When conducting a meta-analysis of clinical trials with binary outcomes, a normal approximation for the summary treatment effect measure in each trial is inappropriate in the common situation where some of the trials in the meta-analysis are small, or the observed risks are close to 0 or 1. This problem can be avoided by making direct use of the binomial distribution within trials. A fully Bayesian method has already been developed for random effects meta-analysis on the log-odds scale using the BUGS implementation of Gibbs sampling. In this paper we demonstrate how this method can be extended to perform analyses on both the absolute and relative risk scales. Within each approach we exemplify how trial-level covariates, including underlying risk, can be considered. Data from 46 trials of the effect of single-dose ibuprofen on post-operative pain are analysed and the results contrasted with those derived from classical and Bayesian summary statistic methods. The clinical interpretation of the odds ratio scale is not straightforward. The advantages and flexibility of a fully Bayesian approach to meta-analysis of binary outcome data, considered on an absolute risk or relative risk scale, are now available.

Substantial reductions in mortality and the need for subsequent MV were associated with NIV in acute respiratory failure, especially in the COPD subgroup. Hospital length of stay was variably affected. Heterogeneity of treatment effects was observed.

Meta-analysis can be considered a multilevel statistical problem, since information within studies is combined in the presence of potential heterogeneity between studies. Here a general multilevel model framework is developed for meta-analysis to combine either summary data or individual patient outcome data from each study, and to include either study or individual level covariates that might explain heterogeneity. Classical and Bayesian approaches to estimation are contrasted. These methods are applied to a meta-analysis of trials of thrombolytic therapy after myocardial infarction. Subgroups within the trials were available, categorized by the time delay until treatment, so that a three-level random effects model that includes time delay as a covariate is proposed. In addition it was desired to represent the treatment effect as an absolute risk reduction, rather than the conventional odds ratio. We show how this can be achieved within a Bayesian analysis, while still recognizing the binary nature of the original outcome data.

A Bayesian hierarchical modelling approach to the analysis of cluster randomized trials has advantages in terms of allowing for full parameter uncertainty, flexible modelling of covariates and variance structure, and use of prior information. Previously, such modelling of binary outcome data required use of a log-odds ratio scale for the treatment effect estimate and an approximation linking the intracluster correlation (ICC) to the between-cluster variance on a log-odds scale. In this paper we develop this method to allow estimation on the absolute risk scale, which facilitates clinical interpretation of both the treatment effect and the between-cluster variance. We describe a range of models and apply them to data from a trial of different interventions to promote secondary prevention of coronary heart disease in primary care. We demonstrate how these models can be used to incorporate prior data about typical ICCs, to derive a posterior distribution for the number needed to treat, and to consider both cluster and individual level covariates. Using these methods, we can benefit from the advantages of Bayesian modelling of binary outcome data at the same time as providing results on a clinically interpretable scale.