Estimating the contribution of studies in network meta-analysis: paths, flows and streams

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 proportion 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 proportion 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 proportion 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 proportion contributions of direct evidence from individual studies to network treatment effects.

Keywords: flow networks; indirect evidence; projection matrix; proportion contributions.

Figure 1.. Network plot for the network of topical antibiotics without steroids for chronically discharging ears (a), comparison graph corresponding to the h ^xy row of H matrix (b), flows f _uv with respect to the ‘ x versus y’ network meta-analysis treatment effect are indicated along the edges), streams (c) and proportion contributions of each direct comparison (d).

x, no treatment; y, quinolone antibiotic; u, non-quinolone antibiotic; v, antiseptic.

Figure 2.. Illustration of the steps of the algorithm for approximating proportion contributions per comparison in the network of topical antibiotics without steroids for chronically discharging ears focusing on the comparison ‘ x versus y’.

Treatment labels: x, no treatment; y, quinolone antibiotic; u, non-quinolone antibiotic; v, antiseptic.

Figure 3.. Bar plot showing the study proportion contributions of direct comparisons with low (green), moderate (yellow) and high (red) risk of bias.

The bar plot has been produced in CINeMA (Confidence In Network Meta-Analysis) software . Studies are synthesized using the random effects model. x, no treatment; y, quinolone antibiotic; u, non-quinolone antibiotic; v, antiseptic.

Figure 4.. Network plot for the network of antimanic drugs.

ASE, asenapine; ARI, aripiprazole; PLA, placebo; HAL, haloperidol; QUE, quetiapine; LITH, lithium; ZIP, ziprasidone; OLA, olanzapine; DIV, divalproex; RIS, risperidone; CARB, carbamazepine; LAM, lamotrigine; PAL, paliperidone; TOP, topiramate; ASE, asenapine.