Abstract
In network meta-analysis, it is important to assess the influence of the limitations or other characteristics of individual studies on the estimates obtained from the network. The percentage contribution matrix, which shows how much each direct treatment effect contributes to each treatment effect estimate from network meta-analysis, is crucial in this context. We use ideas from graph theory to derive the percentage that is contributed by each direct treatment effect. We start with the 'projection' matrix in a two-step network meta-analysis model, called the H matrix, which is analogous to the hat matrix in a linear regression model. We develop a method to translate H entries to percentage contributions based on the observation that the rows of H can be interpreted as flow networks, where a stream is defined as the composition of a path and its associated flow. We present an algorithm that identifies the flow of evidence in each path and decomposes it into direct comparisons. To illustrate the methodology, we use two published networks of interventions. The first compares no treatment, quinolone antibiotics, non-quinolone antibiotics and antiseptics for underlying eardrum perforations and the second compares 14 antimanic drugs. We believe that this approach is a useful and novel addition to network meta-analysis methodology, which allows the consistent derivation of the percentage contributions of direct evidence from individual studies to network treatment effects.
Highlights
Decision making around multiple alternative healthcare interventions is increasingly based on meta-analyses of a network of relevant studies, which contribute direct and indirect evidence to different treatment comparisons[1,2]
Taking advantage of previous findings on how the flow of evidence can be considered in NMA4,6, we present an algorithm to decompose the flow in a network and subsequently approximate the percentage contributions of direct effect estimates for each network metaanalysis (NMA) effect estimate
Indirect effects are not weighted averages, we find this approximation to be a pragmatic approach that reasonably reflects the amount that each comparison contributes to network effects
Summary
Decision making around multiple alternative healthcare interventions is increasingly based on meta-analyses of a network of relevant studies, which contribute direct and indirect evidence to different treatment comparisons[1,2]. A relative treatment effect from NMA (hereafter the NMA effect estimate) is estimated as a linear combination of the available direct estimates of the treatment effect (i.e. the results from pairwise meta-analyses) and the indirect evidence on the treatment effect. Salanti et al suggested that in order to assess the impact of study deficiencies on an NMA effect estimate, the limitations of studies contributing to direct estimates should be considered jointly, taking into account their relative contribution to the overall NMA effect estimate[3]. The absolute contributions of direct effects to an NMA effect is the projection matrix from a two-step NMA model[4,5]. All direct effects are derived from pairwise meta-analyses. The NMA effect estimates are produced as a linear combination of the derived direct effects. The elements in the H matrix can be viewed as generalized weights from pairwise meta-analysis, but they do not add up to 1 and depend on the precision of the available studies, the degree of between-study heterogeneity and the network structure
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