Predictive distributions for between-study heterogeneity and simple methods for their application in Bayesian meta-analysis

Abstract

Numerous meta-analyses in healthcare research combine results from only a small number of studies, for which the variance representing between-study heterogeneity is estimated imprecisely. A Bayesian approach to estimation allows external evidence on the expected magnitude of heterogeneity to be incorporated. The aim of this paper is to provide tools that improve the accessibility of Bayesian meta-analysis. We present two methods for implementing Bayesian meta-analysis, using numerical integration and importance sampling techniques. Based on 14,886 binary outcome meta-analyses in the Cochrane Database of Systematic Reviews, we derive a novel set of predictive distributions for the degree of heterogeneity expected in 80 settings depending on the outcomes assessed and comparisons made. These can be used as prior distributions for heterogeneity in future meta-analyses. The two methods are implemented in R, for which code is provided. Both methods produce equivalent results to standard but more complex Markov chain Monte Carlo approaches. The priors are derived as log-normal distributions for the between-study variance, applicable to meta-analyses of binary outcomes on the log odds-ratio scale. The methods are applied to two example meta-analyses, incorporating the relevant predictive distributions as prior distributions for between-study heterogeneity. We have provided resources to facilitate Bayesian meta-analysis, in a form accessible to applied researchers, which allow relevant prior information on the degree of heterogeneity to be incorporated.

Keywords: Bayesian methods; heterogeneity; meta-analysis; prior distributions.

Figure 1

Conventional (DerSimonian and Laird, marked D + L) and Bayesian random-effects meta-analyses combining odds ratios (ORs) from example 1: four studies of ticlopidine plus aspirin versus oral anticoagulants for prevention of major bleeding events following coronary stenting; 95% confidence intervals and % weight in meta-analysis shown.

Figure 2

Histograms (a) and (b) show prior and posterior distributions respectively for heterogeneity variance τ²in Example 1. Histograms (c) and (d) show prior and posterior distributions for τ²in Example 2. Distributions obtained using MCMC methods.

Figure 3

Conventional (DerSimonian and Laird, marked D + L) and Bayesian random-effects meta-analyses combining odds ratios (ORs) from Example 2: four studies examining withdrawal from cocaine dependence treatment: acupuncture versus sham acupuncture; 95% confidence intervals and % weight in meta-analysis shown.