The Positive Stable Frailty Model And Its Application To Dependent Survival Data

There are many statistical models available to time-to-event data, in which thestudy objects are from homogeneous population and the failure event occurs onceeach one. In medical follow-up study, the recurrent event data is common and moreinteresting, which is called when the same event occurs in the same object for severaltimes in a period of time. A patient with cancer diagnosis may relapse several timesduring his/her treatment. The individual lifetimes among a group or in the same objectare non-independent, such time-to-event data is called multivariate time-to-event data,whose data structure is sophisticate. Not only the repeated measure data feature iscontained, but also the failure hazard of event is different in different time, namelyheterogeneous, in multivariate time-to-event data. The classical survival analysisapproach such as the Cox proportional hazard model cannot make use of theinformation of the data structure. The frailty model takes the random effect into theCox proportional hazard model, regard the dependent within group and heterogeneitybetween group in data. The shared frailty model is a common frailty model and iswide used in the multivariate survival analysis, in which the assumption is the severalobservation in the same individual or the individuals in the same cluster share thesame unobserved frailty and the frailty follows a special distribution. This definition offrailty assumes that each individual is born at a certain level of relative frailty and stays at thislevel all its life.To analyze multivariate time-to-event, it must result in information loss if onlyincorporate the data simply, ignoring the complex data structure. In the research, thepositive stable shared frailty model is introduced, which assume the frailty follows thepositive stable distribution. According to whether the shape of the baseline hazard isassumed, the parametric and semi-parametric frailty models are constructed separately.In parametric model, the MMLE is the parameter estimate method, and asemi-parametric EM algorithm based on a profile likelihood is applied tosemi-parametric model.Then two data cases, a cluster survival data and a recurrent event data, areanalyzed through conducting the positive stable shared frailty model. Meanwhile,model comparison is carried out among the Cox proportional hazard model, thepositive stable frailty model and the gamma frailty model. The personal prognosisindex containing the frailty into the formula is first introduced in this research. Theconclusions are as follows:The positive stable frailty model can make full use of the information the dataoffers, in which the frailty is a positive stable distribution, and the parameterestimators are more accurate than the result of the Cox proportional hazard model. In the case analysis, the gamma distribution as the frailty distribution cannot explain thereal data structure, thus, to the two data, the positive stable frailty model is moresuitable to analyze them.The prognosis index evaluation plays a much more important role and has a keyapplying value in survival follow-up study. We first contain the frailty into theformula of the personal prognosis index to fulfill the prognosis index evaluation in thepositive stable frailty model. The result shows the prognosis index considering thefrailty can distinguish the patient more explicitly than the prognosis index withoutconsidering the frailty. As a result, regarding the frailty the prognosis index evaluationis more precise.To sum up, the positive stable frailty model is one of the effective methods. Theparameter estimate method of the MMLE or the EM algorithm is right and exact andcan finish in the SAS or R easily and quickly. Thus, it provide a new approach in therecurrent event data analysis, the familial disease study and the multi-center trial inmedical practice. The prognosis index considering the frailty is first introduced intothe frailty model in this research and the result is more reasonable. Thus the method isworth being recommended to the non-independent survival data analysis.