Bayesian inference for heterogeneity in meta-analysis

A generalized marginal random effects model is described that enables exact Bayesian inference using either the Jeffreys or Berger-Bernardo non-informative prior distributions without the need for Markov Chain Monte Carlo sampling, requiring only numerical integrations. This contribution focuses on inference for the heterogeneity parameter, often called "dark uncertainty" and denoted tau in this contribution. The proposed models are used for consensus building in meta-analyses of measurement results for the Newtonian constant of gravitation, G, and for the effectiveness of anti-retroviral pre-exposure prophylaxis in preventing HIV infection. The estimates of tau that seventeen alternative different methods produce, including those that we propose, were also compared. The relative impact (gauged in terms of the ratio of the range of estimates to their median) that model choice had on the estimate of tau was very substantial: 79 % for G and 87 % for HIV prophylaxis. For G, the estimates of tau ranged from 0.0009x10(-10)m(3)kg(-1)s(-2) to 0.0013x10(-10) m(3) kg(-1) s(-2). For Truvada they ranged from 0.49 to 0.92. Since the estimate of tau impacts the quality of the estimate of the measurand substantially, we recommend the Bayesian approaches to estimate tau because they take the whole posterior distribution of tau into account, hence the corresponding uncertainty, rather than using a single value and pretending that it is known with certainty. In the case of the measurement results for G, we found that the model with Student's t random effects and the Jeffreys prior provides the best fit, while for the Truvada data the normal marginal random effects model, also with Jeffreys prior, produces an estimate of tau closest to classical estimators like DerSimonian-Laird, but offers the advantage of recognizing and propagating the uncertainty associated with tau, which the classical procedures ignore.