Degrees of freedom and inconsistency-generating mechanisms in global inconsistency testing: a simulation comparison of DBTI and Loop models in NMA

The design-by-treatment interaction (DBTI) and Loop models assess global inconsistency in frequentist network meta-analysis using different parameterizations. This study examined their theoretical relationship and compared their performance across network configurations and inconsistency-generating mechanisms. We characterized the relationship between the models by matching their design-specific mean effects. We defined the differences in degrees of freedom between the two models as Δ d f = d f DBTI − d f Loop and constructed simulation scenarios with varying values of Δ d f in networks with three, five, and eight treatments. Binary data were generated under DBTI-based and Loop-based inconsistency specifications. Both models were fitted using restricted maximum likelihood, and global inconsistency was assessed using joint Wald tests. Performance was evaluated through empirical size, power, and bias in heterogeneity variance estimation. When Δ d f = 0, the models were equivalent up to reparameterization under the same covariance assumptions and produced nearly identical results. When Δ d f > 0, the Loop model imposed additional constraints on the inconsistency patterns represented by the DBTI model. Under the Loop-based generating mechanism, power differences were generally small, with no consistent advantage for the Loop model. Under the selected DBTI-based configurations, the DBTI test had higher rejection rates, whereas the Loop model exhibited upward bias in heterogeneity variance estimates at larger inconsistency values. Empirical size was elevated in some scenarios, complicating power comparisons. Fewer degrees of freedom for inconsistency did not consistently yield higher empirical power in the simulated networks. The selection and interpretation of global inconsistency tests should consider both model complexity and the inconsistency structures each model can represent, alongside network configuration and heterogeneity estimates.