Evidence contribution has no definition inside a Bayesian fit
Statement
The proportion contribution matrix, evidence flow and per-study importance are all defined from the frequentist two-stage projection matrix H, and no analogue to H exists in Bayesian network meta-analysis; the derivation states that limitation itself and offers the construction as a source of useful approximations rather than as a Bayesian quantity. This matters beyond bookkeeping: CINeMA’s per-comparison within-study-bias and indirectness judgments are computed from that matrix, so the GRADE-style certainty machinery the field relies on is frequentist-only. Bayesian synthesis is the framework used for most health technology assessment submissions and the only one available for multilevel network meta-regression and component models, so contributions there are obtained by refitting the network frequentist-style and importing the answer. Nothing validates that the imported number approximates how the Bayesian posterior actually depends on each study, and no study characterizes the discrepancy as a function of the heterogeneity prior, network sparsity or prior informativeness. The nearest Bayesian-native quantity is a posterior-to-prior variance ratio for a disconnected contrast, which measures how far the data move the prior but does not attribute the movement to studies or comparisons. Every subsequent contribution method, including shortest-path and leave-one-out contribution analysis for component networks, extends the frequentist construction rather than replacing it. There is therefore no posterior-native, validated study or comparison contribution measure, and no reported column for the prior, which in a Bayesian fit is itself a source of the estimate.
Why it is open
H exists because the frequentist network estimate is a fixed linear map of the direct estimates; a posterior is not a linear function of the data, so there is no matrix to decompose and no unique way to divide the posterior among studies. Under a random-effects model with a prior on heterogeneity the weight a study receives is itself a random quantity, so any contribution figure is a posterior summary and requires a prior decision about which functional is being summarized. The candidate replacements, namely leave-one-study-out posterior distance, Kullback-Leibler divergence and prior-to-posterior variance decomposition, each answer a different question, none is on the same zero-to-one-hundred-percent scale as the contribution matrix, and none has published operating characteristics. Meanwhile the frequentist import is cheap and produces a number CINeMA accepts, so nothing in practice forces the harder derivation. The problem is also invisible in output: a contribution table computed by refitting looks identical to one computed from the model that produced the reported estimates.
What has been tried
| Work | What it contributes |
|---|---|
| Papakonstantinou et al. 2018, Estimating the contribution of studies in network meta-analysis: paths, flows and streams, F1000Research | Derives the proportion contribution matrix from the frequentist hat matrix and states that deriving the contribution of evidence sources in a Bayesian network meta-analysis is not straightforward because no analogue to H exists, offering the method only as a source of useful approximations. |
| Papakonstantinou et al. 2020, In network meta-analysis, most of the information comes from indirect evidence: empirical study, Journal of Clinical Epidemiology | Applies the contribution matrix across 213 published network meta-analyses, establishing it as the field’s working provenance quantity; the computation is frequentist throughout. |
| Nikolakopoulou et al. 2020, CINeMA: An approach for assessing confidence in the results of a network meta-analysis, PLOS Medicine | Builds per-comparison confidence judgments for within-study bias and indirectness on the contribution matrix, which is what makes the frequentist-only construction consequential for certainty assessment. |
| Rucker et al. 2020, The statistical importance of a study for a network meta-analysis estimate, BMC Medical Research Methodology | Defines study importance for a network estimate through the change induced by removing a study, again from the frequentist design rather than from a posterior. |
| Beliveau and Gustafson 2021, A Theoretical Investigation of How Evidence Flows in Bayesian Network Meta-Analysis of Disconnected Networks, Bayesian Analysis | Supplies the closest Bayesian-native evidence-flow quantity, the ratio of posterior to prior variance for a disconnected contrast, and shows evidence cannot flow across a true disconnection under weak priors; it measures prior versus data influence and does not attribute information to individual studies. |
| Yang et al. 2026, Evidence contributions in component network meta-analysis from the shortest-path approach, BMC Medical Research Methodology | Extends evidence-contribution tracing to component network meta-analysis by adapting the shortest-path decomposition, keeping the frequentist projection as the foundation. |
| Mao et al. 2025, A leave-one-out algorithm for contribution analysis in component network meta-analysis, BMC Medical Research Methodology | Gives a leave-one-out route to contribution in component networks, showing the alternative constructions the field is exploring are still defined on the frequentist fit. |
| Donegan et al. 2018, Graphs of study contributions and covariate distributions for network meta-regression, Research Synthesis Methods | Displays study contributions alongside covariate distributions for network meta-regression, the reporting form a Bayesian contribution measure would have to reproduce. |
Probable solution or research direction
Define contribution as a functional of the posterior rather than of a design matrix: for each decision contrast report the movement of the posterior when a study is removed or downweighted, anchored so that removing nothing scores zero and removing all direct evidence for that contrast scores one, using a symmetric divergence rather than Kullback-Leibler alone. The standardized information-sharing metrics that anchor a borrowing measure between splitting and lumping supply a working template for that anchoring and can be adapted from population sharing to study-level attribution within a network. Validate by simulating networks in which the frequentist contribution is known, computing the posterior-based quantity, and reporting the discrepancy as a function of the heterogeneity prior, network sparsity and prior informativeness, which also tells current CINeMA users how much error the frequentist import carries. Add an explicit prior column, since in a Bayesian fit the prior contributes information and the frequentist matrix has no cell for it.
Verification
Verdict. Confirmed open. One reader proposed this and the reviewer accepted it on the ground that the quoted paper states the absence of an H analogue and characterizes what it offers as an approximation rather than a Bayesian contribution calculation. The evidence is a single methods paper, which is a weaker basis than a survey, although the frequentist-only construction is visible in every contribution method published since.
Source
Derived from reading:nma-31 of the reviewed corpus.