Meta-analysis methods are used to synthesize results of multiple studies on the same topic. The most frequently used statistical model in meta-analysis is the random-effects model containing parameters for the overall effect, between-study variance in primary study’s true effect size, and random effects for the study-specific effects. We propose Bayesian hypothesis testing and estimation methods using the marginalized random-effects meta-analysis (MAREMA) model where the study-specific true effects are regarded as nuisance parameters which are integrated out of the model. We propose using a flat prior distribution on the overall effect size in case of estimation and a proper unit information prior for the overall effect size in case of hypothesis testing. For the between-study variance (which can attain negative values under the MAREMA model), a proper uniform prior is placed on the proportion of total variance that can be attributed to between-study variability. Bayes factors are used for hypothesis testing that allow testing point and one-sided hypotheses. The proposed methodology has several attractive properties. First, the proposed MAREMA model encompasses models with a zero, negative, and positive between-study variance, which enables testing a zero between-study variance as it is not a boundary problem. Second, the methodology is suitable for default Bayesian meta-analyses as it requires no prior information about the unknown parameters. Third, the proposed Bayes factors can even be used in the extreme case when only two studies are available because Bayes factors are not based on large sample theory. We illustrate the developed methods by applying it to two meta-analyses and introduce easy-to-use software in the R package BFpack to compute the proposed Bayes factors.

Meta分析方法用于综合同一主题的多项研究结果。Meta 分析中最常用的统计模型是随机效应模型，其中包含总体效应参数、主要研究真实效应大小的研究间方差和研究特定效应的随机效应。我们提出了使用边缘化随机效应荟萃分析 (MAREMA) 模型的贝叶斯假设检验和估计方法，其中特定于研究的真实效应被视为集成到模型之外的有害参数。我们建议在估计的情况下对整体效应大小使用平坦的先验分布，在假设检验的情况下对整体效应大小使用适当的单位信息先验。对于研究间方差（在 MAREMA 模型下可以达到负值），一个适当的统一先验放置在可归因于研究间变异性的总方差的比例上。贝叶斯因子用于允许测试点和单边假设的假设检验。所提出的方法具有几个吸引人的特性。首先，提出的 MAREMA 模型包含具有零、负和正研究间方差的模型，这使得能够测试零研究间方差，因为它不是边界问题。其次，该方法适用于默认的贝叶斯元分析，因为它不需要有关未知参数的先验信息。第三，由于贝叶斯因子不是基于大样本理论，因此建议的贝叶斯因子甚至可以在只有两项研究可用的极端情况下使用。BFpack来计算建议的贝叶斯因子。