Background:
Decision making about multiple alternative healthcare interventions often considers network meta-analysis (NMA) summaries. Limitations in the design and conduct of individual studies synthesized in NMA will reduce the confidence in the results: a given treatment comparison in the network may be directly or indirectly informed by studies at high risk of bias. It has been suggested that in order to infer upon study limitations in NMA effects, judgements on observed direct effect should be considered jointly with their contribution in each NMA effect. The key instrument in this approach is the percentage contribution matrix that shows how much each direct treatment effect contributes to each NMA treatment effect.

Objectives:
We aim to derive the percentage that each direct treatment effect contributes to each NMA treatment effect.

Methods:
To achieve this aim we use ideas from graph theory. We start with the ‘projection’ matrix in a two-step NMA model, called H matrix, which is analogous to the hat matrix in a regression model. A previous attempt to translate H entries to percentage contributions by normalising them was incorrect (Salanti 2014 PLOS One). We developed a novel method to translate H entries to percentage contributions based on the observation that the rows of H can be interpreted as flow networks. We present an algorithm that identifies the flow of evidence in each path and decomposes it to direct comparisons.

Results:
We use several published networks of interventions to illustrate the methodology and we show how important it is to consider evidence that is directed via more than one intermediate comparator. We exemplify how to infer further about risk of bias in NMA treatment effects. For instance, in a network of interventions of topical antibiotics without steroids for underlying eardrum perforations, the percentage contribution matrix in conjunction with 'Risk of bias' judgments produced Figure 1 where evidence at high risk of bias contributed more than 50% in the estimation of the ‘non-quilone antibiotic versus antiseptic’ NMA comparison.

Conclusions:
Derivation of percentage contributions of direct evidence to NMA treatment effects is important when examining the impact of trial characteristics to the results from NMA.

Patient or healthcare consumer involvement:
Not involved.