Quantifying the contributions, or weights, of comparisons or single studies to the estimates in a network meta-analysis (NMA) is an active area of research. We extend this to the contributions of paths to NMA estimates. We present a general framework, based on the path-design matrix, that describes the problem of finding path contributions as a linear equation. The resulting solutions may have negative coefficients. We show that two known approaches, called shortestpath and randomwalk, are special solutions of this equation, and both meet an optimization criterion, as they minimize the sum of absolute path contributions. In general, there is an infinite space of solutions, which can be identified using the generalized inverse (Moore-Penrose pseudoinverse). We consider two further special approaches. For complex networks we find that shortestpath is superior with respect to run time and variability, compared to the other approaches, and is thus recommended in practice. The path-weights framework also has the potential to answer more general research questions in network meta-analysis.

翻译：量化网络荟萃分析（NMA）中比较或单项研究对估计值的贡献（即权重）是当前活跃的研究领域。本文将这一概念拓展至路径对NMA估计值的贡献。我们基于路径设计矩阵提出了一个通用框架，将寻找路径贡献问题描述为线性方程。所得解可能包含负系数。我们证明两种已知方法——最短路径法和随机游走法——是该方程的特解，且均满足优化准则，即最小化路径贡献绝对值之和。一般情况下，存在无穷解空间，可通过广义逆（Moore-Penrose伪逆）进行识别。我们进一步探讨了两种特殊解法。针对复杂网络分析发现，与其他方法相比，最短路径法在运行时间和变异性方面更具优势，因此推荐实际应用。该路径权重框架同样具有解决网络荟萃分析中更广泛研究问题的潜力。