Rank-based inference for the accelerated failure time model in the presence of interval censored data

Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

* Corresponding author: M. Karimi

Received  April 2016 Revised  November 2016 Published  February 2017

Semiparametric analysis and rank-based inference for the accelerated failure time model are complicated in the presence of interval censored data. The main difficulty with the existing rank-based methods is that they involve estimating functions with the possibility of multiple roots. In this paper a class of asymptotically normal rank estimators is developed which can be acquired via linear programming for estimating the parameters of the model, and a two-step iterative algorithm is introduced for solving the estimating equations. The proposed inference procedures are assessed through a real example. The results of applying the proposed methodology on the breast cancer data show that the algorithm converges after three iterations, and the estimations of model parameter based on Log-rank and Gehan weight functions are fairly close with small standard errors.

Citation: Mostafa Karimi, Noor Akma Ibrahim, Mohd Rizam Abu Bakar, Jayanthi Arasan. Rank-based inference for the accelerated failure time model in the presence of interval censored data. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 107-112. doi: 10.3934/naco.2017007
References:
[1]

M. BouadoumouY. Zhao and Y. Lu, Jackknife empirical likelihood for the accelerated failure time model with censored data, Communications in Statistics-Simulation and Computation, 44 (2014), 1818-1832.  doi: 10.1080/03610918.2013.833234.

[2]

S. H. ChiouS. Kang and J. Yan, Rank-based estimating equations with general weight for accelerated failure time models: an induced smoothing approach, Statistics in Medicine, 34 (2015), 1495-1510.  doi: 10.1002/sim.6415.

[3]

M. ChungQ. Long and B. A. Johnson, A tutorial on rank-based coefficient estimation for censored data in small-and large-scale problems, Statistics and computing, 23 (2013), 601-614.  doi: 10.1007/s11222-012-9333-9.

[4]

D. R. Cox, Regression models and life-tables, Journal of the Royal Statistical Society, 34 (1972), 187-220. 

[5]

D. M. Finkelstein and R. A. Wolfe, A semiparametric model for regression analysis of interval-censored failure time data, Biometrics, 41 (1985), 933-945.  doi: 10.2307/2530965.

[6]

M. Fygenson and Y. Ritov, Monotone estimating equations for censored data, The Annals of Statistics, 22 (1994), 732-746.  doi: 10.1214/aos/1176325493.

[7]

E. A. Gehan, A generalized Wilcoxon test for comparing arbitrarily single-censored samples, Biometrika, 52 (1965), 203-223. 

[8]

Z. JinD. Y. LinL. J. Wei and Z. Ying, Rank-based inference for the accelerated failure time model, Biometrika, 90 (2003), 341-353.  doi: 10.1093/biomet/90.2.341.

[9]

J. D. Kalbfleisch and R. L. Prentice, The Statistical Analysis of Failure Time Data Wiley, New York, 1980.

[10]

T. L. Lai and Z. Ying, Large sample theory of a modified Buckley-James estimator for regression analysis with censored data, The Annals of Statistics, 19 (1991a), 1370-1402.  doi: 10.1214/aos/1176348253.

[11]

T. L. Lai and Z. Ying, Rank regression methods for left-truncated and right-censored data, The Annals of Statistics, 19 (1991b), 531-556.  doi: 10.1214/aos/1176348110.

[12]

D. Y. LinL. J. Wei and Z. Ying, Accelerated failure time models for counting processes, Biometrika, 85 (1998), 605-618.  doi: 10.1093/biomet/85.3.605.

[13]

N. Mantel, Evaluation of survival data and two new rank order statistics arising in its considerations, Cancer Chemotherapy Reports, 50 (1966), 163-170. 

[14]

J. NingJ. Qin and Y. Shen, Semiparametric accelerated failure time model for length-biased data with application to dementia study, Statistica Sinica, 24 (2014), 313-333. 

[15]

L. Peng and J. P. Fine, Regression modeling of semicompeting risks data, Biometrics, 63 (2007), 96-108.  doi: 10.1111/j.1541-0420.2006.00621.x.

[16]

R. L. Prentice, Linear rank tests with right censored data, Biometrika, 65 (1978), 167-179.  doi: 10.1093/biomet/65.1.167.

[17]

Y. Ritov, Estimation in a linear regression model with censored data, The Annals of Statistics, 18 (1990), 303-328.  doi: 10.1214/aos/1176347502.

[18]

A. A. Tsiatis, Estimating regression parameters using linear rank tests for censored data, The Annals of Statistics, 18 (1990), 354-372.  doi: 10.1214/aos/1176347504.

[19]

Y. G. Wang and L. Fu, Rank regression for accelerated failure time model with clustered and censored data, Computational Statistics and Data Analysis, 55 (2011), 2334-2343.  doi: 10.1016/j.csda.2011.01.023.

[20]

L. J. WeiZ. Ying and D. Y. Lin, Linear regression analysis of censored survival data based on rank tests, Biometrika, 77 (1990), 845-851.  doi: 10.1093/biomet/77.4.845.

[21]

H. XueK. F. LamB. J. Cowling and F. de Wolf, Semi-parametric accelerated failure time regression analysis with application to intervalcensored HIV/AIDS data, Statistics in medicine, 25 (2006), 3850-3863.  doi: 10.1002/sim.2486.

[22]

Z. Ying, A large sample study of rank estimation for censored regression data, The Annals of Statistics, 21 (1993), 76-99.  doi: 10.1214/aos/1176349016.

[23]

J. Zhang and Y. Peng, Semiparametric estimation methods for the accelerated failure time mixture cure model, Journal of the Korean Statistical Society, 41 (2012), 415-422.  doi: 10.1016/j.jkss.2012.01.003.

show all references

References:
[1]

M. BouadoumouY. Zhao and Y. Lu, Jackknife empirical likelihood for the accelerated failure time model with censored data, Communications in Statistics-Simulation and Computation, 44 (2014), 1818-1832.  doi: 10.1080/03610918.2013.833234.

[2]

S. H. ChiouS. Kang and J. Yan, Rank-based estimating equations with general weight for accelerated failure time models: an induced smoothing approach, Statistics in Medicine, 34 (2015), 1495-1510.  doi: 10.1002/sim.6415.

[3]

M. ChungQ. Long and B. A. Johnson, A tutorial on rank-based coefficient estimation for censored data in small-and large-scale problems, Statistics and computing, 23 (2013), 601-614.  doi: 10.1007/s11222-012-9333-9.

[4]

D. R. Cox, Regression models and life-tables, Journal of the Royal Statistical Society, 34 (1972), 187-220. 

[5]

D. M. Finkelstein and R. A. Wolfe, A semiparametric model for regression analysis of interval-censored failure time data, Biometrics, 41 (1985), 933-945.  doi: 10.2307/2530965.

[6]

M. Fygenson and Y. Ritov, Monotone estimating equations for censored data, The Annals of Statistics, 22 (1994), 732-746.  doi: 10.1214/aos/1176325493.

[7]

E. A. Gehan, A generalized Wilcoxon test for comparing arbitrarily single-censored samples, Biometrika, 52 (1965), 203-223. 

[8]

Z. JinD. Y. LinL. J. Wei and Z. Ying, Rank-based inference for the accelerated failure time model, Biometrika, 90 (2003), 341-353.  doi: 10.1093/biomet/90.2.341.

[9]

J. D. Kalbfleisch and R. L. Prentice, The Statistical Analysis of Failure Time Data Wiley, New York, 1980.

[10]

T. L. Lai and Z. Ying, Large sample theory of a modified Buckley-James estimator for regression analysis with censored data, The Annals of Statistics, 19 (1991a), 1370-1402.  doi: 10.1214/aos/1176348253.

[11]

T. L. Lai and Z. Ying, Rank regression methods for left-truncated and right-censored data, The Annals of Statistics, 19 (1991b), 531-556.  doi: 10.1214/aos/1176348110.

[12]

D. Y. LinL. J. Wei and Z. Ying, Accelerated failure time models for counting processes, Biometrika, 85 (1998), 605-618.  doi: 10.1093/biomet/85.3.605.

[13]

N. Mantel, Evaluation of survival data and two new rank order statistics arising in its considerations, Cancer Chemotherapy Reports, 50 (1966), 163-170. 

[14]

J. NingJ. Qin and Y. Shen, Semiparametric accelerated failure time model for length-biased data with application to dementia study, Statistica Sinica, 24 (2014), 313-333. 

[15]

L. Peng and J. P. Fine, Regression modeling of semicompeting risks data, Biometrics, 63 (2007), 96-108.  doi: 10.1111/j.1541-0420.2006.00621.x.

[16]

R. L. Prentice, Linear rank tests with right censored data, Biometrika, 65 (1978), 167-179.  doi: 10.1093/biomet/65.1.167.

[17]

Y. Ritov, Estimation in a linear regression model with censored data, The Annals of Statistics, 18 (1990), 303-328.  doi: 10.1214/aos/1176347502.

[18]

A. A. Tsiatis, Estimating regression parameters using linear rank tests for censored data, The Annals of Statistics, 18 (1990), 354-372.  doi: 10.1214/aos/1176347504.

[19]

Y. G. Wang and L. Fu, Rank regression for accelerated failure time model with clustered and censored data, Computational Statistics and Data Analysis, 55 (2011), 2334-2343.  doi: 10.1016/j.csda.2011.01.023.

[20]

L. J. WeiZ. Ying and D. Y. Lin, Linear regression analysis of censored survival data based on rank tests, Biometrika, 77 (1990), 845-851.  doi: 10.1093/biomet/77.4.845.

[21]

H. XueK. F. LamB. J. Cowling and F. de Wolf, Semi-parametric accelerated failure time regression analysis with application to intervalcensored HIV/AIDS data, Statistics in medicine, 25 (2006), 3850-3863.  doi: 10.1002/sim.2486.

[22]

Z. Ying, A large sample study of rank estimation for censored regression data, The Annals of Statistics, 21 (1993), 76-99.  doi: 10.1214/aos/1176349016.

[23]

J. Zhang and Y. Peng, Semiparametric estimation methods for the accelerated failure time mixture cure model, Journal of the Korean Statistical Society, 41 (2012), 415-422.  doi: 10.1016/j.jkss.2012.01.003.

Figure 1.  Line chart of Gehan and Log-rank estimating functions versus failure time
Table 1.  Accelerated failure time analysis for the breast cancer data
WeightCovariateParameterStandardConfidence
functionestimateerrorinterval
Log-ranktreatment-0.8210.210(-1.232, -0.409)
Gehantreatment-0.6770.196(-1.061, -0.293)
WeightCovariateParameterStandardConfidence
functionestimateerrorinterval
Log-ranktreatment-0.8210.210(-1.232, -0.409)
Gehantreatment-0.6770.196(-1.061, -0.293)
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