Computation of reference Bayesian inference for variance components in longitudinal studies |
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Authors: | Miao-Yu Tsai Chuhsing K Hsiao |
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Institution: | (1) Institute of Statistics and Information Science, College of Science, National Changhua University of Education, Changhua, Taiwan;(2) Department of Public Health and Institute of Epidemiology, College of Public Health, National Taiwan University, No.17, Xu-Zhou Road, Room 523, Taipei, 100, Taiwan |
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Abstract: | Generalized linear mixed models (GLMMs) have been applied widely in the analysis of longitudinal data. This model confers
two important advantages, namely, the flexibility to include random effects and the ability to make inference about complex
covariances. In practice, however, the inference of variance components can be a difficult task due to the complexity of the
model itself and the dimensionality of the covariance matrix of random effects. Here we first discuss for GLMMs the relation
between Bayesian posterior estimates and penalized quasi-likelihood (PQL) estimates, based on the generalization of Harville’s
result for general linear models. Next, we perform fully Bayesian analyses for the random covariance matrix using three different
reference priors, two with Jeffreys’ priors derived from approximate likelihoods and one with the approximate uniform shrinkage
prior. Computations are carried out via the combination of asymptotic approximations and Markov chain Monte Carlo methods.
Under the criterion of the squared Euclidean norm, we compare the performances of Bayesian estimates of variance components
with that of PQL estimates when the responses are non-normal, and with that of the restricted maximum likelihood (REML) estimates
when data are assumed normal. Three applications and simulations of binary, normal, and count responses with multiple random
effects and of small sample sizes are illustrated. The analyses examine the differences in estimation performance when the
covariance structure is complex, and demonstrate the equivalence between PQL and the posterior modes when the former can be
derived. The results also show that the Bayesian approach, particularly under the approximate Jeffreys’ priors, outperforms
other procedures. |
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Keywords: | Bayesian GLMM Jeffreys’ prior PQL Reference prior Uniform shrinkage prior Variance component |
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