Impact of covariate omission on the estimation of regression coefficient parameters in the Cox PH model, the accelerated failure time model and the frailty model

This project is embedded in the study and the understanding of survival analysis. The history of survival analysis dates back to 1662 when survival rates were presented by John Graunt. Nowadays, survival analysis has become a major type of analysis performed in biology, medicine but also agriculture, engineering or sociology. When a survival analysis is performed, the aim is to study the occurrence of an event as well as the time at which it will occur (if it does). In sociology, for instance, one could be interested in the time to first employment after graduation. In medicine, survival analysis is often used to study the impact of a treatment on the time before relapse or death of a patient. Nonetheless, the outcome is not necessarily negative and can be the time before full recovery. In this document, we will majorly focus on survival analyses with application in the medical research field. We will study the estimation of regression coefficients in survival regression models.\\ In survival analysis, specific regression models where developed such as the Cox proportional hazard model or the accelerated failure time model. In these models, the omission of a covariate is known to have an impact on the estimation of the other regression coefficients. However, the estimation bias varies in the different types of model and the choice of the most adequate model to moderate the impact of covariate omission on estimation of a coefficient of particular interest (say $\beta_1$) is still unclear (see for instance, Keiding et al.(2003)). In this work, we aim to compare the impact of the omission of one covariate in three different survival regression models : the Cox proportional hazard model, the accelerated failure time model and the frailty model.\\ To address this issue, a literature review was done and core results on the three types of models were gathered. Based on these results, simulation scenarios were developed, data were simulated and analyzed for the three models. For each of them, two types of regression models were fit to the data: a full regression model which contains two variables and a restricted regression model which contains only one variable. Then the estimations of the regression coefficient $\beta_1$ were compared between the full and the restricted model. The simulation of the data was implemented in a way to reproduce a randomized control trial framework and to address the impact of the omission of covariate with a binary treatment variable (representing for example the treatment arm) and an additional normally distributed covariate. The results showed that the amount of censure impacts positively the amount of bias observed in case of omission of a covariate. In addition, the results shed light on the relative importance of the effect size of the omitted covariate over the effect size of the included covariate. Finally and more importantly, the results revealed that the frailty model was the survival model which offered the most adequate estimations when one covariate is omitted.