Evidence inconsistency degrees of freedom in Bayesian network meta-analysis

Network meta-analysis (NMA) is a popular tool to synthesize direct and indirect evidence for simultaneously comparing multiple treatments, while evidence inconsistency greatly threatens its validity. One may use the inconsistency degrees of freedom (ICDF) to assess the potential that an NMA might suffer from inconsistency. Multi-arm studies provide intrinsically consistent evidence and complicate the ICDF's calculation; they commonly appear in NMAs. The existing ICDF measure may not feasibly handle multi-arm studies. Motivated from the effective numbers of parameters of Bayesian hierarchical models, we propose new ICDF measures in generic NMAs that may contain multi-arm studies. Under the fixed- or random-effects setting, the new ICDF measure is the difference between the effective numbers of parameters of the consistency and inconsistency NMA models. We used artificial NMAs created based on an illustrative example and 39 empirical NMAs to evaluate the performance of the existing and new measures. In NMAs with two-arm studies only, the proposed ICDF measure under the fixed-effects setting was nearly the same with the existing measure. Among the empirical NMAs, 27 (69%) contained at least one multi-arm study. The existing measure was not applicable to them, while the proposed measures led to interpretable ICDFs in all NMAs.