# A baseline-risk bridge carries design nuisance across a disconnected
network as if it were prognosis
Ahmad Sofi-Mahmudi
2026-09-23

# Abstract

**Background.** Disconnected networks are sometimes linked by assuming
that study baseline risks are exchangeable across subnetworks. Catalog
problem DIS-03 asks whether design differences between subnetworks, such
as endpoint definitions, pass into the cross-gap contrast and whether
design covariates can separate them from prognosis.

**Methods.** Two subnetworks of 5 or 10 trials; study baselines with
prognostic variation, possible prognostic drift between subnetworks (0
or 0.2 on the log odds scale) and a design nuisance of 0, 0.2 or 0.5
attached to a binary design covariate that either coincided with the
subnetwork or was used by a quarter of the other subnetwork’s trials: 24
scenarios, 2000 replicates each. Exchangeable bridge with and without
adjustment for the design covariate.

**Results.** When the design covariate coincided with the subnetwork,
adjustment was impossible and the bridge carried the nuisance one for
one (bias 0.207 and 0.500 for nuisance 0.2 and 0.5; coverage 0.084).
With a quarter of trials crossing over, adjustment removed the nuisance
(bias -0.000) but not prognostic drift, which stayed at 0.204.

**Conclusion.** A baseline bridge transports whatever makes the
subnetworks’ baselines differ. Design covariates separate nuisance from
prognosis only if they vary within subnetworks; prognostic differences
between subnetworks remain unidentified. Report how much of each
cross-gap contrast the bridge supplies and treat it as an assumption,
not an estimate.

# The problem

Study baselines are $\mu_s = \mu + b_s + \nu_s$: prognosis $b_s$, which
the bridge is meant to carry, plus design nuisance $\nu_s$ from endpoint
definition, follow-up or ascertainment. The likelihood sees only
$\mu_s$, and the contrast that would separate $b$ from $\nu$ spans the
gap. A design covariate identifies $\nu$ only where it varies within a
subnetwork.

# Design

Registered protocol: `protocol.md`; ADEMP structure
([1](#ref-morris2019)). Subnetwork 1: A versus B; subnetwork 2: C versus
D. Arm-level log odds $\mu_s + d_t$ with arm-level SE 0.15;
$d_C - d_A = 0.2$. The exchangeable bridge estimates $d_C - d_A$ as the
difference in mean control-arm log odds between subnetworks; the
adjusted bridge regresses control-arm log odds on subnetwork and the
design covariate.

# Results

<div id="fig-bias">

![](figures/fig1-bias.png)

Figure 1: Bias of the cross-gap contrast by design nuisance, with and
without adjustment, ten trials per subnetwork. The dotted line is bias
equal to the nuisance.

</div>

Without crossover the unadjusted bias lay on the identity line
(<a href="#fig-bias" class="quarto-xref">Figure 1</a>), plus any
prognostic drift. With crossover the unadjusted bias was diluted but not
removed, and adjustment removed exactly the part attributable to the
design covariate. Prognostic drift is by definition a baseline
difference with no design signature, so no covariate in the network
could remove it.

# What this does not answer

Study-level summaries on the log odds scale rather than an arm-based
Bayesian model; no commensurate prior; one design covariate. Peer review
has not been done.

# References

<div id="refs" class="references csl-bib-body">

<div id="ref-morris2019" class="csl-entry">

<span class="csl-left-margin">1.
</span><span class="csl-right-inline">Tim P. Morris, Ian R. White,
Michael J. Crowther. Using simulation studies to evaluate statistical
methods. Statistics in Medicine. 2019;38(11):2074–102.
doi:[10.1002/sim.8086](https://doi.org/10.1002/sim.8086)</span>

</div>

</div>