Modelling Issues in Three-state Progressive Processes
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This dissertation focuses on several issues pertaining to three-state progressive stochastic processes. Casting survival data within a three-state framework is an effective way to incorporate intermediate events into an analysis. These events can yield valuable insights into treatment interventions and the natural history of a process, especially when the right censoring is heavy. Exploiting the uni-directional nature of these processes allows for more effective modelling of the types of incomplete data commonly encountered in practice, as well as time-dependent explanatory variables and different time scales. In Chapter 2, we extend the model developed by Frydman (1995) by incorporating explanatory variables and by permitting interval censoring for the time to the terminal event. The resulting model is quite general and combines features of the models proposed by Frydman (1995) and Kim <i>et al</i>. (1993). The decomposition theorem of Gu (1996) is used to show that all of the estimating equations arising from Frydman's log likelihood function are self-consistent. An AIDS data set analyzed by these authors is used to illustrate our regression approach. Estimating the standard errors of our regression model parameters, by adopting a piecewise constant approach for the baseline intensity parameters, is the focus of Chapter 3. We also develop data-driven algorithms which select changepoints for the intervals of support, based on the Akaike and Schwarz Information Criteria. A sensitivity study is conducted to evaluate these algorithms. The AIDS example is considered here once more; standard errors are estimated for several piecewise constant regression models selected by the model criteria. Our results indicate that for both the example and the sensitivity study, the resulting estimated standard errors of certain model parameters can be quite large. Chapter 4 evaluates the goodness-of-link function for the transition intensity between states 2 and 3 in the regression model we introduced in chapter 2. By embedding this hazard function in a one-parameter family of hazard functions, we can assess its dependence on the specific parametric form adopted. In a simulation study, the goodness-of-link parameter is estimated and its impact on the regression parameters is assessed. The logistic specification of the hazard function from state 2 to state 3 is appropriate for the discrete, parametric-based data sets considered, as well as for the AIDS data. We also investigate the uniqueness and consistency of the maximum likelihood estimates based on our regression model for these AIDS data. In Chapter 5 we consider the possible efficiency gains realized in estimating the survivor function when an intermediate auxiliary variable is incorporated into a time-to-event analysis. Both Markov and hybrid time scale frameworks are adopted in the resulting progressive three-state model. We consider three cases for the amount of information available about the auxiliary variable: the observation is completely unknown, known exactly, or known to be within an interval of time. In the Markov framework, our results suggest that observing subjects at just two time points provides as much information about the survivor function as knowing the exact time of the intermediate event. There was generally a greater loss of efficiency in the hybrid time setting. The final chapter identifies some directions for future research.