The Comparison of Time Functions in the Extended Cox Regression Model

Abstract

Extended Cox regression model by using any form of time function is one of the alternative methods to the Cox regression model in non-proportional hazards case or time-dependent covariate problem. It is a key concern which time function should be used in which case for an extended Cox regression model. In this study, a comparison of the most commonly used time functions for the extended Cox regression model to obtain the effects of variables not satisfying the proportional hazard assumption is carried out. This simulation study assesses the ability of the time functions for the extended Cox regression model in modeling non-proportional hazards according to sample sizes, censoring rate, and prevalence ratio of the binary covariate. The results indicate that the linear time function (t) is more biased than the logarithmic time function (log(t)), which is a frequently used time function in modeling the hazard ratio. Also, it is shown that the use of time function 1/t has better results in most situations.

Keywords

• Extended Cox model
• Time function
• Non-proportional hazards
• Time dependent covariate
• Proportional hazards

References

1. [1] Cox, D.R., “Regression models and life‐tables”, Journal of the Royal Statistical Society: Series B (Methodological), 34: 187-202, (1972). DOI: https://doi.org/10.1111/j.2517-6161.1972.tb00899.x
2. [2] Fisher, L.D., and Lin, D.Y., “Time-dependent covariates in the Cox proportional-hazards regression model”, Annual Review of Public Health, 20: 145-157, (1999). DOI: https://doi.org/10.1146/annurev.publhealth.20.1.145
3. [3] Pettitt, A., and Daud, I.B., “Investigating time dependence in Cox's proportional hazards model”, Journal of the Royal Statistical Society: Series C (Applied Statistics), 39: 313-329, (1990). DOI: https://doi.org/10.2307/2347382
4. [4] Henderson, R., and Oman, P., “Effect of frailty on marginal regression estimates in survival analysis”, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61: 367-379, (1999). DOI: https://doi.org/10.1111/1467-9868.00182
5. [5] David, G.K., and Mitchel, K., Survival Analysis: A Self‐Learning Text, Spinger, (2012).
6. [6] Therneau, T.M., and Grambsch, P.M., Modeling Surivival Data: Extending the Cox Model, Springer -Verlag, (2000).
7. [7] Sauerbrei, W., Royston, P., and Look, M., “A new proposal for multivariable modelling of time‐varying effects in survival data based on fractional polynomial time‐transformation”, Biometrical Journal, 49: 453-473, (2007). DOI: https://doi.org/10.1002/bimj.200610328
8. [8] Persson, I., and Khamis, H.J., “A comparison of statistical tests for assessing the proportional hazards assumption in the Cox model”, Journal of Statistics and Applications, 3(1): 135, (2008).

9. [9] Cai, Z., and Sun, Y., “Local linear estimation for time‐dependent coefficients in Cox's regression models”, Scandinavian Journal of Statistics, 30: 93-111, (2003). DOI: https://doi.org/10.1111/1467-9469.00320
10. [10] Platt, R.W., Joseph, K., Ananth, C.V., Grondines, J., Abrahamowicz, M., and Kramer, M.S., “A proportional hazards model with time-dependent covariates and time-varying effects for analysis of fetal and infant death”, American Journal of Epidemiology, 160: 199-206, (2004). DOI: https://doi.org/10.1093/aje/kwh201
11. [11] Wang, W., “Proportional hazards regression models with unknown link function and time-dependent covariates”, Statistica Sinica, 885-905, (2004).
12. [12] Sparling, Y.H., Younes, N., Lachin, J.M., and Bautista, O.M., “Parametric survival models for interval-censored data with time-dependent covariates”, Biostatistics, 7: 599-614, (2006). DOI: https://doi.org/10.1093/biostatistics/kxj028
13. [13] Zhang, H., and Huang, C., “Nonparametric survival analysis on time-dependent covariate effects in case-cohort sampling design”, Statistica Sinica, 267-285, (2006).
14. [14] Kremers, W.K., “Concordance for survival time data: fixed and time-dependent covariates and possible ties in predictor and time”, Mayo Foundation, (2007).
15. [15] Heinze, G., and Dunkler, D., “Avoiding infinite estimates of time‐dependent effects in small‐sample survival studies”, Statistics in Medicine, 27: 6455-6469, (2008). DOI: https://doi.org/10.1002/sim.3418
16. [16] Bower, H., Crowther, M.J., Rutherford, M.J., Andersson, T.M.L., Clements, M., Liu, X.R., and Lambert, P.C., “Capturing simple and complex time-dependent effects using flexible parametric survival models: A simulation study”, Communications in Statistics-Simulation and Computation, 50(11): 3777-3793, (2019). DOI: https://doi.org/10.1080/03610918.2019.1634201
17. [17] Thernau, T., Crowson, C., and Atkinson, E., “Using time dependent covariates and time dependent coefficients in the Cox model”, Survival Vignettes, 2(3): 1-25, (2017).
18. [18] Husain, H., Thamrin, S.A., Tahir, S., Mukhlisin, A., and Apriani, M.M., “The application of extended Cox proportional hazard method for estimating survival time of breast cancer”, Journal of Physics: Conference series, Vol. 979, No. 1, p. 012087, IOP Publishing, (2018). DOI: 10.1088/1742-6596/979/1/012087
19. [19] Wu, D., and Li, C., “Joint analysis of multivariate interval-censored survival data and a time-dependent covariate”, Statistical Methods in Medical Research, 30: 769-784, (2021). DOI: https://doi.org/10.1177/0962280220975064
20. [20] Suresh, K., Taylor, J.M., and Tsodikov, A., “A copula‐based approach for dynamic prediction of survival with a binary time‐dependent covariate”, Statistics in Medicine, 40(23): 4931-4946, (2021). DOI: https://doi.org/10.1002/sim.9102
21. [21] Austin, P.C., Fang J., and Lee D.S., “Using fractional polynomials and restricted cubic splines to model non-proportional hazards or time-varying covariate effects in the Cox regression model”, Statistics in Medicine, 41(3):612-624, (2022). DOI: https://doi.org/10.1002/sim.9259
22. [22] Geraili, Z., Hajian-Tilaki, K., Bayani, M., Hosseini, S.R., Khafri, S., Ebrahimpour, S., and Shokri, M., “Evaluation of time-varying biomarkers in mortality outcome in COVID-19: An application of extended cox regression model”, Acta Informatica Medica, 30(4): 295-301, (2022). DOI: https://doi.org/10.5455/aim.2022.30.295-301
23. [23] Maharela, I.A., Fletcher, L., and Chen, D.G., “Modified Cox Models: A Simulation Study on Different Survival Distributions, Censoring Rates, and Sample Sizes”, Mathematics, 12(18):2903, (2024). DOI: https://doi.org/10.3390/math12182903
24. [24] Collett, D., Modelling survival data in medical research, Springer, pp.53-106, (1994). DOI: https://doi.org/10.1201/9781003282525
25. [25] Schemper, M., “Cox analysis of survival data with non‐proportional hazard functions”, Journal of the Royal Statistical Society: Series D (The Statistician), 41: 455-465, (1992). DOI: https://doi.org/10.2307/2349009
26. [26] Ata, N., and Demirhan, H., “Weighted estimation in Cox regression model: An application to breast cancer data”, Gazi University Journal of Science, 26: 63-72, (2013).
27. [27] Persson, I., and Khamis, H., “Bias of the Cox model hazard ratio”, Journal of Modern Applied Statistical Methods, 4(1): 10, (2005). DOI: https://doi.org/10.22237/jmasm/1114906200
28. [28] Andersen, P.K., Borgan, O., Gill, R.D., and Keiding, N., Statistical models based on counting processes, Springer Science & Business Media, (2012).
29. [29] Șafak, Ç., and Tutkun, N.A., “Analyzing the factors influencing the duration of breastfeeding: an example of Turkish Republic of Northern Cyprus”, İzmir Dr Behçet Uz Çocuk Hastanesi Dergisi, 5: 167-176, (2015). DOI: https://doi.org/10.5222/buchd.2015.167
30. [30] Austin, P.C., “Statistical power to detect violation of the proportional hazards assumption when using the Cox regression model”, Journal of Statistical Computation and Simulation, 88: 533-552, (2018). DOI: https://doi.org/10.1080/00949655.2017.1397151
31. [31] Abrahamowicz, M., and MacKenzie, T.A., “Joint estimation of time‐dependent and non‐linear effects of continuous covariates on survival”, Statistics in Medicine, 26: 392-408, (2007). DOI: https://doi.org/10.1002/sim.2519