Semi-parametric Mixture Models For The Analysis Of Survival Data With Long-Term Survivors

In cancer clinical trials, a group of patients response favorably to thetreatment,they usually have long-term censored survival times and tend to beregarded as cured. In such experiments with laboratory animals, a substantialproportion of the animals at some toxicant levels do not die by the end of theexperiment. These observations are called as long-term survivors or curedindividuals. Examples of such kind can be found in many disciplines, includingbiomedical science, economics, sociology and engineering science . Long-term survivors or cured individuals mean those who are not subject tothe event under study —death or relapse of a disease, or a return to prison,having been released from it. The first approach to survival data analysis is totest for the presence of immunes in the data. If the statistical decision is thatnone are present, or that the precision in the data is insufficient for us to beconfident of their presence, we recommend proceeding with ordinary survivalanalysis. However, if we decide that long-term survivors are present andfollow-up has been sufficient, ordinary survival analysis are inappropriate,because they consider long-term survivors as censored observations, obviouslythat is not reasonable. If a set of survival data is fitted by an ordinary survivalanalysis, we may explain the result uneasily, even may obtain wrong ormisleading answers. The primary objective of the study is to develop a new method—semi-parametric mixture models. Semi- parametric mixture models have some types,such as a mixture model combining logistic regression with proportional hazardsregression, semi-parametric transformation mixture models and semi-parametric cure models with fixed coefficient 1 and so on. The study primarilyintroduces the third kind of cure models which combines logistic regression andCox's proportional hazards regression with the inclusion of an additionalcovariate with fixed coefficient 1: Cox's proportional hazards regression modelsthe survival times while logistic regression models the cure fraction. Theapproach to estimate the parameters in semi-parametric cure models is based onrecognizing that the EM algorithm consists of fitting a proportional hazardsmodel and a logistic regression model , respectively. This estimation procedureis implemented in many standard statistical packages. Furthermore an attractivefeature of this model is that it allows easy extensions of this model, whichmakes it possible to apply this model in broader settings. Three clinical follow-up survival data with long-term survivors arecollected and semi-parametric cure models with fixed coefficient 1 and Cox'sproportional hazards regression model are fitted and compared. The result showssemi-parametric mixture models have been proved to be more advantageousthan Cox's proportional hazards regression model in analyzing survival datawith a cured fraction. Not only is the form of the model simpler and theexplanation of estimated parameters is more reasonable, but also it providesmore valuable information from many aspects. It is an enrichment andenhancement of ordinary survival analysis and it is a more applicable andpracticable statistical method.