First Advisor

Jong Sung Kim

Term of Graduation

Spring 2023

Date of Publication


Document Type


Degree Name

Doctor of Philosophy (Ph.D.) in Mathematical Sciences


Mathematics and Statistics




non-Markov, semi-parametric regression, transition probability



Physical Description

1 online resource (xiii, 99 pages)


A multi-state model is a graphical tool widely used to illustrate a transitional relationship between states in many applications. We will study the transition probabilities of an illness-death model, which is an example of a multi-state model. We will investigate transition probabilities using a counting process approach. Aalen-Johansen estimator is the gold-standard in estimating a transition probability. However, Aalen-Johansen estimator may be biased when the Markov assumption is violated. Therefore, Aalen-Johansen estimator is an unreliable estimator when the Markov assumption is violated. Several papers have published non-parametric estimators that accommodate for non-Markov models using a counting process approach.

Furthermore, there are few existing work in creating a regression model for transition probabilities in the non-Markov setting. Our goal is to contribute to the few existing work of regression models that accommodate non-Markov behavior. In creating the regression model, we use the jackknife method, pseudo-observations. In finding parameter estimates, generalized estimation equation(GEE) will be used. An important requirement in using pseudo-observations is that we need an unbiased estimator. Aalen-Johansen estimator would be a unreliable choice since it is susceptible to bias. We propose in using Titman estimator as an alternative estimator to create the pseudo-observation for the regression model. Titman estimator is shown to be unbiased from Titman (2015) [28]. It also can be used in time-irreversible and time-reversible models. This feature of Titman estimator allows practitioners to find the transition probability of recovering from an illness in the illness-death model.

In a simulation study, we will compare the results when creating pseudo-observations by using Titman estimator and Aalen-Johansen estimator. We will illustrate the regression model using the illness-death model when recovery is not assumed and illness-death model when recovery is assumed. We will study when the model is "pathologically" non-Markov and the model has a frailty effect. Both cases violate the Markov assumption. Finally, we will analyze the liver cirrhosis dataset using our proposed method.


©2023 Michael Gray

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