Bayesian Analysis Of Recurrent Events Data

Recurrent events data refer to the events which may occur repeatedly and sequentially in the period of observation. There are two status of recurrent event data:single type and multi-type which can be distinguished by the number of event types of interest. Researches regarding the two kinds of recurrent event data are mostly on the basis of large sample theory. Bayesian approach, especially objective Bayesian approach, is rarely applied and would be considered throughout the dissertation.For single type recurrent event data, nonhomogeneous Poisson process and renewal process are the two most popular processes utilized to characterize the pattern of events occurring. As a compromise between them, modulated power law process is more appro-priate to model it since both renewal type behavior and time trend are presented in such process. In this article, objective Bayesian method are proposed to analyze the modulat-ed power law process. Seven reference priors are derived, one of which is Jeffreys prior. However, only four of them are taken into consideration. Propriety of the posterior densi-ties considering the four reference priors are proved. Predictive distribution of the future failure time is obtained additionally. For the purpose of comparison, the simulation work and real data analysis are carried out based on both the objective Bayesian and maximum likelihood approaches. Study results show that the objective Bayesian estimation and pre-diction possess much better statistical properties in a frequentist context and, therefore, outperforms the maximum likelihood method even with small or moderate sample sizes.When competing risks are incorporated into recurrent event data, multi-type re- current events data model can be established. In this dissertation, Bayesian reference analysis is proceeded to make inference about such model and subjects experiencing the recurrent events are assumed to be multiple. Cox model is applied to depict the relation among individuals. The sampling distribution of the observations due to some type is modeled through a proportional intensity homogeneous Poisson process and is dependen-t on some time-fixed covariates additionally. The goal is to estimate the cause-specific intensity functions so that intensity for different event types can be compared. Three reference priors are derived according to this model. Their associated posteriors are also discussed. The simulation work and real data application based on the reference priors and the classical likelihood approach show the efficiency of reference analysis.For the multi-type recurrent events data, event types would sometimes be masked in many clinical studies and software reliability experiments because of the limitation of monitoring. Under the assumption that masking probabilities are irrelevant with event categories, inferences for the cause-specific intensity functions are independent of the masking probabilities and such problem is investigated from both frequentist and Bayesian perspective in the article. The sampling distribution of the recurrent events is modeled via a multivariate nonhomogeneous Poisson process with Weibull (power law) intensity. The time-fixed covariates are also considered for multiple individuals via Cox model. Data augmentation technique is introduced, based on which EM algorithm and Gibbs sampling are utilized to obtain the maximum likelihood estimates of the parameters and to generate samples from posterior distributions, respectively. The simulation results indicate that both classical maximum likelihood and Bayesian methods give accurate estimation to the intensity functions. However, Bayesian estimates perform slightly better than the latter one in meeting the target coverage probability with small sample size.Moreover, under the assumption that masking probabilities are dependent on event types, inferences of the cause-specific intensity are accordingly dependent on the masking probabilities. It is a more general case than that mentioned above. For this problem, Cox model is still utilized and the corresponding covariates are time-fixed. No further assumptions are made for the baseline intensity so that semiparametric Bayesian analysis is executed in the dissertation. Cumulative baseline intensity functions are the key ele-ments serving to proceed the Bayesian analsyis and priors of them are assumed as Gamma processes. Data augmentation is also conducted to facilitate the Bayesian inference. Sim-ulation work show the good performance of Bayesian approach in estimating the masking probabilities and cumulative cause-specific intensity functions.