Contrast-Space Projection for Network Meta-Analysis: An Exact and Invariant Study-Based Decomposition of Direct and Indirect Contributions

Abstract

Network meta-analysis (NMA) combines direct and indirect comparisons across a treatment network, but exact contribution decompositions that reproduce NMA estimates are lacking, especially for multi-arm trials with correlated contrasts. We develop a contrast-space projection formulation of NMA that expresses the estimator as a linear mapping of observed pairwise contrasts onto the consistency-constrained contrast space. Building on this representation, we define direct and indirect evidence through a canonical within-study reduction that removes algebraic redundancy and yields a unique, invariant study-level decomposition. The resulting covariance-aware weights exactly reconstruct the NMA estimator and can be further resolved into indirect path-level components. Under fixed effects, the same projection also represents the generalized Cochran Q decomposition into within-design heterogeneity and between-design inconsistency. The framework yields diagnostic and graphical tools, including forest plots, tension plots, and path-based visualizations. Applications to empirical networks illustrate how the approach provides a reproducible and interpretable account of evidence contributions in NMA.

arXiv comment43 pages, 6 figures. Revised version with expanded theoretical development, clarification of the representation-invariant study-based direct and indirect decomposition, additional results on canonical reduction and path construction, revised generalized Cochran Q diagnostics, and updated empirical illustrations and exposition