基于Cox回归模型的约束估计问题的研究

The Constraint Estimation Based on Cox Regression Model

摘要: 具有约束条件的右删失Cox回归模型的研究经常是基于偏似然函数，而该似然函数忽略了未知的基准风险率函数。文章致力于解决两个问题：基于完全似然函数得到回归参数和累积风险函数的极大似然估计；把有限制条件的优化问题转化为无限制条件的优化问题。ADMM算法能把限制条件引入到目标函数中，从而把有条件的优化问题转化为无条件的优化问题，MM算法在解决优化问题方面具有可分离参数的优点。因此，文章首先利用ADMM算法把有限制条件的优化问题转化为无限制条件的优化问题，然后将MM算法应用于极大化新的目标函数，实现了参数和非参数的分离，进而解决了回归参数和累积风险函数的非参数估计问题。同时，利用不等式放缩把高维优化问题转化为一维优化问题，避开了矩阵求逆的困难。

Abstract: The research on the right-censored Cox regression model with constraints is often based on the partial likelihood function, whereas the likelihood function ignores the unknown baseline hazard rate function. We devote to solving two problems. One is to obtain the maximum likelihood estimations of regression parameters and cumulative hazard function based on the complete likelihood function. Another is to transform the optimization problem with restrictions into the unrestricted optimization problem. The ADMM algorithm can introduce constraints into the objective function so as to transform the conditional optimization problem into an unconditional optimization problem. The MM algorithm has the advantage of separating parameters in solving optimization problems. Therefore, we first use the ADMM algorithm to transform the optimization problem with restrictions into an unconditional optimization problem. Then we apply the MM algorithm to maximize the new objective function in order to separate the parameters and the non-parameters, which is helpful to solve the problems of estimating the regression parameters and nonparametric cumulative hazard function. Meanwhile, the use of inequality which can transform the high-dimensional function into a sum of one-dimensional functions avoids the difficulty of matrix inversion.