Diana Fisher

Date of Award

2005

Degree Name

Mathematics

College

College of Science

Type of Degree

M.A.

Document Type

Thesis

First Advisor

Dr. Alfred Akinsete, Committee Chairman

Second Advisor

Dr. Yulia Dementieva

Third Advisor

Dr. Laura Adkins

Abstract

Computational infeasibility of exact methods for solving genetic linkage analysis problems has led to the development of a new collection of stochastic methods, all of which require the use of Markov chains. The purpose of this work is to investigate the complexities of missing data in pedigree analysis using the Monte Carlo Markov Chain (MCMC) method as compared to the exact results. Also, we attempt to determine an association between missing data in a familial pedigree and the convergence to stationarity of a descent graph Markov chain implemented in the stochastic method for parametric linkage analysis.

In particular, we will implement the stochastic method to solve a pedigree problem for a disease trait, in order to look at the associated problems with missing data from the pedigree, and investigate the deviation between the MCMC method and the exact results. Using the method for maximum autocorrelation and bounding of the second largest eigenvalue, we will study the effects of missing data on the convergence rate and the accuracy of the MCMC method in solving the pedigree analysis problem. Finally, we will use the computational implementation of SimWalk2 to study the convergence rate and accuracy of the MCMC method for the disease Episodic Ataxia.

The implementation of the MCMC method through SimWalk2 for the disease gene Episodic Ataxia found evidence to suggest that both the efficiency and accuracy of the method may be severely reduced by an increase in missing data in the pedigree. Certain variations of model parameters influenced the ability of the method to produce accurate results, but the most crucial of the variables studied was the level of missing information from the pedigree itself. This can be seen as a detriment to the implementation, as pedigree information is very often missing from the model. Further research in this topic would need to include the implementation of this method on more genetic parameters and differing pedigree variations. Also, it might be of interest to look into possible ways to combat the effects of missing data on the MCMC method.

Subject(s)

Monte Carlo method.

Markov processes.

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