Piecewise survival models

The Cox regression model is a widely used method in survival analysis, allowing researchers to effectively examine the duration between the start and end points. The dependent variable represents the duration between these two points, while the endpoint is generally defined as the occurrence of an event of interest, such as death, recurrence, or failure. However, this model is based on the proportional hazards assumption, which suggests that hazard ratios remain constant throughout the observation period. If this assumption is not satisfied, the use of the model becomes inappropriate, prompting the development of alternative survival models. In this study, piecewise survival models that can better reflect time-varying hazard ratios are discussed. Piecewise survival models are based on the assumption that survival time can be divided into specific intervals, with the hazard ratio remaining constant within each interval. In other words, the hazard ratio differs across different time periods, but it is assumed to be constant within any given time interval. This model can be used as an alternative to the Cox regression model, parametric survival models, and discrete survival models. Since there is no requirement to define the general form of the hazard ratio in advance, as in parametric models, it has a more flexible structure compared to parametric survival models. This model can be used to observe how the hazard ratio changes over time and to use this information in selecting an appropriate parametric model. The thesis analyzes exponential piecewise, Weibull piecewise and piecewise models with frailty terms in detail. In the application section of the study, the applicability of the models was demonstrated using lung cancer data from the literature. The dataset was analyzed by applying piecewise survival models, the Cox regression model, exponential and Weibull parametric survival models, and discrete survival models, with the results being interpreted accordingly. The results of the study show that, based on the data used, the discrete logit survival model is more suitable than the binomial log-log survival model. Additionally, the piecewise Weibull model has been selected as the most suitable model for reflecting time-varying hazard ratios. Among the parametric models, the Weibull distribution has shown a better fit to the dataset compared to the exponential distribution. In the piecewise models with Gamma and inverse-Gaussian frailty term, the effect of the frailty terms was not found to be statistically significant. These findings demonstrate that piecewise survival models, which consider the duration factor, provide an effective and flexible approach for survival analysis. However, they also show that frailty terms are not always significant. Keywords: Discrete survival models, exponential, frailty, lung cancer, parametric survival models, piecewise survival models, Weibull.