Bayesian Linear Transformation Cure Models Using Bernstein Polynomials

In this paper,we mainly discuss the linear transformation mixture cure model with right censored data.Because the linear transformation model has different forms under different transformation families G,this paper only discusses the situation of G as a logarithmic transformation family,and hopes to compare it with the traditional Cox proportional hazards model(PH)and the proportional odds model(PO),PH and PO models are two special cases of the linear transformation model.We use the Bernstein polynomials to approximate the baseline hazard function,which can not only achieve the precision of approximation,but also can easily impose various shape constraints through the Bernstein polynomials,which is more flexible and more convenient to execute at the same time.We use the shape information as the prior information in the Bayesian method by imposing constraints on the parameters,we use the maximum entropy priori and the non-information priori.In the numerical calculation of the Bayesian method,we use the MCMC algorithm,in particular,the method of combining the Gibbs sampling and the MH algorithm to sample from the posterior distribution,and then further statistical inference is made by using the sampling result.Model selection we use the CPO and the DIC statistics.In simulation,for the evaluation criteria of the curve estimation,we use the integral absolute error(IAE)to compare the estimated results directly through the box diagram.By the simulation under different sample sizes,we obtain that with the increase of the sample size,the parameters and curves are gradually approaching the real situation,and the fluctuation is becoming smaller.It can be seen from the simulation experiments that the constrained maximum likelihood method and the Bayesian method have achieved similar results under large sample sizes,and the results are good.Through the CPO and the DIC statistics,we show that the linear transformation model used in this paper is superior to the traditional PO and PH models,because the linear transformation model can not only get the same results as the real model,the estimation of the regulation parameter ρ can also be obtained,so that we have a better understanding of the possible real model.In the real data analysis,we analyze a piece of brain cancer data from the SEER database and get results that were more in line with the literature.