Variable Selection In Additive Hazard Model For Right Censored Data

In addition to Cox proportional hazard model,the additive hazard model is another important model which assumes the hazard function is the sum of base-line risk function and a regression function of covariates.Variable selection is a significant foundation of statistics rmodeling.In survival analysis,variable selection usually can be achieved by two major types of methods,best subset selection and regularization.Under the assumption of a.dditive hazard model,we proposed a vari-able selection method for right censored survival data with multiple covaria.tes by borrowing strength from the above mentioned methods.BIC is more cautious about whether variables can enter the model which makes it performance better in consistency of variable selection.For this reason,we pro-posed a method by a.pproximating BIC to perform variable selection and coefficient estimation simultaneously.In generally,variable selection,where BIC is employed,can be achieved by solving a penalized likelihood problem.Under the assumption of additive hazard model,we replace the likelihood function with a least squares loss function in the BIC,and define the corresponding BIC in the context of our research.We try to use the defined criterion to impleme.nt variable selection and coefficient estimation.Because of discreteness of the L₀ norm,it is hard to achieve variable selection by solving the objective function directly.The best subset method firstly fits all candidate models with the least square regression or maximum likelihood estimation,and then achieves variable selection by solving the objective function in BIC,which is a NP-hard problem.When p is considerable large,the method is infeasible.In order to achieve sparse estimation directly by solving the objective function of BIC,regularization methods make a convex relaxation of the L₀ norm,which makes the objective function continuous and convex.Then it transforms the problem into a continuous and convex optimization problem.Inspired by the idea of regularization,we use a continuous or smooth unit function to approximate the L₀ norm,and then make the objective function to be a continuous and smoothing func-tion in BIC.Although the objective function can be solved by the above mentioned approximation,it still could not achieve sparse estimation.We use a reparame-terization procedure to the regression coefficient which helps us to achieve variable selection.This proposed method borrows strength from best subset selection and regularization,both of which make it free of tuning parameter and computation efficient.About the tuning parameter,we fix it to be ln?n₀?beca.use of the proposed methods is a a.pproximation of BIC,where n₀ is the number of uncensored indi-viduals.The objective function here is continuous and non-convex which leads to a local optimal solution.At the same time,the oracle property of our method is established.Simulation study and real data analysis are provided to illustrate and assess the proposed procedure.