Proportional hazards model under ranked set sampling scheme using censored data of coronary heart disease

Abstract

The proportional hazards model is one of the most common model for analyzing survival data. Only proportional hazards assumption is required to apply this model. Using appropriate sampling methods is an important part of modelling data and estimation of parameters. In literature there is a few studies based on sampling methods in survival analysis and most of them are related with non-parametric estimations of survival functions, sample size calculation etc.  The main innovation of our approach is to examine the sampling methods for the proportional hazards model. This paper describes usage of ranked set sampling design in the proportional hazards model. In order to analyze the performance of our methods, we use a real data and conduct a simulation study. We conclued that ranked set sampling is more efficient than simple random sampling.

Keywords

• Cox regression,likelihood function,censored data,ranked set sampling

References

1. [1] A.I. Al-Omari, Ratio estimation of the population mean using auxiliary information in simple random sampling and median ranked set sampling, Statist. Probab. Lett. 82, 1883-1890, 2012.
2. [2] M.F. Al-Saleh and A.I. Al-Omari, Multistage ranked set sampling, J. Statist. Plann. Inference 102, 273286, 2002.
3. [3] D. G. Altman, B.L. De Stavola, S.B. Love and K. A. Stepniewska, Review of survival analyses published in cancer journals, British Journal of Cancer 72(2), 511, 1995.
4. [4] N. Ata Tutkun, N. Koyuncu and U. Karabey, Discrete-time survival analysis under ranked set sampling: an application to Turkish motor insurance data, J. Stat. Comput. Simul. 89(4), 660-667, 2019.
5. [5] S.K. Ashour and M.S. Abdallah, New distribution function estimators and tests of perfect ranking in concomitant-based ranked set sampling, Comm. Statist. Simulation Comput. 1-26, 2019.
6. [6] M. J. Bradburn, T. G. Clark, S. B. Love, and D. G. Altman, Survival analysis part II: multivariate data analysisan introduction to concepts and methods, British Journal of Cancer 89(3), 431-436, 2003.
7. [7] J. Borucka, Methods of handling tied events in the Cox proportional hazard model, Studia Oeconomica Posnaniensia 2(2), 91-106, 2014.
8. [8] N.E. Breslow, Covariance analysis of censored survival data, Biometrics 30, 89-99, 1974.

9. [9] H. Che, Cutoff sample size estimation for survival data: a simulation study, Unpublished Master Thesis, Department of Statistics, Uppsala University, Sweden, 2013.
10. [10] D. Collett, Modelling survival data in medical research, Chapman and Hall, UK, 1994.
11. [11] D.R. Cox, Regression models and life tables (with discussion), J. R. Stat. Soc. Ser. B. Stat. Methodol. 34,187-220, 1972.
12. [12] B. Efron, The efficiency of Coxs likelihood function for censored data, J. Amer. Statist. Assoc. 76, 312-319, 1977.
13. [13] J. Frey, Nonparametric mean estimation using partially ordered sets, Environmental and Ecological Statistic 19, 309-326, 2012.
14. [14] N.M. Gemayel, E.A. Stasny, J.A. Tackett and D. A. Wolfe, Ranked set sampling: An auditing application. Review of Quantitative Finance and Accounting 39, 413-422, 2012.
15. [15] F.Y. Hsieh and P. W. Lavori, Sample-size calculations for the Cox proportional hazards regression model with nonbinary covariates, Controlled Clinical Trials 21,552560, 2000.
16. [16] A.A. Jemain and A.I. Al-Omari, Multistage median ranked set samples for estimating the population mean, Pakistan Journal of Statistics 22,195207, 2006.
17. [17] A.A. Jemain, A.I. Al-Omari and K. Ibrahim, Multistage extreme ranked set sampling for estimating the population mean, J. Stat. Theory Appl. 6(4),456471, 2007.
18. [18] J.D. Kalbfleisch and R.L. Prentice, The statistical analysis of failure time data, Wiley, New York, 1980.
19. [19] M. Mahdizadeh, and E. Zamanzade, Smooth estimation of a reliability function in ranked set sampling, Statistics 52, 750-768, 2018.
20. [20] M. Mahdizadeh, and E. Zamanzade, Interval estimation of $P(X < Y)$ in ranked set sampling, Comput. Statist. 33, 1325-1348, 2018.
21. [21] M. Mahdizadeh, and E. Zamanzade, Efficient body fat estimation using multistage pair ranked set sampling, Stat. Methods Med. Res. 28: 223-234, 2019.
22. [22] M. G. Marmot, M. J. Shipley and G. Rose, Inequalities in deathspecific explanations of a general pattern, The Lancet 323(8384), 1003-1006, 1984.
23. [23] G.A. McIntyre, A method for unbiased selective sampling, using ranked sets, Australian Journal of Agricultural Research 3, 385390, 1952.
24. [24] M. Moerbeek, Sufficient sample sizes for discrete-time survival analysis mixture models, Structural Equation Modelling: A Multidisipliniary Journal 21(1), 63-67, 2014.
25. [25] O. Ozturk, Sampling from partially rank-ordered sets, Environmental and Ecological Statistics 18, 757-779, 2011.
26. [26] H. M. Samawi, A. Helu, H. Rochani, J. Yin, L. Yu and R. Vogel, Reducing sample size needed for accelerated failure time model using more efficient sampling methods, J. Stat. Theory Pract. 12(3), 530-541, 2018.
27. [27] D. Schoenfeld, Sample-size formula for the proportional-hazards regression model, Biometrics 39,499503, 1983.
28. [28] P. Royston, G. Ambler, and W. Sauerbrei, The use of fractional polynomials to model continuous risk variables in epidemiology, International Journal of Epidemiology, 28(5), 964-974, 1999.
29. [29] S. Wang, J. Zhang and W. Lu, Sample size calculation for the proportional hazards model with a time-dependent covariate, Comput. Statist. Data Anal. 74, 217-227, 2014.
30. [30] E. Zamanzade and M. Vock, Variance estimation in ranked set sampling using a concomitant variable, Statist. Probab. Lett. 105, 1-5, 2015.
31. [31] E. Zamanzade and M. Mahdizadeh, A more efficient proportion estimator in ranked set sampling Statist. Probab. Lett. 129, 28-33, 2017.
32. [32] E. Zamanzade and M. Mahdizadeh, Distribution function estimation using concomitant-based ranked set sampling, Hacet. J. Math. Stat. 47(3), 755-761, 2018.
33. [33] E. Zamanzade and M. Mahdizadeh, Estimating the population proportion in pair ranked set sampling with application to air quality monitoring, J. Appl. Stat. 45(3), 426-437, 2018.
34. [34] E. Zamanzade and M. Mahdizadeh, Using ranked set sampling with extreme ranks in estimating the population proportion, Stat. Methods Med. Res. 29 (1), 165-177, 2020.