Log-linearity for Cox's regression model

Abstract

Cox's regression model is one of the most applied methods in medical research. This method finds also applications in other fields such as econometrics, demography, insurance etc. This method is based on two crucial assumptions that (i) the method assumes log-linearity in covariates, and (ii) that the hazard ratio of two individuals are proportional. In survival analysis data, both numeric and binary covariates are typically encountered. There is no issue with the log-linearity assumption when working with binary covariates, however, the issue may arise when numeric covariates are involved. This thesis, thus, studies methods that are used to check assumption (i). For this purpose, there have been proposed a number of graphical procedures and formal test procedures in the literatures. This thesis in particularly aims to give a systematic review of the various test procedures and formal tests, and also to assess how the test procedures perform. All the proposed test procedures will be illustrated using publicly available data. To study the performance of these procedures, both real and simulated data (using the Monte Carlo method) will be used. In the simulation studies, first we must find a general formula for how to generate survival data on the computer. That is done through the fundamental relation between hazard rate and survival function. It is shown how the Weibull distribution function can be used to generate appropriate survival data on the computer.

Cox's regression model is one of the most applied methods in medical research. This method finds also applications in other fields such as econometrics, demography, insurance etc. This method is based on two crucial assumptions that (i) the method assumes log-linearity in covariates, and (ii) that the hazard ratio of two individuals are proportional. In survival analysis data, both numeric and binary covariates are typically encountered. There is no issue with the log-linearity assumption when working with binary covariates, however, the issue may arise when numeric covariates are involved. This thesis, thus, studies methods that are used to check assumption (i). For this purpose, there have been proposed a number of graphical procedures and formal test procedures in the literatures. This thesis in particularly aims to give a systematic review of the various test procedures and formal tests, and also to assess how the test procedures perform. All the proposed test procedures will be illustrated using publicly available data. To study the performance of these procedures, both real and simulated data (using the Monte Carlo method) will be used. In the simulation studies, first we must find a general formula for how to generate survival data on the computer. That is done through the fundamental relation between hazard rate and survival function. It is shown how the Weibull distribution function can be used to generate appropriate survival data on the computer.