First Advisor

Jong Sung Kim

Term of Graduation

Spring 2023

Date of Publication

6-7-2023

Document Type

Dissertation

Degree Name

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

Department

Mathematics and Statistics

Language

English

Subjects

Mathematical models, Regression analysis

DOI

10.15760/etd.3577

Physical Description

1 online resource (xiii, 99 pages)

Abstract

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). 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.

Rights

©2023 Michael Gray

In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).

Persistent Identifier

https://archives.pdx.edu/ds/psu/40442

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