# Sharing one heterogeneity parameter across a bridged network cost little; the interval method mattered more Ahmad Sofi-Mahmudi 2026-09-23 # Abstract **Background.** A bridged network contrast pools two subnetworks, and a single between-study SD $\tau$ lends one subnetwork’s heterogeneity to the other. Catalog problem CMP-16 asks whether that breaks the bridged contrast’s interval and whether a stratified $\tau$ does better at realistic study counts. **Methods.** A bridged contrast formed as the sum of two subnetworks’ pooled effects; heterogeneity ratio 1, 2 or 5 in the bridging or the other subnetwork; 3, 6 or 12 studies each; shared, stratified and shrunk REML heterogeneity with Wald intervals, and Hartung-Knapp intervals; 2000 replicates per cell. **Results.** With the non-bridging subnetwork five times as heterogeneous, the shared model’s Wald interval covered 0.906, 0.911 and 0.932 at 3, 6 and 12 studies per subnetwork, against 0.917, 0.927 and 0.943 for the stratified model: at most 0.016 apart. Every Wald interval undercovered at few studies, and Hartung-Knapp intervals covered 0.974 to 0.997 at three studies, at 1.5 times the width. **Conclusion.** For a contrast that sums both subnetworks, a shared $\tau$ overstates one part’s variance and understates the other’s, and the errors largely offset. The undercoverage comes from ignoring uncertainty in $\tau$, not from sharing it. # The problem A shared $\tau$ inflates the homogeneous subnetwork’s variance and deflates the heterogeneous one’s. For a contrast that uses only one subnetwork this misstates its interval directly; for a bridged contrast that sums both, the two errors enter with opposite signs. Stratifying $\tau$ fixes the structure but estimates two poorly determined variances instead of one. # Design Registered protocol: `protocol.md`. $K$ studies of component A against its backbone (effect $-0.3$) and $K$ of B against A (0.1); the target is their sum. Within-study SE uniform on 0.1 to 0.25; between-study SDs 0.05 in one subnetwork and $0.05R$ in the other. Shared, stratified and shrunk (weight $K/(K+4)$ toward the shared value) REML heterogeneity; Wald intervals, and Hartung-Knapp intervals ([1](#ref-hartung2001)) for shared and stratified. # Results
![](figures/fig1-coverage.png) Figure 1: Coverage of the bridged contrast at a 5:1 heterogeneity ratio.
Table 1: Coverage of 95% intervals, 2000 replicates per cell (MCSE at most 0.007). | ratio | heterogeneous | studies each | shared | stratified | shrunk | shared, Hartung-Knapp | stratified, Hartung-Knapp | |---:|----|---:|---:|---:|---:|---:|---:| | 1 | bridging | 3 | 0.955 | 0.960 | 0.958 | 0.997 | 0.997 | | 2 | bridging | 3 | 0.944 | 0.951 | 0.948 | 0.991 | 0.992 | | 5 | bridging | 3 | 0.897 | 0.906 | 0.903 | 0.978 | 0.979 | | 2 | other | 3 | 0.941 | 0.957 | 0.948 | 0.997 | 0.997 | | 5 | other | 3 | 0.906 | 0.917 | 0.913 | 0.974 | 0.974 | | 1 | bridging | 6 | 0.951 | 0.957 | 0.957 | 0.979 | 0.978 | | 2 | bridging | 6 | 0.943 | 0.950 | 0.947 | 0.976 | 0.973 | | 5 | bridging | 6 | 0.925 | 0.939 | 0.932 | 0.967 | 0.966 | | 2 | other | 6 | 0.950 | 0.961 | 0.958 | 0.978 | 0.977 | | 5 | other | 6 | 0.911 | 0.927 | 0.922 | 0.964 | 0.960 | | 1 | bridging | 12 | 0.954 | 0.955 | 0.955 | 0.968 | 0.968 | | 2 | bridging | 12 | 0.950 | 0.951 | 0.952 | 0.968 | 0.967 | | 5 | bridging | 12 | 0.930 | 0.935 | 0.934 | 0.957 | 0.948 | | 2 | other | 12 | 0.942 | 0.945 | 0.947 | 0.961 | 0.961 | | 5 | other | 12 | 0.932 | 0.943 | 0.942 | 0.965 | 0.956 |
The registered rule returned mixed: in the registered cells (non-bridging subnetwork heterogeneous) the shared model never covered below 0.90, and the shrunk model was not nominal where the shared one fell short (Table 1); the lowest shared coverage, 0.897, was with the bridging subnetwork heterogeneous. Across the grid the three structures differed by at most 0.016 in coverage and a few percent in width (Figure 1). The gap to nominal was driven by the heterogeneity ratio and the study count, not by the structure: at a 5:1 ratio and three studies per subnetwork every Wald interval covered 0.90 to 0.92, rising to 0.93 to 0.94 at twelve. The shared estimate of $\tau$ sat between the two true values, as it must, and the bridged contrast’s variance, which sums both subnetworks, was close to right on average. # What this does not answer The bridged contrast here sums both subnetworks; a contrast drawing on one subnetwork only would carry the shared $\tau$’s error without the offset. Frequentist REML without priors; no residual-variance arm, IPD studies or sparse network. Peer review has not been done. # References
1. Joachim Hartung, Guido Knapp. On tests of the overall treatment effect in meta-analysis with normally distributed responses. Statistics in Medicine. 2001;20(12):1771–82. doi:[10.1002/sim.791](https://doi.org/10.1002/sim.791)