Authors:
Selvam N.,Lakshmanan Babu,Shaik Fayaz Ahamed,Ponnuraja Chinnaiyan,DOI NO:
https://doi.org/10.26782/jmcms.2026.05.00009Keywords:
Survival analysis,Cox model,Frailty model,Breast cancer,Tumour grade,Heterogeneity,Abstract
Background: Breast cancer survival is influenced by multiple clinical and pathological factors, and appropriate modelling is required to obtain reliable prognostic estimates while accounting for unobserved heterogeneity. Methodology: A population-based retrospective survival analysis was conducted among women diagnosed with primary breast cancer. Survival time from diagnosis to death or event was analysed using proportional hazards (PH) and accelerated failure time (AFT) models across multiple parametric distributions. Shared gamma frailty models were fitted at the age-group level to account for unobserved heterogeneity. Results: Higher tumour grade and lymph node ratio (LNR) were the strongest predictors of poor survival. Compared with grade 1 tumours, grade 3 tumours were associated with substantially shorter survival times (time ratio ?0.55 – 0.59) and increased hazard (hazard ratio ?1.8 - 1.9). Patients with LNR > 0.68 experienced markedly earlier events (time ratio ? 0.33 – 0.38) and higher hazard (hazard ratio ? 3.1). Advanced age showed the largest adverse effect, with patients older than 78.5 years experiencing events approximately three to four times earlier (time ratio ? 0.26 – 0.29). Hormone receptor-negative tumours were associated with reduced survival (time ratio ? 0.71 – 0.86). Flexible AFT models, particularly the generalized gamma distribution, demonstrated superior fit. Frailty modelling revealed moderate unobserved heterogeneity (?? 0.30), with attenuation of effect sizes but preserved inference. Conclusion: Key prognostic factors for breast cancer survival remained robust across modelling frameworks and after accounting for unobserved heterogeneity. The combined use of PH, AFT, and frailty models provides clinically interpretable and reliable survival estimatesRefference:
I. Altman, Douglas G. Practical statistics for medical research. Chapman and Hall/CRC, 1990. 10.1201/9780429258589
II. Brandt, Jasmine, et al. "Age at diagnosis in relation to survival following breast cancer: a cohort study." World journal of surgical oncology 13.1 (2015): 33. 10.1186/s12957-014-0429-x
III. Chang, Yao-Jen, et al. "Recursive partitioning analysis of lymph node ratio in breast cancer patients." Medicine 94.1 (2015): e208. 10.1097/MD.0000000000000208
IV. Collett, David. Modelling survival data in medical research. Chapman and Hall/CRC, 2023. 10.1201/9781003282525
V. Cox, David R. "Regression models and life?tables." Journal of the royal statistical society: Series B (methodological) 34.2 (1972): 187-202. 10.1007/978-1-4612-4380-9_37
VI. Dou, He, et al. "Estrogen receptor-negative/progesterone receptor-positive breast cancer has distinct characteristics and pathologic complete response rate after neoadjuvant chemotherapy." Diagnostic Pathology 19.1 (2024): 5. 10.1186/s13000-023-01433-6
VII. Duchateau, Luc, and Paul Janssen. The frailty model. New York, NY: Springer New York, 2008. 10.1007/978-0-387-72835-3_4
VIII. Duchateau, Luc, Paul Janssen, and Steven Abrams. "Frailty Model, The." International Encyclopedia of Statistical Science. Berlin, Heidelberg: Springer Berlin Heidelberg, 2025. 982-992. 10.1007/978-3-662-69359-9_694
IX. Elston, Christopher W., Ian O. Ellis, and Sarah E. Pinder. "Pathological prognostic factors in breast cancer." Critical reviews in oncology/hematology 31.3 (1999): 209-223. 10.1016/S1040-8428(99)00034-7
X. Faradmal, Javad, et al. "Survival analysis of breast cancer patients using Cox and frailty models.” Journal of research in health sciences 12.2 (2012). https://pubmed.ncbi.nlm.nih.gov/23241526/
XI. Goldhirsch, Aron, et al. “Strategies for subtypes—dealing with the diversity of breast cancer: highlights of the St Gallen International Expert Consensus on the Primary Therapy of Early Breast Cancer 2011.” Annals of oncology 22.8 (2011): 1736-1747. 10.1093/annonc/mdr304
XII. Gorfine, Malka, and David M. Zucker. "Shared frailty methods for complex survival data: a review of recent advances." Annual Review of Statistics and Its Application 10.1 (2023): 51-73. https://doi.org/10.1146/annurev-statistics-032921-021310
XIII. Hurley, Margaret Anne. "A reference relative time-scale as an alternative to chronological age for cohorts with long follow-up." Emerging themes in epidemiology 12.1 (2015): 18. https://doi.org/10.1186/s12982-015-0043-6
XIV. Alotaibi, Refah Mohammed, and Chris Guure. “Bayesian and Frequentist Analytical Approaches Using Log-Normal and Gamma Frailty Parametric Models for Breast Cancer Mortality.” Computational and mathematical methods in medicine vol. 2020 9076567. 8 Feb. 2020, 10.1155/2020/9076567.
XV. Kleinbaum, D.G. and Klein, M. (2012) Survival Analysis: A Self-Learning Text. 3rd Edition, Springer, New York. https://doi.org/10.1007/978-1-4419-6646-9
XVI. Liu, Dechun, et al. "Lymph node ratio and breast cancer prognosis: a meta-analysis." Breast Cancer 21.1 (2014): 1-9. 10.1007/s12282-013-0497-8
XVII. Rakha, Emad A., et al. "Prognostic significance of Nottingham histologic grade in invasive breast carcinoma." Journal of clinical oncology 26.19 (2008): 3153-3158. 10.1200/JCO.2007.15.5986
XVIII. Rakha, Emad A., et al. "Breast cancer prognostic classification in the molecular era: the role of histological grade." Breast cancer research 12.4 (2010): 207. 10.1186/bcr2607
XIX. Putter H, Van Houwelingen HC. Frailties in multi-state models: Are they identifiable? Do we need them?. Statistical methods in medical research. 2015 Dec;24(6):675-92. 10.1177/0962280211424665
XX. Surveillance, Epidemiology, and End Results (SEER) Program (www.seer.cancer.gov) Research Data (1973-2013), National Cancer Institute, DCCPS, Surveillance Research Program, Surveillance Systems Branch, released April 2016, based on the November 2015 submission. https://seer.cancer.gov/
XXI. Yazdani, Akram et al. “Investigation of Prognostic Factors of Survival in Breast Cancer Using a Frailty Model: A Multicenter Study.” Breast cancer : basic and clinical research vol. 13 1178223419879112. 29 Sep. 2019, doi:10.1177/1178223419879112
XXII. Sung, Hyuna, et al. "Global cancer statistics 2020: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries." CA: a cancer journal for clinicians 71.3 (2021): 209-249. 10.3322/caac.21660
XXIII. Therneau, Terry M., and Patricia M. Grambsch. "The cox model." Modeling survival data: extending the Cox model. New York, NY: Springer New York, 2000. 39-77. 10.1007/978-1-4757-3294-8_3
XXIV. Vaupel, James W., Kenneth G. Manton, and Eric Stallard. "The impact of heterogeneity in individual frailty on the dynamics of mortality." Demography 16.3 (1979): 439-454. 10.2307/2061224
XXV. Wei, Lee-Jen. "The accelerated failure time model: a useful alternative to the Cox regression model in survival analysis." Statistics in medicine 11.14?15 (1992): 1871-1879. 10.1002/sim.4780111409
XXVI. Wienke, Andreas. Frailty models in survival analysis. Chapman and Hall/CRC, 2010. 10.1201/9781420073911
XXVII. Yazdani, Akram, et al. "Application of Frailty Quantile Regression Model to investigate of factors survival time in breast Cancer: a Multi-center Study." Health Services Research and Managerial Epidemiology 10 (2023): 23333928231161951. 10.1177/23333928231161951