# Which evidence drives a network contrast changes across the plausible range of heterogeneity Ahmad Sofi-Mahmudi 2026-09-23 # Abstract **Background.** Contribution tables attribute a network meta-analysis contrast to its studies and comparisons ([1](#ref-papakonstantinou2018)) and feed certainty assessments. They are computed at the point estimate of the between-study SD $\tau$. Catalog problem DEC-17 asks whether they are stable across its plausible range. **Methods.** Three-treatment star and loop networks of 4, 8 or 16 two-arm studies, with equal or 5:1 within-study variances and true $\tau$ of 0, 0.1 or 0.25; absolute hat-matrix contribution shares for the indirect-or-mixed contrast, evaluated across the REML profile interval for $\tau$; 1000 networks per cell. **Results.** In the loop network with four studies and unequal within-study variances, the leading contributing comparison changed across the $\tau$ interval in 38% to 42% of networks, and some study’s share moved by 0.152 to 0.159 on average. With equal within-study variances shares did not move at all. The REML estimate of $\tau$ was zero in 30% to 67% of four-study networks and the interval’s upper end reached the grid’s cap in up to 62%. **Conclusion.** A contribution table is conditional on one value of $\tau$. With few studies of unequal precision it should be reported across the plausible range of $\tau$, or stated as conditional on the point estimate. # The problem Contribution shares are functions of the weights $(V + \tau^2I)^{-1}$. As $\tau$ grows the weights flatten, so precise studies lose share and imprecise ones gain, and the leading study or comparison can change. With few studies $\tau$ is poorly determined, so the share reported at $\hat\tau$ is one point in a wide range. With equal within-study variances the weights are proportional at every $\tau$ and shares cannot move. # Design Registered protocol: `protocol.md`. Treatments A, B, C; two-arm studies of AB and AC (star) or AB, AC and BC (loop), allocated in turn; within-study variances 0.04, or 0.02 to 0.10 assigned at random; target contrast C versus B. $\tau^2$ by REML on a grid over $[0, 1]$ with its 95% profile-likelihood interval. A study’s share is its absolute entry in the target row of the hat matrix over their sum; a comparison’s share sums its studies’. In the star network the target needs both comparisons with coefficient one, so comparison-level shares are exactly 50% at every $\tau$ and only study-level shares are informative (the comparison-level entries for the star in `results/decision.md` are ties broken by rounding and are not results). # Results
![](figures/fig1-leading.png) Figure 1: Share of networks whose leading contributor changes across the $\tau$ interval, 5:1 within-study variance spread.
Table 1: 5:1 within-study variance spread; with equal variances no share moved in any cell. 1000 networks per cell. | network | studies | true $\tau$ | leading comparison changes | leading study changes | mean largest share movement | leading study at $\hat\tau$ differs from true $\tau$ | $\hat\tau = 0$ | |----|---:|---:|---:|---:|---:|---:|---:| | star | 4 | 0.00 | tie | 0% | 0.125 | 0% | 65% | | loop | 4 | 0.00 | 42% | 42% | 0.159 | 6% | 67% | | star | 8 | 0.00 | tie | 15% | 0.095 | 2% | 62% | | loop | 8 | 0.00 | 35% | 31% | 0.104 | 7% | 61% | | star | 16 | 0.00 | tie | 14% | 0.050 | 2% | 58% | | loop | 16 | 0.00 | 1% | 10% | 0.058 | 3% | 56% | | star | 4 | 0.10 | tie | 0% | 0.126 | 0% | 57% | | loop | 4 | 0.10 | 41% | 41% | 0.157 | 11% | 55% | | star | 8 | 0.10 | tie | 18% | 0.096 | 5% | 48% | | loop | 8 | 0.10 | 35% | 30% | 0.103 | 10% | 50% | | star | 16 | 0.10 | tie | 16% | 0.054 | 4% | 36% | | loop | 16 | 0.10 | 2% | 13% | 0.061 | 5% | 35% | | star | 4 | 0.25 | tie | 0% | 0.122 | 0% | 37% | | loop | 4 | 0.25 | 38% | 38% | 0.152 | 15% | 36% | | star | 8 | 0.25 | tie | 18% | 0.093 | 8% | 19% | | loop | 8 | 0.25 | 31% | 27% | 0.101 | 10% | 17% | | star | 16 | 0.25 | tie | 16% | 0.048 | 4% | 4% | | loop | 16 | 0.25 | 0% | 6% | 0.055 | 1% | 5% |
The registered primary was confirmed (Figure 1, Table 1). Instability was greatest where $\tau$ is least determined and precision most unequal: four- and eight-study loop networks. With sixteen studies the interval for $\tau$ narrowed and the leading comparison was stable in almost every network, but the leading study still changed in 6% to 13% of them. Taking shares at $\hat\tau$ rather than at the true $\tau$ changed individual shares by only 0.01 to 0.05 on average; the problem is not that $\hat\tau$ is badly biased but that the data cannot distinguish values of $\tau$ that attribute the contrast differently. # What this does not answer One contribution measure (absolute hat-matrix row), not the shortest-path or random-walk flow decompositions; three treatments, two-arm studies; frequentist only, so no posterior distribution of contributions, information-matrix interval or bootstrap. Peer review has not been done. # References
1. Theodoros Papakonstantinou, Adriani Nikolakopoulou, Gerta Rücker, Anna Chaimani, Guido Schwarzer, Matthias Egger, Georgia Salanti. Estimating the contribution of studies in network meta-analysis: Paths, flows and streams. F1000Research. 2018;7:610.