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Res Synth Methods. 2012 Dec;3(4):312-24. doi: 10.1002/jrsm.1058. Epub 2012 Sep 25.

Network meta-analysis, electrical networks and graph theory.

Author information

1
Institute of Medical Biometry and Medical Informatics, University Medical Center Freiburg, Freiburg, Germany. ruecker@imbi.uni-freiburg.de.

Abstract

Network meta-analysis is an active field of research in clinical biostatistics. It aims to combine information from all randomized comparisons among a set of treatments for a given medical condition. We show how graph-theoretical methods can be applied to network meta-analysis. A meta-analytic graph consists of vertices (treatments) and edges (randomized comparisons). We illustrate the correspondence between meta-analytic networks and electrical networks, where variance corresponds to resistance, treatment effects to voltage, and weighted treatment effects to current flows. Based thereon, we then show that graph-theoretical methods that have been routinely applied to electrical networks also work well in network meta-analysis. In more detail, the resulting consistent treatment effects induced in the edges can be estimated via the Moore-Penrose pseudoinverse of the Laplacian matrix. Moreover, the variances of the treatment effects are estimated in analogy to electrical effective resistances. It is shown that this method, being computationally simple, leads to the usual fixed effect model estimate when applied to pairwise meta-analysis and is consistent with published results when applied to network meta-analysis examples from the literature. Moreover, problems of heterogeneity and inconsistency, random effects modeling and including multi-armed trials are addressed.

KEYWORDS:

Laplacian matrix; Moore-Penrose pseudoinverse; electrical network; graph theory; network meta-analysis

PMID:
26053424
DOI:
10.1002/jrsm.1058

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