Fully Bayesian hierarchical modelling in two stages, with application to meta-analysis

David Lunn*, Jessica Barrett, Michael Sweeting, Simon Thompson

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

39 Citations (Scopus)


Meta-analysis is often undertaken in two stages, with each study analysed separately in stage 1 and estimates combined across studies in stage 2. The study-specific estimates are assumed to arise from normal distributions with known variances equal to their corresponding estimates. In contrast, a one-stage analysis estimates all parameters simultaneously. A Bayesian one-stage approach offers additional advantages, such as the acknowledgement of uncertainty in all parameters and greater flexibility. However, there are situations when a two-stage strategy is compelling, e.g. when study-specific analyses are complex and/or time consuming. We present a novel method for fitting the full Bayesian model in two stages, hence benefiting from its advantages while retaining the convenience and flexibility of a two-stage approach. Using Markov chain Monte Carlo methods, posteriors for the parameters of interest are derived separately for each study. These are then used as proposal distributions in a computationally efficient second stage. We illustrate these ideas on a small binomial data set; we also analyse motivating data on the growth and rupture of abdominal aortic aneurysms. The two-stage Bayesian approach closely reproduces a one-stage analysis when it can be undertaken, but can also be easily carried out when a one-stage approach is difficult or impossible.

Original languageEnglish
Pages (from-to)551-572
Number of pages22
JournalJournal of the Royal Statistical Society. Series C: Applied Statistics
Issue number4
Publication statusPublished - Aug 2013
Externally publishedYes


  • Abdominal aortic aneurysm
  • BUGS
  • Bayesian hierarchical modelling
  • Markov chain Monte Carlo methods
  • Random-effects meta-analysis


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