Survival analysis: an algorithmic approach to lifetime prediction

Clinical trials often collect and assess data of survival (or time-to-event) with the objective of comparing different treatments or identifying risk factors that are linked to individuals risk rate of experiencing an event, that can be death, tumor progression or any other meaningful clinical outcome. When dealing with these types of survival data, the most popular method is the Cox proportional hazards regression model, used to explore the relationship between survival experience and characteristics of patients. The standard outcome of the Cox model is the hazard ratio, a relative measure that informs on the rank of patients' risk among others, but does not meaningfully inform on individual patients. However, especially in the context of personalized medicine, it is of interest to identify an accurate model for lifetime prediction on an individual level. The aim of this work is to inspect if it is possible to obtain an estimate of survival time for a new patient, starting from a validated Cox model and known regression coefficients. First of all two functions of the "survival" package in the software R are examined, namely "predict.coxph" and "survfit.coxph", in order to identify which methods can be used for the purpose of individual survival time prediction. The first function turns out to provide risk predictions, that are relative measures not adequate to describe individual survival times; the second one provides instead individualized survival curves, that are absolute measures and from which two different values can be extrapolated: median survival time and restricted mean survival time. These can be used for prediction but rely on an estimate of the survival function and are thus classified as parametric approaches. Since the Cox proportional hazards model is classified as a semi-parametric model, not requiring the hazard and the survival function to be specified, the aim would be to preserve this feature and find a non-parametric approach for the estimation of survival times. In particular, it is questioned whether it is possible to invert the equation characterizing the Cox model and solve it for the survival time of a single individual, once the estimated coefficients are known. At this scope, a kind of inversion of the model is proposed, where the partial likelihood used in the estimation of the regression coefficients is exploited. It is shown that, if the regression coefficients are known, the model can be inverted somehow and a range of survival times can be obtained for a new patient; practically, there are some issues when the real values of the regression coefficients are not known and the main limitations are underlined. Finally, the results obtained with the parametric approaches of median and restricted mean survival time are analyzed over two example data sets and compared with the novel algorithmic approach in terms of predictive performance.