Estimations Of Additive-multiplicative Hazard Models With Survival Data

Survival time often acts as the duration time between the initial event and the terminal event.As no research can be followed up until all observations' terminal events occur,and other uncontrolled factors will inevitably affect the occurrence of the terminal events in the tracking,there usually exists right censoring in survival data.It is called left truncated and doubly censored data,while the initial event suffers interval censoring and the terminal event suffers left truncation and right censoring.In survival analysis,the interest often relies on the influence of associated hazard factors on survival time,including both form and extent of the influence.The hazard-based additivemultiplicative model contains additive and multiplicative relationship between covariates and hazard function.Its distinguished advantage is its ability to estimate the degrees of the influence coming from the covariates of different forms at the same time.Nevertheless,its estimation procedure is more complicated than the single multiplicative model or additive model.Meanwhile,according to whether the parameters of covariates in the model depends on time,several different forms of additive-multiplicative model can be chosen.In the Lin-Ying additive-multiplicative model,the parameters of covariates are all constant,but in Cox-Aalen additive-multiplicative hazard model,the parameters with the multiplicative effect are constant,while the parameters with the additive effect are functions of time.However,time-varying additive-multiplicative hazard model,both the parameters with the multiplicative and additive effect are functions of time.This paper mainly studies the estimation and practical analysis of the additivemultiplicative hazard model with left truncated and doubly censored data,as well as random right censored data.First,the estimation of the Lin-Ying additive-multiplicative hazard model with left truncated and doubly censored data is conducted.Firstly,a two-step estimation procedure is promoted to estimate the parameters of the model,which includes the initial estimators without interval censored data and the revised estimators with interval censored data.Secondly,simulation studies of limited samples are conducted to verify the estimators' accuracy and show the revised estimators perform better than the initial estimators.Finally,the proposed method is applied to the data relating to survival of patients after operation for malignant melanoma.Both the simulation and practical studies confirm that the two-step estimation with interval censored information is more effective than estimation without the interval censored observations.Second,the estimation of the Cox-Aalen additive-multiplicative hazard model with left truncated and doubly censored data is conducted.The two-step estimation procedure here is similar with the method in estimating the Lin-Ying additive-multiplicative hazard model.However,the initial estimator is obtained from estimating equation in Lin-Ying model,while in Cox-Aalen model is obtained from score equation derived by maximum likelihood function.Meanwhile,simulation studies of limited samples are conducted to verify the estimators' accuracy and show the revised estimators perform better than the initial estimators.The proposed method applied to the data relating to survival of patients after operation for malignant melanoma confirms the two-step estimation more reliable and effective.Third,the estimation of the time-varying additive-multiplicative hazard model with random right censored data is studied.Firstly,based on the local constant coefficient expansion,the local likelihood function is derived to estimate the coefficient.Secondly,large sample properties of the proposed estimator are established,including the consistency and asymptotic normality.Finally,the stability and effectivity of the estimators can be shown through a simulation of limited samples.