Application of Proportional Hazard and Additive Models  in the Survival Analysis of Breast Cancer Patients

Muhammad Bayu Nirwana; Tiara Fitri Adani; Kayla Argya Puruhita; Andreas Rony Wijaya; Hasih Pratiwi; Silvina Rosita Yulianti

doi:10.31764/jtam.v10i3.37028

Application of Proportional Hazard and Additive Models in the Survival Analysis of Breast Cancer Patients

Muhammad Bayu Nirwana, Tiara Fitri Adani, Kayla Argya Puruhita, Andreas Rony Wijaya, Hasih Pratiwi, Silvina Rosita Yulianti

Abstract

Breast cancer is the most common type of cancer among women and one of the highest causes of death among other types of cancer. This study aims to evaluate the methodological advantages of additive hazard models over the multiplicative Cox model in identifying temporal risk factors for breast cancer survival. Using secondary data from 1458 patients and 10 covariates, applying three methods, Cox proportional hazards model, Lin-Ying additive hazard model, and Aalen additive hazard model. The proportional hazard assumption test indicated that Cox regression model did not fully satisfy the assumption; therefore, the Lin–Ying and Aalen additive models were applied. In the Lin–Ying models, hormonal therapy, radiotherapy, the Nottingham Prognostic Index (NPI), and tumor size were identified as significant predictors of survival, whereas in the Aalen model, significant factors also included age and chemotherapy in addition to those four covariates. These findings highlight that while the Cox model provides efficient estimation and interpretable hazard ratios, the Lin–Ying and Aalen models offer more robust alternatives when the proportional hazard assumption is violated. The Aalen model was selected based on the results of the Aalen plot. Overall, risk control efforts in breast cancer patients should focus on managing NPI scores and tumor size as well as ensuring appropriate therapies, particularly hormonal therapy and radiotherapy, which have been demonstrated to provide protective effects.

Keywords

Breast Cancer; Survival Analysis; Cox Proportional Hazards Model; Additive Hazard Model.

Full Text:

DOWNLOAD [PDF]

References

Ayodele, V., Sherry, E., & Mazo-Canola, M. (2024). Abstract PO5-12-03: Cardiovascular Toxicities in Breast Cancer Survivors. Cancer Research, 84(9_Supplement), PO5-12-03-PO5-12–03. https://doi.org/10.1158/1538-7445.SABCS23-PO5-12-03

Başar, E. (2017). Aalen’s Additive, Cox Proportional Hazards and the Cox-Aalen Model: Application to Kidney Transplant Data. Sains Malaysiana, 46(3), 469–476. https://doi.org/10.17576/jsm-2017-4603-15

Brueckner, M., Titman, A., & Jaki, T. (2019). Instrumental Variable Estimation in Semi-Parametric Additive Hazards Models. Biometrics, 75(1), 110–120. https://doi.org/10.1111/biom.12952

Bruijn, I. De, Kundra, R., Mastrogiacomo, B., Tran, T. N., Sikina, L., Mazor, T., Li, X., Ochoa, A., Zhao, G., Lai, B., Abeshouse, A., Baiceanu, D., Ciftci, E., Dogrusoz, U., Du, A., Erkoc, Z., Lara, E. G., Fu, Z., Gross, B., … Schultz, N. (2023). Analysis and Visualization of Longitudinal Genomic and Clinical Data from the AACR Project GENIE Biopharma Collaborative in cBioPortal. Cancer Research, 83(23), 3861–3867. https://doi.org/10.1158/0008-5472.CAN-23-0816

BioPortal for Cancer Genomics. (2016). Breast Cancer (METABRIC, Nature 2012 & Nat Commun 2016).

Cho, H., Lee, S., Sim, S. H., Park, I. H., Lee, K. S., Kwak, M. H., & Kim, H. J. (2020). Cumulative incidence of chemotherapy-induced cardiotoxicity during a 2-year follow-up period in breast cancer patients. Breast Cancer Research and Treatment, 182(2), 333–343. https://doi.org/10.1007/s10549-020-05703-5

Deo, S. V. S., Sharma, J., & Kumar, S. (2022). GLOBOCAN 2020 Report on Global Cancer Burden: Challenges and Opportunities for Surgical Oncologists. Annals of Surgical Oncology, 29(11), 6497–6500. https://doi.org/10.1245/s10434-022-12151-6

Dizaji, P. A., Farahani, M. V., Sheikhaliyan, A., & Biglarian, A. (2020). Application of additive hazards models for analyzing survival of breast cancer patients. Journal of Research in Medical Sciences, 25(1), 99. https://doi.org/10.4103/jrms.JRMS

Dudley, W. N., Wickham, R., & Coombs, N. (2016). An Introduction to Survival Statistics: Kaplan-Meier Analysis. Journal of the Advanced Practitioner in Oncology, 7(1), 91–100. https://doi.org/10.6004/jadpro.2016.7.1.8

Dumas, E., & Stensrud, M. J. (2025). How Hazard Ratios Can Mislead and Why It Matters in Practice. European Journal of Epidemiology, 40(6), 603–609. https://doi.org/10.1007/s10654-025-01250-9

Elhafeez, S. A., Arrigo, D., Leonardis, D., Fusaro, M., Tripepi, G., & Roumeliotis, S. (2021). Methods to Analyze Time-to-Event Data: The Cox Regression Analysis. Oxidative Medicine and Cellular Longevity, 2021(1), 1–6. https://doi.org/10.1155/2021/1302811

Fathoni, M. I. A., Gunardi, Adi-Kusumo, F., Hutajulu, S. H., & Purwanto, I. (2022). Cox Proportional Hazard Regression Interaction Model and Its Application to Determine The Risk of Death in Breast Cancer Patients after Chemotherapy. International Journal of Statistics in Medical Research, 11(13), 105–113. https://doi.org/10.6000/1929-6029.2022.11.13

Hazra, A., & Gogtay, N. (2017). Biostatistics series module 9: survival analysis. Indian journal of dermatology, 62(3), 251-257. https://doi.org/10.4103/ijd.IJD

In, J., & Lee, D. K. (2018). Survival analysis: Part I — analysis of time-to-event. Korean Journal of Anesthesiology, 71(3), 182–191. https://doi.org/10.4097/kja.d.18.00067

In, J., & Lee, D. K. (2019). Survival Analysis: Part II – Applied Clinical Data Analysis. Korean Journal of Anesthesiology, 72(5), 441–457. https://doi.org/10.4097/kja.19183

Jeong, S. H., Chun, S. M., Lee, H., Kim, M., Choi, M., & Leigh, J.-H. (2025). Association between chemotherapy and the risk of developing breast cancer-related lymphedema: a nationwide retrospective cohort study. Supportive Care in Cancer, 33(2), 143. https://doi.org/10.1007/s00520-025-09169-3

Jin, L., Han, B., Siegel, E., Cui, Y., Giuliano, A., & Cui, X. (2018). Breast cancer lung metastasis: Molecular biology and therapeutic implications. Cancer Biology & Therapy, 19(10), 858–868. https://doi.org/10.1080/15384047.2018.1456599

Kristina, S. A., Krisnadewi, K. I., Trung, V. Q., Ramadhani, A., Verdiana, A. S., & Aryasatya, M. D. (2025). Cancer Burden Across The South East Asia Nation (ASEAN) in 2022. Asian Pacific Journal of Cancer Prevention, 26(9), 3377–3388. https://doi.org/10.31557/APJCP.2025.26.9.3377

Kuitunen, I., Ponkilainen, V. T., Uimonen, M. M., Eskelinen, A., & Reito, A. (2021). Testing the Proportional Hazards Assumption in Cox Regression and Dealing with Possible Non-Proportionality in Total Joint Arthroplasty Research: Methodological Perspectives and Review. BMC Musculoskeletal Disorders, 22(1), 1–7. https://doi.org/10.1186/s12891-021-04379-2

Lefebvre, F., & Giorgi, R. (2021). A Strategy for Optimal Fitting of Multiplicative and Additive Hazards Regression Models. BMC Medical Research Methodology, 21(1), 1–17. https://doi.org/10.1186/s12874-021-01273-2

Liu, Y., He, M., Zuo, W., Hao, S., Wang, Z., & Shao, Z.-M. (2021). Tumor Size Still Impacts Prognosis in Breast Cancer With Extensive Nodal Involvement. Front Oncol, 11(21), 236–242. https://doi.org/10.3389/fonc.2021.585613

Madadizadeh, F., Ghanbarnejad, A., Ghavami, V., Mohammad, Bandamiri, Z., & Mohammadianpanah, M. (2017). Applying Additive Hazards Models for Analyzing Survival in Patients with Colorectal Cancer in Fars Province, Southern Iran. Asian Pacific Journal of Cancer Prevention, 18(4), 1077–1083. https://doi.org/10.22034/APJCP.2017.18.4.1077

Obeagu, E. I., & Obeagu, G. U. (2024). Breast Cancer: A Review of Risk Factors and Diagnosis. Medicine, 103(3), 1–6. https://doi.org/http://dx.doi.org/10.1097/MD.0000000000036905

Oller, R., & Gómez Melis, G. (2023). Survival Analysis with Censored Data: A Further Twist on Ignorability Conditions. Statistics, 57(6), 1529–1550. https://doi.org/10.1080/02331888.2023.2283091

Pereira, B., Chin, S., Rueda, O. M., Vollan, H. M., Provenzano, E., Bardwell, H. A., Pugh, M., Jones, L., Russell, R., Sammut, S., Tsui, D. W. Y., Liu, B., Dawson, S., Abraham, J., Northen, H., Peden, J. F., Mukherjee, A., Rosenfeld, N., Murphy, L., … Caldas, C. (2016). The somatic mutation profiles of 2,433 breast cancers refines their genomic and transcriptomic landscapes. Nature Communications, 7(1), 11479. https://doi.org/10.1038/ncomms11479

Perera, M., & Tsokos, C. (2018). A Statistical Model with Non-Linear Effects and Non-Proportional Hazards for Breast Cancer Survival Analysis. Advances in Breast Cancer Research, 7(1), 65–89. https://doi.org/10.4236/abcr.2018.71005

Rueda, O. M., Sammut, S., Seoane, J. A., Chin, S., Caswell-jin, J. L., Callari, M., Batra, R., Pereira, B., Ali, H. R., Provenzano, E., Liu, B., Parisien, M., Gillett, C., Mckinney, S., Green, A. R., Murphy, L., Purushotham, A., Ian, O., Pharoah, P. D., … Aparicio, S. A. J. R. (2019). Dynamics of breast-cancer relapse reveal late-recurring ER-positive genomic subgroups. Nature, 567(7748), 399–404. https://doi.org/10.1038/s41586-019-1007-8.Dynamics

Schober, P., & Vetter, T. R. (2018). Survival analysis and interpretation of time-to-event data: the tortoise and the hare. Anesthesia & Analgesia, 127(3), 792-798. https://doi.org/10.1213/ANE.0000000000003653

Siegel, R. L., Kratzer, T. B., Giaquinto, A. N., Sung, H., & Jemal, A. (2025). Cancer Statistics, 2025. A Cancer Journal for Clinicians, 75(1), 10–45. https://doi.org/10.3322/caac.21871

Stevens, N. A., Lydon, M., Marshall, A. H., & Taylor, S. (2020). Identification of Bridge Key Performance Indicators Using Survival Analysis for Future Network-Wide Structural Health Monitoring. Sensors, 20(23), 1–15. https://doi.org/10.3390/s20236894

Sung, H., Ferlay, J., Siegel, R. L., Laversanne, M., Soerjomataram, I., Jemal, A., & Bray, F. (2021). GLOBOCAN Estimates of Incidence and Mortality Worldwide for 36 Cancers in 185 Countries. Global Cancer Statistics 2020, 71(3), 209–249. https://doi.org/10.3322/caac.21660

Tamirisa, N., Lin, H., Shen, Y., Shaitelman, S. F., Sri Karuturi, M., Giordano, S. H., Babiera, G., & Bedrosian, I. (2020). Association of Chemotherapy With Survival in Elderly Patients With Multiple Comorbidities and Estrogen Receptor–Positive, Node-Positive Breast Cancer. JAMA Oncology, 6(10), 1548. https://doi.org/10.1001/jamaoncol.2020.2388

Tuelah, G. I. R., Lubis, H. S., Feriyawati, L., & Harahap, F. Y. (2025). Relationship of Histopathological Grading and Age of HER2-type Breast Cancer Patients at Adam Malik Hospital in January 2022 – June 2024. SCRIPTA SCORE Scientific Medical Journal, 7(1), 50–56. https://doi.org/https://doi.org/10.32734/scripta.v7i1.20761

Turkson, A. J., Ayiah-mensah, F., & Nimoh, V. (2021). Handling Censoring and Censored Data in Survival Analysis : A Standalone Systematic Literature Review. 2021(1), 1–16. https://doi.org/10.1155/2021/9307475

Zafar, A., Khatoon, S., Khan, M. J., Abu, J., & Naeem, A. (2025). Advancements and Limitations in Traditional Anti-Cancer Therapies: A Comprehensive Review of Surgery, Chemotherapy, Radiation Therapy, and Hormonal Therapy. Discover Oncology, 16(1), 607. https://doi.org/10.1007/s12672-025-02198-8

Zhang, M., Deng, H., Hu, R., Chen, F., Dong, S., Zhang, S., Guo, W., Yang, W., & Chen, W. (2025). Patterns and Prognostic Implications of Distant Metastasis in Breast Cancer Based on SEER Population Data. Scientific Reports, 15(1), 26717. https://doi.org/https://doi.org/10.1038/s41598-025-12883-x

Zhang, Z. (2016). Semi-Parametric Regression Model for Survival Data: Graphical Visualization with R. Annals of Translational Medicine, 4(23), 1–8. https://doi.org/10.21037/atm.2016.08.61

Zhu, S., Li, Y., Chen, W., Fei, X., Shen, K., & Chen, X. (2021). Molecular Subtype May Be More Associated With Prognosis and Chemotherapy Benefit Than Tumor Size in T1N0 Breast Cancer Patients: An Analysis of 2,168 Patients for Possible De-Escalation Treatment. Frontiers in Oncology, 11(7), 71–79. https://doi.org/10.3389/fonc.2021.636266

DOI: https://doi.org/10.31764/jtam.v10i3.37028