Inference about the fixed and random effects in a mixed-effects linear model: an approximate Bayesian approach

dc.contributor.advisor David A. Harville Zimmermann, Alan
dc.contributor.department Statistics 2018-08-23T15:58:17.000 2020-06-30T07:03:38Z 2020-06-30T07:03:38Z Fri Jan 01 00:00:00 UTC 1993 1993
dc.description.abstract <p>An approximate Bayesian analysis is considered for data that follow a mixed-effects linear model of the form y = X[beta] + Zs + e, where X and Z are known matrices, [beta] is a vector of unknown parameters, and s and e are statistically independent random vectors whose distributions are multivariate normal with null mean vectors and variance-covariance matrices [sigma][subscript]sp12 I and [sigma][superscript]2 I, respectively. It is assumed that a priori [beta], [sigma][subscript]sp12, and [sigma][superscript]2 are statistically independent, that the distribution of [beta] is noninformative, and that 1/[sigma][subscript]sp12 and 1/[sigma][superscript]2 have gamma distributions;The problem considered is that of inferences about a linear combination of the fixed and random effects, say w = [lambda][superscript]'[beta] + [delta][superscript]' s. Specifically, consideration is given to inferences about w that based on a normal approximation to the posterior density p([beta], s ǁ y) of [beta] and s. The use of this normal approximation (in making inferences about w) does not require numerical integration, whereas the use of the exact posterior distribution of w requires numerical integration;Two approaches to the approximation of p([beta], s ǁ y) are discussed. One is centered at ([beta],\ s) where \ s is the maximum of p( s ǁ y) and [beta] is the maximum of p([beta] ǁ s = \ s, y), and the other is centered at the values of [beta] and s that maximize p([beta], s ǁ y). The possible multimodality of p( s ǁ y) and of p([beta], s ǁ y) is discussed. The numerical problem of finding the maxima of p( s ǁ y) or p([beta], s ǁ y) is considered. It is shown that by adopting an approach that is similar in spirit and technique to the ridge analysis of a response surface, this problem can be reduced to a one-dimensional problem. A "linearized" version of Newton's method is proposed for solving the latter problem, and appropriate starting values for this and other iterative algorithms are described;The approximate and exact posterior distributions of w are compared for some real data sets. The approximate posterior distribution was found to perform relatively well when the prior information is reasonably consistent with the information provided by the data;Some extensions to mixed-effects linear models with more than two variance components are considered.</p>
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dc.identifier archive/
dc.identifier.articleid 11291
dc.identifier.contextkey 6399199
dc.identifier.s3bucket isulib-bepress-aws-west
dc.identifier.submissionpath rtd/10292
dc.language.iso en
dc.source.bitstream archive/|||Fri Jan 14 18:17:40 UTC 2022
dc.subject.disciplines Statistics and Probability
dc.subject.keywords Statistics
dc.title Inference about the fixed and random effects in a mixed-effects linear model: an approximate Bayesian approach
dc.type article
dc.type.genre dissertation
dspace.entity.type Publication
relation.isOrgUnitOfPublication 264904d9-9e66-4169-8e11-034e537ddbca dissertation Doctor of Philosophy
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