Multiplicative-additive Hazard Model And Its Application In Survival Analysis

Study Background In regression analysis, nonparametric regression expands the scope of application of parameter regression and enhances the adaptability of the model because of its applicability and the relaxed model assumptions. However, nonparametric regression has its limitation. When there are many variables in the model to explain but not a large sample size, the fitting results of non-parametric regression were not good, easily lead to a sharp increase of variance. As the increase of dimensions the problem of the variance rapidly expanding is often referred to as "curse of dimensionality". The non-parametric regression is mostly based on kernel estimation and smoothing spline, and its explanation is also a problem. In order to solve these problems, in the 1980's AALEN first proposed additive models, this model estimates an additive approximation to the multi-variable regression equation. Additive approximation has two advantages:(1) As the additive section of an individual is estimated based on a single variable smoothing, therefore "curse of dimensionality" can be avoided; (2) the estimations of individual items explain that how the independent variables change with the changes of dependent variables. This is a very flexible model, and now there are very clear estimates. This model can easily be implemented, but in practice it could easily be overlooked, probably because this model is a non-parameter model, and some methods for statistical inference are not perfect.The intensity formula of AALEN model is:A(t)= Y(t)XT (t)β(t)Additive AALEN model is non-parameter model. Although it is very flexible, this model has difficultly in finding information of the data because the regression coefficient of some variables may be constant. In some cases, the effect of covariates estimated by either multiplicative or additive is not good, then need a combination of both. In 2003, Scheike & Zhang replacedα(t, x) of smooth COX regression model by XT (t)a, this model is known as COX-AALEN model. This model increases an additive structure on the basis of multiplication model, with less elastic damage making the explanation of estimates and baseline levels of covariate effects more easily, and making a reasonable compromise between bias and variance, and a more practical advantage is that classification covariates can be processed at its baseline level. COX-AALEN model is a typical multiplicative-additive hazard model, which is more flexible and useful based on the additive model.It greatly improves the utilization of data and information, reduces the impact of confounding factors. This model can be said to be the product of additive models combined with the COX regression model. Considering the role and time dependent effects of covariates, it is more flexible and more adequate than the COX regression model. It is unlike the generalized additive model which is only applicable to count data, this model is applicable to any form of information, and it does not require stage and meet MARKOV assumption.The estimates, forecasts and fitting methods of multiplicative-additive risk model have been gradually improved in recent years, and methods are gradually accurate.On the survival analysis, we generally select the relevant tests based directly on whether parameters or non-parametric test conditions are met, but in practice the effect of covariates may change over time, of course, may also be multiplicative, or both additive and multiplicative. So, for survival data with multiple covariates, the model should be fit, but the fit should first determine which variables vary with time, which variables don't; which effect is additive or multiplicative. If the effect types of covariates are non-single, the flexible multiplicative-additive risk model should be used to estimate and infer, as well as survival rate estimates.Research Purposes and MethodsBased on in-depth and extensive application of multiplicative-additive risk model in survival data, this study simulated the follow-up data in line with the requirements of co-variables and selected a sub-sample of 500 cases of myocardial infarction patients studied by Jensen, GV, and Torp-Pedersen, C., etc. Then follow-up data of 700 laryngeal-cancer patients in Shanxi Province was used to carry on comparative analysis between the traditional Cox regression model and multiplicative-additive risk model. We hoped that the introduction of multiplicative-additive risk model could overcome the disadvantages of underutilizing data information and small amount of information mining in current multivariable survival analysis. For such data the analysis of multiplicative-additive risk model was more complete and accurate. Simulation analysis was done by SAS software, and all procedures and results of multiplicative-additive models were achieved in the R software.The Major Findings1. SimulationParameter estimation of small sample was unstable. With the increase of sample size, the results became stable, gradually approached the definition of simulation coefficients. The fit was poor when the sample size was small, for example two of four segments fitted poor in 200 cases. With the increase of sample size, the fit effect was getting better, the effect of 1600 cases was best.2. The case study of multiplicative-additive (COX-AALEN) hazards regression modelThe results of statistical inference about age and vf were both p<0.05 rejected the null hypothesis, indicating that the two variables were time-dependent. Sex, diabetes and chf were not time-dependent variables, the same as the results of cumulative regression coefficient map. Age was significant, but vf had no significance in the additive segment of COX-AALEN model. In the multiplicative part, sex, diabetes and chf were all the factors that affected the survival of patients with myocardial infarction, and the three variables were risk factors which could be seen from the sign of regression coefficients. The model of four variables fitted well. With the increasing of survival time, the survival rate of patients with myocardial infarction decreased.3. The comparative analysis between the traditional COX regression model and multiplicative-additive risk modelAll variables were not time-dependent with p values larger than 0.05. The estimates of regression coefficient, relative risk (RR) and standardized regression coefficient were only different in the thousandth. The levels of tumor invasion (arrangement), lymph node metastasis (transfer), radiotherapy were the factors of survival time. The two models fitted well, and the survival function estimates were nearly the same. At the same survival time, the estimation of survival rate of COX-AALEN model was slightly larger than that of COX regression model.The Main Conclusions1. SimulationMultiplicative-additive model can be applied to data which does not meet the proportional hazards assumption. With the increasing of sample cases, the results of parameter estimation and model fitting become stable, gradually approaching the true value.2. The case study of multiplicative-additive (COX-AALEN) hazards regression model The traditional analysis methods to the data of containing time-dependent variables are mostly too complicated or have many drawbacks. Multiplicative-Additive (COX-AALEN) hazards regression model can handle time-dependent covariates simultaneously, provide their effects, and easily estimate the cumulative survival function of survival data that does not meet the proportional hazards assumption.3. Compare with traditional COX regression modelThe flexibility of COX-AALEN model is that the model is not only used to analyze the survival data which does not meet the proportional hazards assumption, but also fit the data which meet. It can be seen that the model is used widely, requests less for information, and can be used to deal with survival data which doesn't meet the proportional hazards assumption without any changes.