Abstract
Many meta‐analyses combine results from only a small number of studies, a situation in which the between‐study variance is imprecisely estimated when standard methods are applied. Bayesian meta‐analysis allows incorporation of external evidence on heterogeneity, providing the potential for more robust inference on the effect size of interest. We present a method for performing Bayesian meta‐analysis using data augmentation, in which we represent an informative conjugate prior for between‐study variance by pseudo data and use meta‐regression for estimation. To assist in this, we derive predictive inverse‐gamma distributions for the between‐study variance expected in future meta‐analyses. These may serve as priors for heterogeneity in new meta‐analyses. In a simulation study, we compare approximate Bayesian methods using meta‐regression and pseudo data against fully Bayesian approaches based on importance sampling techniques and Markov chain Monte Carlo (MCMC). We compare the frequentist properties of these Bayesian methods with those of the commonly used frequentist DerSimonian and Laird procedure. The method is implemented in standard statistical software and provides a less complex alternative to standard MCMC approaches. An importance sampling approach produces almost identical results to standard MCMC approaches, and results obtained through meta‐regression and pseudo data are very similar. On average, data augmentation provides closer results to MCMC, if implemented using restricted maximum likelihood estimation rather than DerSimonian and Laird or maximum likelihood estimation. The methods are applied to real datasets, and an extension to network meta‐analysis is described. The proposed method facilitates Bayesian meta‐analysis in a way that is accessible to applied researchers. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.
Highlights
An inverse-gamma prior is declared for the between-study variance τ 2
The metafor package [1] must be installed for this code to work
We assume that the study data from the meta-analysis are stored as length K vectors of observed intervention effects and corresponding within-study variances
Summary
The intervention effect is highest in magnitude for the studies representing the inverse-gamma(1,1) distribution with the greatest prior mean for τ 2 (Figure S1(b)). The code defines a R function “importance.rma” for performing a fully Bayesian meta-analysis with an inverse-gamma prior for the between-study variance τ 2, using importance sampling techniques. We assume that the study data from the meta-analysis are stored as length K vectors of observed intervention effects (in our case, log odds ratios) and corresponding within-study variances.
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