Extending R-Squared to the Generalized Linear Mixed Model for Longitudinal Data

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  • March 22, 2019
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  • Jaeger, Byron
    • Affiliation: Gillings School of Global Public Health, Department of Biostatistics
Abstract
  • The R2 statistic is a well known measure of association for the linear model that has been extended in various ways to the linear mixed model. Statistical literature suggests multiple measures of R2 must be used to characterize linear mixed models by summarizing goodness-of-fit for both fixed and random effects. Regarding the latter, there are currently no measures of R2 which consistently demonstrate a capacity to avoid over-fitting and under-fitting covariance models. In this dissertation we extend R2 to the generalized linear mixed model and develop new covariance model selection and inference techniques for R2 in the linear mixed model that can also be extended to the generalized linear mixed model. Chapter 2 describes a marginal R2 statistic for the linear mixed model that measures generalized explained variance. Our method utilizes standardized generalized variance to stabilize the estimated denominator degrees of freedom used in the approximate Wald F test. The proposed modification consistently estimates a well-defined population value, exhibits a non-central beta sample distribution, and demonstrates superior performance in a simulation study where R2 statistics are used to assess covariance goodness-of-fit. Chapter 3 introduces a paradigm of conducting statistical tests regarding R2 statistics in the linear mixed model. Simple summary tests of covariance goodness-of-fit for a specific model are explored as well as tests for model selection. This approach is able to compare covariance models that are not hierarchically related (i.e. nested). A simulation study and two applied examples demonstrate the testing procedure’s capacity to fill in the gaps of uncertainty regarding covariance model selection when candidates are non-nested. Chapter 4 discusses a method to extend R2 from the linear mixed model to the generalized linear mixed model. The approach utilizes penalized quasi-likelihood estimation and is the first to enable computation of semi-partial R2 statistics for fixed effects in the generalized linear mixed model. A simulation study assesses the performance of the proposed method. Extensions based on the linear mixed model results from Chapters 2 and 3 are explored.
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  • In Copyright
Advisor
  • Preisser, John
  • Qaqish, Bahjat
  • Golin, Carol E.
  • Kosorok, Michael
  • Edwards, Lloyd
Degree
  • Doctor of Philosophy
Degree granting institution
  • University of North Carolina at Chapel Hill Graduate School
Graduation year
  • 2017
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