Assessing the consistency assumptions underlying network meta-regression using aggregate data

When numerous treatments exist for a disease (Treatments 1, 2, 3, etc), network meta-regression (NMR) examines whether each relative treatment effect (eg, mean difference for 2 vs 1, 3 vs 1, and 3 vs 2) differs according to a covariate (eg, disease severity). Two consistency assumptions underlie NMR: consistency of the treatment effects at the covariate value 0 and consistency of the regression coefficients for the treatment by covariate interaction. The NMR results may be unreliable when the assumptions do not hold. Furthermore, interactions may exist but are not found because inconsistency of the coefficients is masking them, for example, when the treatment effect increases as the covariate increases using direct evidence but the effect decreases with the increasing covariate using indirect evidence. We outline existing NMR models that incorporate different types of treatment by covariate interaction. We then introduce models that can be used to assess the consistency assumptions underlying NMR for aggregate data. We extend existing node-splitting models, the unrelated mean effects inconsistency model, and the design by treatment inconsistency model to incorporate covariate interactions. We propose models for assessing both consistency assumptions simultaneously and models for assessing each of the assumptions in turn to gain a more thorough understanding of consistency. We apply the methods in a Bayesian framework to trial-level data comparing antimalarial treatments using the covariate average age and to four fabricated data sets to demonstrate key scenarios. We discuss the pros and cons of the methods and important considerations when applying models to aggregated data.