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

arXiv: https://arxiv.org/abs/2604.21994 · HTML: https://mlsanity.com/paper/2604.21994/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 25 references that are in the map.
- First posted on arXiv 23 Apr 2026.

## Builds on (most cited first, 1 of 1)

- [Network meta-analysis and random walks](https://mlsanity.com/paper/2107.02886/) (2021)

## Abstract (arXiv)

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.