Network meta-analysis

A network meta-analysis jointly compares several treatments using a graph of direct Randomized controlled trials and indirect treatment comparisons. If effects are measured relative to a reference treatment, consistency means ${\delta }_{ij}={\delta }_{i0}-{\delta }_{j0}$. A connected graph identifies all relative effects under that model. Transitivity in network meta-analysis is the substantive comparability assumption supporting these relations; inconsistency can be assessed when the network has loops.

For independent studies comparing treatments $A$ and $B$ to common control $C$, a consistent log odds ratio comparison uses ${\hat{\delta }}_{AB}={\hat{\delta }}_{AC}-{\hat{\delta }}_{BC}$, with variance $v_{AC}+v_{BC}$. This follows from subtracting two estimates of additive effects relative to the same reference. With overlapping evidence, subtract twice their covariance. Causal comparability across the studies is supplied by transitivity in network meta-analysis, not by the variance calculation.

Transitivity requires the studies of different treatment comparisons to be sufficiently comparable in distributions of effect modifiers, outcome definitions and other design features to support an indirect treatment comparison in one target population. For example, a treatment that works differently by disease severity cannot safely be compared indirectly across trials with systematically different severity distributions. Statistical consistency is an implication of suitable transitivity and modeling assumptions, not a substitute for assessing them.