A Partial Maximum Likelihood Method to Estimate Cox Model for Competing Risk Data with Application

Abstract

Many Research and clinical studies have addressed the occurrence of failure (death). As a result of other (external) factors, which may introduce additional risks that compete with the event under study. The resulting data refers to the complete data on Competing risks, which are affected by time differences. The exact times of occurrence of these risks are known, meaning the times of failure are observed for all observations with certainty. The impact of these risks on the hazard function is estimated based on the Cox proportional hazards model, which is estimated using the partial maximum likelihood method and numerical algorithms for parameter estimation. This includes the effect of variables on the Cox hazard function and the nonparametric part, estimated by assessing the effect of time on the hazard function using the Kaplan-Meier formula and calculating competing risks through the cumulative hazard function.

These methods were applied to experimental data through large-scale simulations of different sizes and parameters and for several arithmetic means and standard deviations models. Moreover, applied to real data from a sample of 80 individuals with breast cancer. Analyzing the simulation results and real data revealed that the Downhill algorithm outperforms the Newton-Raphson algorithm in terms of estimation accuracy and efficiency, based on the statistical criterion for comparison, the root mean square error. In addition, competitive risks explained the effect of common variables in increasing competitive risks beyond the hazard function of the Cox model of the Newton-Raphson estimator. While it is converging to the hazard function of the Cox model for the Downhill estimator .

Paper type: Research paper