Borrowing intensity has no scale that spans sharing mechanisms
Statement
Synthesis models that borrow strength across populations, treatments, outcomes or study designs each impose some degree of sharing between direct and indirect evidence, and the parameters that nominally control it (the power-prior exponent, a mixture weight, a commensurability parameter, a between-class standard deviation) sit on incommensurable scales. A classification of the field sorted the available devices into four core relationships (functional, exchangeability-based, prior-based and multivariate) and recorded that the few studies assessing the degree of sharing consider precision gains only, not the movement of the point estimate, which is what changes a recommendation. Part of this is now solved: three metrics (point-estimate divergence, precision increase and Kullback-Leibler divergence) have been standardized to run from zero at splitting to one at lumping and used to place power priors, commensurate priors and exchangeability models on a common axis in one case study, where odds ratios ranged from 0.55 to 0.90 and incremental cost-effectiveness ratios from about 16,000 to 52,000 pounds per quality-adjusted life year across sharing models. That same work shows why the problem is not closed: the strength of sharing imposed by a given model varies across the three metrics, so there is no single invariant number, and Kullback-Leibler divergence is asymmetric and interpretable only comparatively. The metrics are demonstrated on one adult-to-children borrowing problem, are not defined per contrast within a network, have no published operating characteristics, and nothing requires them to be reported. The direction of borrowing is not guaranteed either: multivariate synthesis can borrow weakness, and across 40 published networks the network estimate was less precise than the direct estimate for 104 of 181 comparisons (57.5%) informed by a single head-to-head study.
Why it is open
Sharing intensity is a property of the model and the data together rather than of a parameter, so any scalar must be computed after fitting and depends on which functional of the posterior is being compared; point-estimate movement and precision gain can point in opposite directions and no argument privileges one of them. Anchoring at splitting and lumping fixes the endpoints but not the path between them, so two metrics can agree at zero and one and disagree everywhere in between, which is what is observed. Reparameterization invariance conflicts with interpretability: a divergence between posteriors is invariant but has no natural units, while a metric on an estimate has units but is scale-dependent. Sharing intensity also varies contrast by contrast with how much direct evidence each contrast carries, so a single model-level number is the wrong resolution for a decision that concerns one comparison. Finally, reporting the quantity can only make a modeling choice look more consequential than it did, so nothing pushes it into submissions.
What has been tried
| Work | What it contributes |
|---|---|
| Nikolaidis et al. 2021, Classifying information-sharing methods, BMC Medical Research Methodology | Classifies information-sharing methods into four core relationships, states that the few studies assessing the degree of sharing typically consider only precision gains rather than movement of the point estimate, and records that multivariate methods sometimes borrow weakness rather than strength. |
| Nikolaidis et al. 2025, Methods for information-sharing in network meta-analysis: Implications for inference and policy, Research Synthesis Methods | Implements the range of sharing methods on an intravenous immunoglobulin sepsis case study and supplies three metrics standardized between splitting and lumping; shows odds ratios from 0.55 to 0.90 and cost-effectiveness ratios from about 16,000 to 52,000 pounds per quality-adjusted life year across models, and that a given model’s sharing strength changes with the metric used to read it. |
| Jackson et al. 2017, Borrowing of strength and study weights in multivariate and network meta-analysis, Statistical Methods in Medical Research | Defines the borrowing of strength statistic as the percentage reduction in uncertainty attributable to indirect evidence, the one borrowing measure that is already defined per contrast in a network. |
| Lin et al. 2019, Borrowing of strength from indirect evidence in 40 network meta-analyses, Journal of Clinical Epidemiology | Applies the borrowing of strength statistic to 40 networks and 915 comparisons: 484 (53%) had no direct evidence, and among 181 comparisons with one direct study, 104 (57.5%) had reduced rather than increased precision under the network model. |
| Chen and Ibrahim 2000, Power prior distributions for regression models, Statistical Science | Defines the power prior, whose exponent is the archetype of a control parameter that quantifies sharing only within its own family. |
| Hobbs et al. 2011, Hierarchical Commensurate and Power Prior Models for Adaptive Incorporation of Historical Information in Clinical Trials, Biometrics | Introduces the commensurability parameter, a second and incompatible control scale for the same underlying quantity. |
| Morita, Thall and Muller 2008, Determining the Effective Sample Size of a Parametric Prior, Biometrics | Gives prior effective sample size, the closest thing the field has to a common unit for borrowing; it is defined for a prior rather than for a synthesis model and is extended to conditionally independent hierarchical models in the companion paper (doi 10.1214/12-BA720). |
| Zhang et al. 2025, Prior Effective Sample Size When Borrowing on the Treatment Effect Scale, Statistics in Medicine | Extends prior effective sample size to borrowing on the treatment-effect scale, showing the unit is being generalized but remains tied to prior-based devices rather than covering exchangeability or multivariate sharing. |
| Achana et al. 2014, Network meta-analysis of multiple outcome measures accounting for borrowing of information across outcomes, BMC Medical Research Methodology | Borrows across outcomes through a multivariate structure, one of the mechanisms whose sharing intensity is not comparable with a prior-based device. |
| Ye et al. 2023, When we shouldn’t borrow information from an existing network of trials for planning a new trial, Frontiers in Pharmacology | Identifies conditions under which borrowing degrades rather than improves an answer, which is what a reported sharing intensity would let a reader check. |
Probable solution or research direction
Treat the splitting-to-lumping anchoring as settled and attack the disagreement between metrics directly: characterize analytically when point-estimate divergence and precision increase order two models differently, and report both rather than continuing to search for one scalar. Define the metrics per contrast rather than per model, since sharing intensity varies with how much direct evidence each contrast carries; the borrowing of strength statistic already does this for network borrowing and supplies the template. Give the metrics operating characteristics: simulate a known discrepancy between the direct and the borrowed source and report how each metric behaves as the discrepancy grows, so that a reported value tells a reader something about the risk being taken rather than only about the model that was fitted. Require the sharing metric alongside the estimate in any submission that uses a prior-based or exchangeability-based device, which is the reporting step that turns a diagnostic into a check.
Verification
Verdict. Partially addressed. One reader proposed this and the reviewer accepted it on the condition that it be framed as the absence of a model-agnostic, contrast-level scale rather than the absence of any borrowing quantity. It is partially addressed because a 2025 case study supplies three standardized sharing metrics anchored at splitting and lumping, including one based on point-estimate movement; what remains is that the metrics disagree, are demonstrated once, are not defined per contrast, and are required by nobody.
Revised by the full-text reading. Applied 2026-07-26 full-text reading of the consolidated library (687 papers).
- Citation corrected
- normalized a citation year: ‘Jackson et al. 2015, Borrowing of strength and study weights in multivariate and network meta-analysis, Statistical Methods in Medical Research’ -> ‘Jackson et al. 2017, Borrowing of strength and study weights in multivariate and network meta-analysis, Statistical Methods in Medical Research’ 10.1177/0962280215611702 is cited with two different years across the registry. CrossRef gives 2015 online and 2017 in print; the issue year is used throughout.
Source
Derived from reading:nma-20 of the reviewed corpus.