Reduce dimension or reduce weights? Comparing two approaches to multi-arm studies in network meta-analysis

Network meta-analysis is a statistical method combining information from randomised trials that compare two or more treatments for a given medical condition. Consistent treatment effects are estimated for all possible treatment comparisons. For estimation, weighted least squares regression that in a natural way generalises standard pairwise meta-analysis can be used. Typically, as part of the network, multi-arm studies are found. In a multi-arm study, observed pairwise comparisons are correlated, which must be accounted for. To this aim, two methods have been proposed, a standard regression approach and a new approach coming from graph theory and based on contrast-based data (Rücker 2012). In the standard approach, the dimension of the design matrix is appropriately reduced until it is invertible ('reduce dimension'). In the alternative approach, the weights of comparisons coming from multi-arm studies are appropriately reduced ('reduce weights'). As it was unclear, to date, how these approaches are related to each other, we give a mathematical proof that both approaches lead to identical estimates. The 'reduce weights' approach can be interpreted as the construction of a network of independent two-arm studies, which is basically equivalent to the given network with multi-arm studies. Thus, a simple random-effects model is obtained, with one additional parameter for a common heterogeneity variance. This is applied to a systematic review in depression.