A new approach to evaluating loop inconsistency in network meta-analysis

In network meta-analysis, studies evaluating multiple treatment comparisons are modeled simultaneously, and estimation is informed by a combination of direct and indirect evidence. Network meta-analysis relies on an assumption of consistency, meaning that direct and indirect evidence should agree for each treatment comparison. Here we propose new local and global tests for inconsistency and demonstrate their application to three example networks. Because inconsistency is a property of a loop of treatments in the network meta-analysis, we locate the local test in a loop. We define a model with one inconsistency parameter that can be interpreted as loop inconsistency. The model builds on the existing ideas of node-splitting and side-splitting in network meta-analysis. To provide a global test for inconsistency, we extend the model across multiple independent loops with one degree of freedom per loop. We develop a new algorithm for identifying independent loops within a network meta-analysis. Our proposed models handle treatments symmetrically, locate inconsistency in loops rather than in nodes or treatment comparisons, and are invariant to choice of reference treatment, making the results less dependent on model parameterization. For testing global inconsistency in network meta-analysis, our global model uses fewer degrees of freedom than the existing design-by-treatment interaction approach and has the potential to increase power. To illustrate our methods, we fit the models to three network meta-analyses varying in size and complexity. Local and global tests for inconsistency are performed and we demonstrate that the global model is invariant to choice of independent loops.