• consistency equations;
  • diagnosis for evidence inconsistency;
  • mixed treatment comparison meta-analysis;
  • multi-arm trials;
  • two-stage model

Mixed treatment comparisons (MTC) meta-analysis synthesises comparative evidence on multiple treatments or other interventions from a collection of randomised controlled trials (RCT) available in a research area, while still respecting the randomisation structure in RCTs. This paper sets out to examine the properties of MTC estimates and elucidate the concept of consistency between direct and indirect evidence in MTC networks. We decompose MTC synthesis into two stages. At the first stage, ordinary meta-analysis is performed in each group of trials that have the same treatment comparators—this provides the ‘direct’ estimates of relative effect parameters. At the second stage, the optimal consistent estimates that minimise the distance between the direct estimates and the consistency hyper-plane can be deduced as the weighted least squares solution to a linear regression model with a specific design matrix that represents the consistency conditions. The consistent MTC estimates can then be represented explicitly as linear combinations of direct estimates, and under normality assumptions the overall evidence consistency can be tested with a likelihood-ratio statistic. This two-stage framework further allows us to use the leverage statistics to diagnose influence of the first-stage evidence and use the regression residuals to assess local inconsistency. The method is illustrated with two examples from medical research. Copyright © 2011 John Wiley & Sons, Ltd.