Continuous-time multi-state models are widely used in modeling longitudinal data of disease processes with multiple transient states, yet the analysis is complex when subjects are observed periodically, resulting in interval censored data. Recently, most studies focused on modeling the true disease progression as a discrete time stationary Markov chain, and only a few studies have been carried out regarding non-homogenous multi-state models in the presence of interval-censored data. In this dissertation, several likelihood-based methodologies were proposed to deal with interval censored data in multi-state models. Firstly, a continuous time version of a homogenous Markov multi-state model with backward transitions was proposed to handle uneven follow-up assessments or skipped visits, resulting in the interval censored data. Simulations were used to compare the performance of the proposed model with the traditional discrete time stationary Markov chain under different types of observation schemes. We applied these two methods to the well-known Nun study, a longitudinal study of 672 participants aged ≥ 75 years at baseline and followed longitudinally with up to ten cognitive assessments per participant. Secondly, we constructed a non-homogenous Markov model for this type of panel data. The baseline intensity was assumed to be Weibull distributed to accommodate the non-homogenous property. The proportional hazards method was used to incorporate risk factors into the transition intensities. Simulation studies showed that the Weibull assumption does not affect the accuracy of the parameter estimates for the risk factors. We applied our model to data from the BRAiNS study, a longitudinal cohort of 531 subjects each cognitively intact at baseline. Last, we presented a parametric method of fitting semi-Markov models based on Weibull transition intensities with interval censored cognitive data with death as a competing risk. We relaxed the Markov assumption and took interval censoring into account by integrating out all possible unobserved transitions. The proposed model also allowed for incorporating time-dependent covariates. We provided a goodness-of-fit assessment for the proposed model by the means of prevalence counts. To illustrate the methods, we applied our model to the BRAiNS study.

Multi-state models are often used to evaluate the effect of death as a competing event to the development of dementia in a longitudinal study of the cognitive status of elderly subjects. In this dissertation, both multi-state Markov model and semi-Markov model are used to characterize the flow of subjects from intact cognition to dementia with mild cognitive impairment and global impairment as intervening transient, cognitive states and death as a competing risk. Firstly, a multi-state Markov model with three transient states: intact cognition, mild cognitive impairment (M.C.I.) and global impairment (G.I.) and one absorbing state: dementia is used to model the cognitive panel data. A Weibull model and a Cox proportional hazards (Cox PH) model are used to fit the time to death based on age at entry and the APOE4 status. A shared random effect correlates this survival time with the transition model. Secondly, we further apply a Semi-Markov process in which we assume that the wait- ing times are Weibull distributed except for transitions from the baseline state, which are exponentially distributed and we assume no additional changes in cognition occur between two assessments. We implement a quasi-Monte Carlo (QMC) method to calculate the higher order integration needed for the likelihood based estimation. At the end of this dissertation we extend a non-parametric “local EM algorithm” to obtain a smooth estimator of the cause-specific hazard function (CSH) in the presence of competing risk. All the proposed methods are justified by simulation studies and applications to the Nun Study data, a longitudinal study of late life cognition in a cohort of 461 subjects.