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Doubly-censored data refers to time to event data for which both the originating and failure times are censored. In studies involving AIDS incubation time or survival after dementia onset, for example, data are frequently doubly-censored because the date of the originating event is interval-censored and the date of the failure event usually is right-censored. The primary interest is in the distribution of elapsed times between the originating and failure events and its relationship to exposures and risk factors. The estimating equation approach [Sun, et al. 1999. Regression analysis of doubly censored failure time data with applications to AIDS studies. Biometrics 55, 909-914] and its extensions assume the same distribution of originating event times for all subjects. This paper demonstrates the importance of utilizing additional covariates to impute originating event times, i.e., more accurate estimation of originating event times may lead to less biased parameter estimates for elapsed time. The Bayesian MCMC method is shown to be a suitable approach for analyzing doubly-censored data and allows a rich class of survival models. The performance of the proposed estimation method is compared to that of other conventional methods through simulations. Two examples, an AIDS cohort study and a population-based dementia study, are used for illustration. Sample code is shown in the appendix.

Mixture cure models (MCMs) have been widely used to analyze survival data with a cure fraction. The MCMs postulate that a fraction of the patients are cured from the disease and that the failure time for the uncured patients follows a proper survival distribution, referred to as latency distribution. The MCMs have been extended to bivariate survival data by modeling the marginal distributions. In this paper, the marginal MCM is extended to multivariate survival data. The new model is applicable to the survival data with varied cluster size and interval censoring. The proposed model allows covariates to be incorporated into both the cure fraction and the latency distribution for the uncured patients. The primary interest is to estimate the marginal parameters in the mean structure, where the correlation structure is treated as nuisance parameters. The marginal parameters are estimated consistently by treating the observations within the cluster as independent. The variances of the parameters are estimated by the one-step jackknife method. The proposed method does not depend on the specification of correlation structure. Simulation studies show that the new method works well when the marginal model is correct. The performance of the MCM is also examined when the clustered survival times share common random effect. The MCM is applied to the data from a smoking cessation study.

Joinpoint models have been applied to the cancer incidence and mortality data with continuous change points. The current estimation method [Lerman, P.M., 1980. Fitting segmented regression models by grid search. Appl. Statist. 29, 77-84] assumes that the joinpoints only occur at discrete grid points. However, it is more realistic that the joinpoints take any value within the observed data range. Hudson [1966. Fitting segmented curves whose join points have to be estimated. J. Amer. Statist. Soc. 61, 1097-1129] provides an algorithm to find the weighted least square estimates of the joinpoint on the continuous scale. Hudson described the estimation procedure in detail for a model with only one joinpoint, but its extension to a multiple joinpoint model is not straightforward. In this article, we describe in detail Hudson's method for the multiple joinpoint model and discuss issues in the implementation. We compare the computational efficiencies of the LGS method and Hudson's method. The comparisons between the proposed estimation method and several alternative approaches, especially the Bayesian joinpoint models, are discussed. Hudson's method is implemented by C++ and applied to the colorectal cancer incidence data for men under age 65 from SEER nine registries.

Standard survival models assume independence between survival times and frailty models provide a useful extension of the standard survival models by introducing a random effect (frailty) when the survival data are correlated. Several estimation methods have been proposed to find the parameters of shared frailty models. Among them, the EM algorithm (Survival Analysis-Techniques for Censored and Truncated Data, 1997) and the penalized likelihood method (Penalized Survival Models and Frailty, Technical Report No. 66, Mayo Foundation, 2000) are two popular ones. However, the variance estimates involve the calculation of matrix inverse, so the current methods are not able to handle the data with a large number of clusters. This paper provides a modified EM algorithm for the shared frailty models. The new method utilizes standard statistical procedures to find the maximum likelihood estimates (MLE) and it can handle data sets with large numbers of clusters and distinct event times. The confidence intervals of the parameters can be constructed by multiple imputation. Simulation studies were carried out to compare different approaches for the frailty models.