Time-Dependent Hazard Modelling of Breast Cancer Survival: An Extended Cox Framework with Proportional Hazards Assessment

Abstract Objectives: To investigate the proportional hazards' assumption in breast cancer survival analysis. Also to build a more flexible version of the Cox regression model that brings in time-dependent covariates, aiming for sharper hazard estimates. Method: A retrospective analysis was conducted using a publicly available dataset comprising 272 breast cancer patients with long-term follow-up. Descriptive statistics, Kaplan-Meier estimation, and multivariable Cox proportional hazards modeling were applied. The proportional hazards assumption was examined through time-dependent covariate interactions. An extended Cox model was subsequently formulated, and model performance was assessed using -2 log-likelihood, Akaike Information Criterion, and likelihood ratio statistics. Findings: Out of all patients, 77 (28.3%) events were observed during the study period. Tumor grade was identified as a significant predictor of survival (HR = 2.382; 95% CI 1.653-3.433). The proportional hazards assumption was violated for age (p<0.001) and tumor diameter (p=0.001), indicating the presence of time-dependent effects. The extended Cox model demonstrated a substantial improvement in fit, with -2 log-likelihood decreasing from 760.128 to 326.791 and AIC from 772.128 to 344.791, confirming enhanced explanatory performance. The findings confirm that incorporating time-dependent covariates improves model adequacy and provides a more reliable and realistic framework for breast cancer survival analysis, particularly in the presence of non-proportional hazards. Novelty: This study presents an integrated modeling approach that combines proportional hazards assessment with a time-dependent Cox regression model. By addressing violations of model assumptions while preserving interpretability, the proposed framework provides a practical and methodologically robust alternative to conventional survival models. Keywords: Breast cancer, Survival analysis, Cox proportional hazards, Time-dependent covariates, Non-proportional hazards, Hazard modeling