A comparison of two models for detecting inconsistency in network meta-analysis

The application of network meta-analysis is becoming increasingly widespread, and for a successful implementation, it requires that the direct comparison result and the indirect comparison result should be consistent. Because of this, a proper detection of inconsistency is often a key issue in network meta-analysis as whether the results can be reliably used as a clinical guidance. Among the existing methods for detecting inconsistency, two commonly used models are the design-by-treatment interaction model and the side-splitting models. While the original side-splitting model was initially estimated using a Bayesian approach, in this context, we employ the frequentist approach. In this paper, we review these two types of models comprehensively as well as explore their relationship by treating the data structure of network meta-analysis as missing data and parameterizing the potential complete data for each model. Through both analytical and numerical studies, we verify that the side-splitting models are specific instances of the design-by-treatment interaction model, incorporating additional assumptions or under certain data structure. Moreover, the design-by-treatment interaction model exhibits robust performance across different data structures on inconsistency detection compared to the side-splitting models. Finally, as a practical guidance for inconsistency detection, we recommend utilizing the design-by-treatment interaction model when there is a lack of information about the potential location of inconsistency. By contrast, the side-splitting models can serve as a supplementary method especially when the number of studies in each design is small, enabling a comprehensive assessment of inconsistency from both global and local perspectives.