Vulnerability Model And Its Parameter Estimation In Epidemiology

With the rapid development of modern society, more and more statistical models are used to solve practical problems in the field of epidemiology. In1972, the proportional hazard models proposed by British statistician Cox are the most common analysis methods of epidemiology. In survival analysis, due to unpredictable covariates, classical Cox models cannot simulate the survival data well. Vapupel et al proposed the frailty models with one single factor, namely adding a random factor Z on the basis of the Cox models. In binary frailty model, the individuals in the same collection have certain relative frailties and an individual has the certain relative frailty when it experienced the same incident again. Accordingly, Clayton (1978) put forward the shared frailty models, namely the individuals of the same group have the same frailties. However, the assumption that each individual has the same frailty factor value is not reasonable. Yashin and Iachine (1995) put forward the correlative frailty models with the assumption that different individuals have different frailty values.After establishing the frailty models, regression coefficients and frailty parameters are necessary to be estimated. The estimation methods commonly used have maximum likelihood estimation, the EM algorithm, MCMC algorithm and so on.Firstly, this paper proposes the theoretical basis of correlative Gamma frailty models and correlative Log-normal frailty models. Then, the likelihood functions are given. The maximum likelihood estimations and MEM estimates are proposed following. The third chapter shows the theory of multiple correlative Gamma frailty models and multiple correlative Log-normal frailty models. Then the parameters are estimated by using MEM algorithm of frailty models. The fourth chapter discusses the asymptotical property of the estimates through K-L information and Jensen’s inequality. In the fifth chapter, according to the central limit theorem, statistics A and B are presented to test the mean of the frailty factor. Then it uses the statistic χ2to test the hypothesis that the frailty obeys the Gamma distribution. In order to compare the frailty models with different frailty distributions, the AIC and BIC are established. Then an example analysis is performed and the likelihood ratio test of survival time is discussed. The sixth chapter discusses the CDM and double CDM. Then, it covers the diagnosis statistics which can distinguish the strong influence points and abnormal points, such as Cook distance, likelihood distance. Finally, it shows a multiply frailty example.