Approximate inferences for Bayesian hierarchical generalised linear regression models

Pub Date : 2024-05-08 DOI:10.1111/anzs.12412
Brandon Berman, Wesley O. Johnson, Weining Shen
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Abstract

Generalised linear mixed regression models are fundamental in statistics. Modelling random effects that are shared by individuals allows for correlation among those individuals. There are many methods and statistical packages available for analysing data using these models. Most require some form of numerical or analytic approximation because the likelihood function generally involves intractable integrals over the latents. The Bayesian approach avoids this issue by iteratively sampling the full conditional distributions for various blocks of parameters and latent random effects. Depending on the choice of the prior, some full conditionals are recognisable while others are not. In this paper we develop a novel normal approximation for the random effects full conditional, establish its asymptotic correctness and evaluate how well it performs. We make the case for hierarchical binomial and Poisson regression models with canonical link functions, for hierarchical gamma regression models with log link and for other cases. We also develop what we term a sufficient reduction (SR) approach to the Markov Chain Monte Carlo algorithm that allows for making inferences about all model parameters by replacing the full conditional for the latent variables with a considerably reduced dimensional function of the latents. We expect that this approximation could be quite useful in situations where there are a very large number of latent effects, which may be occurring in an increasingly ‘Big Data’ world. In the sequel, we compare our methods with INLA, which is a particularly popular method and which has been shown to be excellent in terms of speed and accuracy across a variety of settings. Our methods appear to be comparable to theirs in terms of accuracy, while INLA was faster, for the settings we considered. In addition, we note that our methods and those of others that involve Gibbs sampling trivially handle parameters that are functions of multiple parameters, while INLA approximations do not. Our primary illustration is for a three-level hierarchical binomial regression model for data on health outcomes for patients who are clustered within physicians who are clustered within particular hospitals or hospital systems.

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贝叶斯分层广义线性回归模型的近似推论
摘要广义线性混合回归模型是统计学的基础。对个体共享的随机效应进行建模,可以考虑这些个体之间的相关性。有许多方法和统计软件包可用于使用这些模型分析数据。大多数都需要某种形式的数值或分析近似,因为似然函数通常涉及对潜变量进行难以处理的积分。贝叶斯方法通过对各种参数块和潜在随机效应的全条件分布进行迭代采样,避免了这一问题。根据先验值的选择,一些全条件分布是可识别的,而另一些则不可识别。在本文中,我们开发了一种新的随机效应全条件正态近似值,建立了其渐近正确性,并对其性能进行了评估。我们对具有典型联系函数的分层二项式和泊松回归模型、具有对数联系的分层伽马回归模型以及其他情况进行了论证。我们还为马尔可夫链蒙特卡洛算法开发了一种称为 "充分还原(SR)"的方法,通过用一个大大降低维度的潜变量函数来替代潜变量的全条件,从而对所有模型参数进行推断。我们预计,这种近似方法在存在大量潜变量效应的情况下会非常有用,而这种情况可能会出现在越来越多的 "大数据 "世界中。在接下来的文章中,我们将把我们的方法与 INLA 进行比较,INLA 是一种特别流行的方法,在各种情况下都表现出卓越的速度和准确性。我们的方法在准确性方面似乎与 INLA 不相上下,而在我们考虑的环境中,INLA 的速度更快。此外,我们还注意到,我们的方法和其他涉及吉布斯采样的方法可以轻松处理多个参数的函数参数,而 INLA 近似方法则不行。我们的主要示例是一个三级分层二叉回归模型,该模型针对的是聚集在特定医院或医院系统内的医生的病人健康结果数据。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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