Üstel ve Weibull Parçalı Yaşam Modelleri: COVID-19 Dönemi Akciğer Kanseri Veri Kümesi Üzerine Bir Uygulama

Year 2025, Volume: 12 Issue: 2, 466 - 487, 30.11.2025

Emine Büşra Milayım , Nihal Ata Tutkun

https://doi.org/10.35193/bseufbd.1685519

Abstract

Cox regresyon modeli, olay gerçekleşene kadar geçen süreyi analiz eder ve orantılı tehlikeler varsayımına dayanır. Bu varsayım sağlanmazsa alternatif modeller tercih edilmelidir. Bu çalışmada, zamanla değişen tehlike oranlarını daha iyi yansıtabilecek parçalı yaşam modelleri ele alınmıştır. Parçalı yaşam modelleri, yaşam süresini belirli aralıklara böler ve her aralıkta tehlike oranının sabit olduğu varsayımına dayanır. Bu model, Cox regresyon modeli, parametrik yaşam modelleri ve kesikli yaşam modellerine alternatif olarak kullanılabilir. Parametrik modellere göre daha esnek bir yapıya sahip olan bu model, tehlike oranının zamanla nasıl değiştiğini gözlemlemeye imkân tanır. Çalışmada, üstel, Weibull ve zayıflık terimli parçalı yaşam modelleri incelenmiştir. Akciğer kanseri verisi üzerinde yapılan uygulamada, parçalı Weibull yaşam modelinin, zamanla değişen tehlike oranlarını yansıtma konusunda en uygun model olduğu belirlenmiştir.

Keywords

Parametrik , Parçalı Üstel Yaşam Modeli , Parçalı Weibull Yaşam Modeli , Zayıflık Terimli Yaşam Modeli

Ethical Statement

Bu çalışmanın hazırlanma sürecinde bilimsel ve etik ilkelere uyulduğu ve yararlanılan tüm çalışmaların kaynakçada belirtildiği beyan olunur.

Supporting Institution

Bu araştırmayı desteklemek için dış fon kullanılmamıştır.

Piecewise Survival Models: An Application on the COVID-19 Lung Cancer Dataset

Abstract

The Cox regression model analyzes the time until an event occurs and is based on the proportional hazards assumption. If this assumption is not met, alternative models should be preferred. In this study, we consider piecewise survival models, which can better reflect time-varying hazard rates. Piecewise survival models divide the survival time into specific intervals and assume that the hazard ratio is constant within each interval. This model can be used as an alternative to the Cox regression model, parametric survival models, and discrete survival models. It is more flexible than parametric models and allows for observing how the hazard ratio changes over time. In this study, piecewise survival models with exponential, Weibull, and frailty terms were analyzed. In the application on lung cancer data, the piecewise Weibull survival model was found to be the most appropriate model for reflecting time-varying hazard ratios.

Keywords

Parametric , Piecewise Exponential Survival Models , Parametric , Piecewise Weibull Survival Models , Frailty Survivals Models

Ethical Statement

It is declared that scientific and ethical principles were followed during the preparation of this study and that all studies used are stated in the bibliography.

Supporting Institution

The author(s) acknowledge that they received no external funding to support this research.

References

• Kaplan, E., & Meier, P. (1958). Nonparametric Estimation From Incomplete Observations. Journal of the American Statistical Association, 53(282), 457-481.
• Breslow, N. (1974). Covariance Analysis of Censored Survival Data. Biometrics, 30, 89-99.
• Holford, T. (1976). Life tables with concomitant information. Biometrics, 32, 587-597.
• Allison, P. D. (2010). Survival Analysis Using SAS®: A Practical Guide (2nd ed.). SAS Institute Inc., Cary, NC, 292.
• Holford, T. (1980). The Analysis of Rates and of Survivorship Using Log-linear Models. Biometrics, 36(2), 299-305.
• Laird, N., & Olivier, D. (1981). Covariance Analysis of Censored Survival Data Using Log-linear Analysis Technique. Journal of the American Statistical Association, 76(374), 231-240.
• Friedman, M. (1982). Piecewise Exponential Models for Survival Data with Covariates. The Annals of Statistics, 10(1), 101-113.
• Kubo, J., Cullen, M., Cantley, L., Slade, M., & Tessi, B. (2013). Piecewise Exponential Models to Assess the Influence of Job-specific Experience on the Hazard of Acute Injury for Hourly Factory Workers. BMC Medical Research Methodology, 13(89), 2-10.
• Jenko, J., Ducrocq, V., & Kovac, M. (2013). Comparison of Piecewise Weibull Baseline Survival Models for Estimation of True and Functional Longevity in Brown Cattle Raised in Small Herds. Journal of Animal Science, 91(10), 1583-1591.
• Li, Y., Panagiotou, O., Black, A., Liao, D., & Wacho, S. (2015). Multivariate Piecewise Exponential Survival Modeling. Biometrics, 72(2), 546-553.
• Saroj, R. (2020). Piecewise Hazard Model for Under-five Child Mortality. BLDE University Journal of Health Sciences, 5(1), 40-44.
• Olayinka, A., Abiodun, A., & Ishaq, A. (2020). The Use of Cox and Piecewise Exponential Models in the Determination of Renal Failure. Benin Journal of Statistics, 3, 91-100.
• Woodhouse, B., Laux, W., Trevarton, A., Lasham A., Knowlton N. (2025). Application of landmark analysis and piecewise Cox regression to identify features associated with prognosis:a national retrospective cohort study of New Zealand women. New Zealand Medical Journal, 133(1520), 45–55.
• Heiling, H. M., Rashid, N. U., Li, Q., Peng, X. L., Yeh, J. J., Ibrahim, J. G. (2025). Efficient Computation of High-Dimensional Penalized Piecewise Constant Hazard Random Effects Models. Statistics in Medicine, 44(6), 15.
• Zhao, W., Zhang, X., & Liu, H. (2024). Association of α-klotho concentrations with cardiovascular and all-cause mortality in American adults with depression: a national prospective cohort study, Translational Psychiatry. 14(505).
• Yitagesu E, Alemnew E. (2022). Mortality rate of Boer, Central Highland goat and their crosses in Ethiopia: Nonparametric survival analysis and piecewise exponential model. Vet Med Sci. 8(5), 2183-2193.
• Yitagesu E, Alemnew E. (2022). Mortality rate of Boer, Central Highland goat and their crosses in Ethiopia: Nonparametric survival analysis and piecewise exponential model. Vet Med Sci. 8(5), 2183-2193.
• Bagust A, Beale S. (2014). Survival analysis and extrapolation modeling of time-to-event clinical trial data for economic evaluation: an alternative approach. Med Decis Making. 34(3), 343-351.
• Cooney, P., White, A. (2023). Extending beyond Bagust and Beale: fully parametric piecewise exponential models for extrapolation of survival outcomes in health technology assessment, Value in Health. 26(10), 1510-1517.
• Freeman S.C., Sutton A.J., Cooper N.J., Gasparini A,. Crowther M.J., Hawkins N. (2024). Bayesian pairwise meta-analysis of time-to-event outcomes in the presence of non-proportional hazards: A simulation study of flexible parametric, piecewise exponential and fractional polynomial models. Res Synth Methods. 15(5):780-801.
• Frumento, P. (2024). Piecewise Constant Hazard Models for Censored and Truncated Data (Version 2.1). R Package.
• Mills, M. (2011). Introducing Survival and Event History Analysis. International Sociology Review of Books, 27, 189-198.
• Kleinbaum, D. G., & Klein, M. (2005). Survival Analysis: A Self-learning Approach (2nd ed.). Springer, New York, XV
• Anonim. (2023). Cox Regression. NCSS Statistical Software. https://www.ncss.com. (24.11.2024).
• Cox, D. (1972). Regression Models and Life-tables. Journal of the Royal Statistical Society: Series B (Methodological), 34, 187-220.
• Anonim. (2023a). Cox Regression. NCSS Statistical Software.
• Cox, D., & Oakes, D. (1972). Analysis of Survival Data. Chapman and Hall, London, 201.
• Hougaard, P. (1995). Frailty Models for Survival Data. Lifetime Data Analysis, 1(3), 255-273.
• Ibrahim, J. G., Chen, M. H., & Sinha, D. (2001). Bayesian Survival Analysis. Springer, New York, 300.
• Thamrin, S. A., & Lawi, A. (2014). Estimating Piecewise Exponential Frailty Model With Changing Prior For Baseline. The 7th SEAMS-UGM Conference, 2-5 Eylül, Yogyakarta, 5-12.
• Cox, D. (1972). Regression Models and Life-table. Journal of the Royal Statistical Society: Series B (Methodological), 34, 187-220.
• Clayton, D. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65, 141-151.
• Hougaard, P. (2002). Analysis of Multivariate Survival Data. Springer Science & Business Media, New York, 354.
• Therneau, T. M., & Grambsch, P. M. (2013). Modeling Survival Data: Extending the Cox Model. Springer Science & Business Media, New York, 350.
• Duchateau, L., Janssen, P., Lindsey, P., & Legrand, C. (2002). The shared frailty model and the power for heterogeneity tests in multicenter trials. Computational Statistics & Data Analysis, 40, 603-620.
• Duchateau, L., Janssen, P., Kezic, I., & Fortpie, C. (2003). Evolution of recurrent asthma event rate over time in frailty models. Journal of the Royal Statistical Society: Series C (Applied Statistics), 52, 355-363.
• Duchateau, L., & Janssen, P. (2004). Penalized partial likelihood for frailties and smoothing splines in time to first insemination models for dairy cows. Biometrics, 60, 608-614.
• Jenkins, S. P. (2008). “Survival Analysis.” Unpublished manuscript. Institute for Social and Economic Research, University of Essex, Colchester.
• Meritt, D. (2010). Cox Proportional Hazards Model. Florida State University, Tallahassee.
• Lawless, J. (2003). Statistical Models and Methods for Lifetime Data (Vol. 362). John Wiley & Sons, New York, 296-306.
• Casellas, J. (2007). Bayesian inference in a piecewise Weibull proportional hazards model with unknown change points. Journal of Animal Breeding and Genetics, 124(4), 176–184.
• Eleftheriou, D., & Karlis, D. (2021). Piecewise Survival Models: A Change-Point Analysis on Herpes Zoster Associated Pain Data Revisited and Extended.
• West, H., & Jin, J. (2015). Performance Status in Patients with Cancer. JAMA Oncology, 1(7), 998.