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

Network meta-analysis (NMA) combines direct and indirect comparisons across a connected treatment network to estimate relative treatment effects. However, there is a lack of exact contribution decompositions that reproduce NMA estimates, particularly in the presence of multi-arm trials that induce within-study correlations. We address this reproducibility gap by developing a contrast-space projection formulation of NMA. Working in the space of all estimable pairwise treatment contrasts, we express the NMA estimator as an explicit linear mapping of the observed contrasts onto the consistency-constrained contrast space induced by orthogonal projection. Building on this representation, we introduce a rigorous study-based definition of direct and indirect evidence through a canonical within-study reduction that removes algebraic redundancy and yields a unique, invariant decomposition. This leads to exact covariance-aware decompositions of the NMA estimator into study-level direct and indirect contributions, with indirect evidence further resolved into path-level components. The resulting weights are directly analogous to inverse-variance weights in pairwise meta-analysis and enable, to our knowledge, the first forest-plot representation that exactly reconstructs the NMA estimator. The framework also yields projection-based diagnostic and graphical tools, including forest plots, tension plots, and path-based visualizations. Applications to empirical datasets demonstrate how the proposed approach provides a reproducible and interpretable framework for understanding evidence contributions in network meta-analysis, supporting transparent interpretation and reporting.