Network meta-analysis and random walks

TL;DR

This work reinterprets network meta-analysis (NMA) through a random-walk lens, connecting NMA, electrical networks, and stochastic processes to derive closed-form proportion contributions and evidence streams. By equating hat-matrix evidence flow with random-walk edge traversals on an aggregate network and then on the directed evidence-flow network, the authors obtain an analytical, unambiguous computation of how direct and indirect evidence contribute to network treatment effects, including for multi-arm trials. The method overcomes ambiguities and path-dependence of prior iterative algorithms, demonstrates computational efficiency, and yields results that align with, and improve upon, existing approaches in real data (e.g., depression treatments). These results have practical implications for risk-of-bias assessments and evidence confidence tools (e.g., CINeMA, ROB-MEN) and offer a versatile framework for broader network-based evidence synthesis.

Abstract

Network meta-analysis (NMA) is a central tool for evidence synthesis in clinical research. The results of an NMA depend critically on the quality of evidence being pooled. In assessing the validity of an NMA, it is therefore important to know the proportion contributions of each direct treatment comparison to each network treatment effect. The construction of proportion contributions is based on the observation that each row of the hat matrix represents a so-called 'evidence flow network' for each treatment comparison. However, the existing algorithm used to calculate these values is associated with ambiguity according to the selection of paths. In this work we present a novel analogy between NMA and random walks. We use this analogy to derive closed-form expressions for the proportion contributions. A random walk on a graph is a stochastic process that describes a succession of random 'hops' between vertices which are connected by an edge. The weight of an edge relates to the probability that the walker moves along that edge. We use the graph representation of NMA to construct the transition matrix for a random walk on the network of evidence. We show that the net number of times a walker crosses each edge of the network is related to the evidence flow network. By then defining a random walk on the directed evidence flow network, we derive analytically the matrix of proportion contributions. The random-walk approach, in addition to being computationally more efficient, has none of the associated ambiguity of the existing algorithm.

Figures (8)

• Figure 1: A network of psychological treatments for depression (original data from Linde et al (2013) Linde:2013; presented in Rücker and Schwarzer (2014) Rucker:2014). We use numerical labels from $1$ to $11$, these are the same as in Rucker:2014. Two treatments are connected by an edge if a direct comparison of the two treatments was made in at least one trial; the edge width indicates the number of trials that make the comparison. The network contains one $4$-arm trial (comparing treatments 1-6-7-9), eight $3$-arm trials (3-5-9, 2-6-8, 1-6-11, 1-3-9, 2-6-11, 2-6-8, 3-6-9, and 3-4-9) and $17$$2$-arm trials. Multi-arm trials are not explicitly indicated in the network graph. The data, including the number of trials per comparison, is described in detail in Rücker and Schwarzer (2014) Rucker:2014.
• Figure 2: (a) A fictional example of an aggregate meta-analytic network with edges weighted and labelled by their respective (inverse-variance) weights. (b) The resulting evidence flow network for the comparison 1-2 from the aggregate network in (a); the comparison 1-2 is indicated by depicting these nodes and their labels in blue. Edges are directed according to the sign of the corresponding element of the hat matrix, and are weighted by the absolute value of the hat matrix element. (c) The random walk on the aggregate network in (a) for a walker starting at node 1 and finishing at node 2; edges are labelled by the associated transition probabilities.
• Figure 3: An illustration of the interpretation of current. (a) An electrical network with associated edge resistances. (b) The same network with a battery attached across the edge 1-2 such that a unit current flows into 1 and out of 2. The current in edge $cd$ is labelled $I_{cd}$. Current is measured in ampéres, hence the unit current is labelled as '1 Amp'. The direction of the current induced in the edges is shown. (c) A possible path taken by a random walker starting at node 1 and stopping at node 2. The sequence of nodes visited is $1\xrightarrow{}3\xrightarrow{}4\xrightarrow{}3\xrightarrow{}2$. For this particular realisation of the random walk, the net number of times the walker crosses edges 1-3 and 3-2 is one, while all other edges are crossed net zero times. The expected net number of times the walker crosses an edge is given by the currents shown in (b) for that edge Doyle:2000. The focus on the comparison of nodes 1 and 2 in panels (b) and (c) is indicated by the blue colour of these nodes.
• Figure 4: An illustration of a random walker moving on a network graph. The walker starts its journey from the far left node. The arrows show the path taken by the walker for one realisation of the random walk. The figure indicates the 'current' position of the walker as it hops between two nodes. The solid arrow indicates this transition. The dotted arrows indicate the previous transitions made between nodes by the walker.
• Figure 5: Illustration of evidence flow, streams of evidence and proportion contributions for a network of topical antibiotics without steroids for chronically discharging ears presented in Macfadyen (2005) Macfadyen:2005. Node 1 is no treatment; 2 is quinolone antibiotic; 3 is antiseptic; and 4 is non-quinolone antibiotic. (a) The evidence flow network for comparison 1-2, based on Figure 1, panel (b) in Papakonstantinou et al (2018)Papakon:2018. The edge labels are the entries of the 1-2 row of the hat matrix, their signs are associated with the direction of the arrows. (b) The decomposition of edge flows into flow through paths of evidence as estimated by the algorithm in Papakonstantinou et al. The paths of evidence shown are equivalent to the possible paths taken by a random walker on the evidence flow network. (c) The proportion contributions (expressed as percentages) of each direct treatment effect to the network estimate of the 1-2 relative treatment effect.