Likelihood Theory and Methods for Generalized Linear Mixed Models

Bhaskaran, Aishwarya

Likelihood Theory and Methods for Generalized Linear Mixed Models

Bhaskaran, Aishwarya

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Publication Type:: Thesis
Issue Date:: 2022

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Field	Value	Language
dc.contributor.author	Bhaskaran, Aishwarya
dc.date.accessioned	2023-09-19T02:30:25Z
dc.date.available	2023-09-19T02:30:25Z
dc.date.issued	2022
dc.identifier.uri	http://hdl.handle.net/10453/172194
dc.description	University of Technology Sydney. Faculty of Science.	en_US.UTF-8
dc.description.abstract	Generalized linear mixed models are an essential group of models for analysing many present-day complex data sets, especially those that contain non-normal and correlated response data. Despite the large volume of research concerning this group of models, there is very little theory concerning the statistical properties of maximum likelihood estimators for generalized linear mixed models. Existing theoretical results available for the asymptotic variance-covariance matrix for such estimators contain limits and expectations over the response distribution, hence such results are not in ready-to-use forms when carrying out tasks such as constructing studentized confidence intervals or optimal design determination. In this thesis, we derive precise asymptotic results for likelihood-based generalized linear mixed model analysis. The novel asymptotic normality results are derived for both cases involving either a canonical or noncanonical link function. In our approach, we derive the exact leading term behaviour of the Fisher information matrix when both the number of groups and number of observations within each group diverge. This leads to asymptotic normality results with explicit and simple studentizable forms. The implications of these results in optimal design theory is also explored, leading to simpler and more direct determination of approximate locally D-optimal designs. Towards the end of this thesis, a Thouless-Anderson-Palmer approach is introduced for modern statistical inference for generalized linear mixed models. Such methods have proven to provide accurate approximations to problems arising in machine learning contexts. However, statistical applications such as generalized linear mixed model analysis have not been investigated. Thus, we derive results for implementing the Thouless-Anderson-Palmer frequentist variational approach to generalized linear mixed models and analyse the accuracy of its variational estimates.	en_US.UTF-8
dc.format	Thesis (PhD)
dc.language.iso	en_US	en_US.UTF-8
dc.relation	https://opus.lib.uts.edu.au/bitstream/10453/172194/2/02whole.pdf
dc.rights	info:eu-repo/semantics/openAccess
dc.rights	The author owns the copyright in this thesis including all reproduction and reuse rights for the work. The work may not be altered without the permission of the copyright owner. Attribution is essential when quoting or paraphrasing from this thesis.
dc.rights	© 2022 Aishwarya Bhaskaran
dc.rights	au.edu.uts.lib/cph
dc.title	Likelihood Theory and Methods for Generalized Linear Mixed Models	en_US.UTF-8
dc.type	Thesis
utslib.copyright.status	open_access	*

Abstract:

Generalized linear mixed models are an essential group of models for analysing many present-day complex data sets, especially those that contain non-normal and correlated response data. Despite the large volume of research concerning this group of models, there is very little theory concerning the statistical properties of maximum likelihood estimators for generalized linear mixed models. Existing theoretical results available for the asymptotic variance-covariance matrix for such estimators contain limits and expectations over the response distribution, hence such results are not in ready-to-use forms when carrying out tasks such as constructing studentized confidence intervals or optimal design determination. In this thesis, we derive precise asymptotic results for likelihood-based generalized linear mixed model analysis. The novel asymptotic normality results are derived for both cases involving either a canonical or noncanonical link function. In our approach, we derive the exact leading term behaviour of the Fisher information matrix when both the number of groups and number of observations within each group diverge. This leads to asymptotic normality results with explicit and simple studentizable forms. The implications of these results in optimal design theory is also explored, leading to simpler and more direct determination of approximate locally D-optimal designs. Towards the end of this thesis, a Thouless-Anderson-Palmer approach is introduced for modern statistical inference for generalized linear mixed models. Such methods have proven to provide accurate approximations to problems arising in machine learning contexts. However, statistical applications such as generalized linear mixed model analysis have not been investigated. Thus, we derive results for implementing the Thouless-Anderson-Palmer frequentist variational approach to generalized linear mixed models and analyse the accuracy of its variational estimates.

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