Federal Survival Analysis Based On Ordinary Differential Equations

In the field of clinical research,individual hospitals usually have limited data,leading to unreliable statistical inference and even the inability to conduct statistical analysis.In addition,due to ethical and legal reasons,individual data cannot be merged for analysis across different hospitals.Therefore,federated learning methods have received increasing attention in recent years because they can utilize all data information without requiring access to individual data.The success of federated learning methods depends on the separability of the objective function with respect to samples/variables.However,in survival analysis,the commonly used partial likelihood method for effective parameter estimation of the Cox proportional hazards model is not separable with respect to samples,rendering existing federated learning methods inapplicable.In this paper,we propose a new federated learning method based on the maximum likelihood method using ordinary differential equations as a tool,which avoids the inseparability of the partial likelihood function.Convential maximum likelihood methods are limited to specific parameter models such as the accelerated failure time model and are not applicable to semi-parametric models like the Cox proportional hazards model.Therefore,this paper utilizes ordinary differential equations and nonparametric spline tools to flexibly estimate the baseline hazard function.Theoretical results including consistency and asymptotic normality of the proposed estimates are established in this paper.Furthermore,the proposed method’s estimation results are shown to be equivalent with probability 1 to the estimation results obtained by merging all individual data and those obtained by the partial likelihood method.Furthermore,it is given theoretically that the numerical approximation error is negligible for the numerical solution of the ordinary differential equation of order r when its step’s convergence is O(n-1/r-c).The effectiveness of the proposed method was verified through extensive numerical simulations and a real data analysis.This method enables joint analysis of data while preserving personal privacy,thereby enhancing the accuracy of data analysis.As such,it provides a novel approach and methodology for data analysis and application in survival analysis.