Trials and triangles: Network meta-analysis of multi-arm trials with correlated arms

For network meta-analysis (NMA), we usually assume that the treatment arms are independent within each included trial. This assumption is justified for parallel design trials and leads to a property we call consistency of variances for both multi-arm trials and NMA estimates. However, the assumption is violated for trials with correlated arms, for example, split-body trials. For multi-arm trials with correlated arms, the variance of a contrast is not the sum of the arm-based variances, but comes with a correlation term. This may lead to violations of variance consistency, and the inconsistency of variances may even propagate to the NMA estimates. We explain this using a geometric analogy where three-arm trials correspond to triangles and four-arm trials correspond to tetrahedrons. We also investigate which information has to be extracted for a multi-arm trial with correlated arms and provide an algorithm to analyze NMAs including such trials.