Overdispersed counts and recurrent events as total disease burden
Overdispersion leaves the transported rate ratio unbiased and makes the Poisson interval too narrow (coverage down to 0.61); a sandwich or negative binomial restores it. A structural-zero fraction that differs between source and target biases the transported absolute rate in every count model, including zero-inflated Poisson, by up to 0.56 on the log scale; the rate ratio is protected only without effect modification. Recalibrating the zero part to the target’s reported zero proportion repairs the absolute rate when the source has structural zeros to learn from. Read the study below.
Statement
The population-adjusted implementations the auditors checked fit Poisson models for counts and offer no negative-binomial, Poisson-lognormal, hurdle, zero-inflated, recurrent-event or frailty family, so those options are unavailable rather than merely uncommon. The consequence is narrower than the source said: where the conditional mean is correctly specified the Poisson estimating score remains unbiased under negative-binomial variance, so residual overdispersion invalidates likelihood-based uncertainty while leaving rate-ratio point estimates consistent, and transport bias instead requires misspecification of the conditional mean, the zero mechanism, the frailty or the exposure weighting. Time to first event and total event burden remain different estimands that can move in different directions, and general count meta-analysis evidence shows pooled effects and precision differ by whether a rate ratio, Poisson, negative-binomial, dichotomized risk ratio or time-to-first-event analysis is used (doi:10.1186/s13643-015-0144-x). The two auditors disagreed about how much of the source’s mechanism survives, and that disagreement is recorded rather than resolved.
Why it is open
No population-adjusted implementation the auditors checked offers an overdispersed, zero-inflated or recurrent-event likelihood, so an analyst facing exacerbation or relapse counts either accepts Poisson uncertainty or leaves the population-adjusted framework. On the survival side, transporting a recurrent-event effect requires transporting a frailty distribution that is itself population-dependent, and aggregate-data trials report first-event Kaplan-Meier curves or annualized rates rather than the full event history, so the joint and dispersion structure cannot be recovered from what is published. A structural-zero subgroup makes the population a mixture whose mixing proportion can vary across trial populations, which is a transport problem and not only a fitting problem. Whether the closed-form-likelihood barrier that blocks non-standard count likelihoods in ML-NMR has already been removed is an open question the literature auditor flagged but did not check.
What has been tried
| Work | What it contributes |
|---|---|
| Do alternative methods for analysing count data produce similar estimates? Implications for meta-analyses, Systematic Reviews 2015 | shows pooled effects and precision differ according to whether counts are analyzed as a rate ratio, Poisson, negative binomial, dichotomized risk ratio or time to first event, supporting both the Poisson-restriction claim and the claim that first-event and total-burden analyses are different estimands |
| multinma (CRAN), nma() likelihoods | the literature auditor verified from the nma() reference that the supported count and rate likelihoods are binomial, Bernoulli, Poisson and normal plus the survival families, with no negative-binomial, Poisson-lognormal, hurdle or zero-inflated option |
| cpaic, unreleased development package on GitHub (not on CRAN), commit 9d150e9 | cmlnmr() accepts only binomial, Gaussian, Poisson and survival families, with no negative-binomial, Poisson-lognormal, hurdle, zero-inflated, recurrent-event or frailty path in NAMESPACE or the Stan models; the code auditor quotes its note that under overdispersion the rate-ratio point estimates stay consistent while the uncertainty does not |
| Don’t Disregard the Data for Lack of a Likelihood: Bayesian Synthetic Likelihood for Enhanced Multilevel Network Meta-Regression (arxiv:2603.11019) | named by the literature auditor as a lead they did not read, so its contents are unverified here; if it does what its title states it would remove the closed-form-likelihood barrier that currently blocks non-standard count likelihoods in ML-NMR |
Probable solution or research direction
Extend component ML-NMR to negative-binomial and Poisson-lognormal likelihoods so overdispersion is modeled rather than left to invalidate the reported interval, and add hurdle and zero-inflated formulations whose zero and count components are permitted, not required, to carry different effect-modification structure. Develop recurrent-event and joint frailty models for population-adjusted network synthesis so total burden and time to first event can both be estimated and standardized to the target population, and connect them to the count models for the same underlying process. Check the lead the literature auditor named but did not read, Bayesian synthetic likelihood for multilevel network meta-regression (arxiv:2603.11019), before rebuilding integration machinery, since if it does what its title says it removes the closed-form-likelihood constraint.
Our study
Overdispersed and zero-inflated counts transported by G-computation
The question put to it. For count outcomes transported to a target, does overdispersion bias the estimate or only its interval, and when the fraction of structural zeros differs between populations, what does a count model transport wrongly and what repairs it?
What it found. Overdispersion leaves the transported rate ratio unbiased and makes the Poisson interval too narrow (coverage down to 0.61); a sandwich or negative binomial restores it. A structural-zero fraction that differs between source and target biases the transported absolute rate in every count model, including zero-inflated Poisson, by up to 0.56 on the log scale; the rate ratio is protected only without effect modification. Recalibrating the zero part to the target’s reported zero proportion repairs the absolute rate when the source has structural zeros to learn from.
Results.
- Poisson rate-ratio coverage 0.846 to 0.610 under overdispersion with bias at most 0.010; sandwich 0.926 to 0.948, so the registered rule (0.93 everywhere) was missed by 0.004 in one cell with MCSE 0.012.
- Differing structural-zero fractions: absolute log-rate bias 0.40 to 0.56 in absolute value for Poisson, negative binomial and zero-inflated Poisson, coverage 0.000 to 0.028.
- Rate ratio with differing zero fractions: bias at most 0.007 without effect modification, up to 0.11 with it.
- Zero part recalibrated to the target’s zero proportion: absolute-rate bias at most 0.04 when the source had structural zeros, 0.25 when it had none.
Design.
- Estimand
- Log rate ratio, A versus C, and log rate under A, in the target population
- Methods compared
- Poisson, Poisson with sandwich variance, negative binomial, zero-inflated Poisson, zero-inflated Poisson with recalibrated zero part
- Data-generating mechanism
- G-computation from an individual-data trial with negative-binomial or zero-inflated Poisson counts, 14 cells separating dispersion from structural-zero differences, with and without effect modification; Poisson, sandwich, negative binomial, zero-inflated and recalibrated zero-inflated fits. 500 replicates per cell.
- Replicates
- 500 per scenario, 7,000 in total
What this does not answer. One covariate; frailty, exposure weighting, recurrent events and zero-inflated negative binomial not run. Peer review has not been done.
Peer review. Not yet done. The result stands on the study alone until it is.
Read it: PDF · Markdown · OpenDocument · Code and data: studies/OUT-08-overdispersion-breaks-the-interval-not-the-estimate · Protocol: registered before the run
Verification
Verdict. Unverifiable. Auditors disagreed and the disagreement was evidence-bearing on both sides.
The source said. “Repeated clinical events such as exacerbations or relapses are analyzed either as Poisson counts, which forces variance to equal the mean and ignores excess zeros, or as time to first event, which discards the recurrent process entirely. Neither negative-binomial and zero-inflated component ML-NMR nor recurrent-event and joint frailty models exist for population-adjusted network synthesis.”
What was wrong with it. The source said a mis-specified count likelihood distorts the transported effect and not only its standard error; the code auditor showed this is false as a general rule, since a correctly specified conditional mean leaves Poisson point estimates consistent under overdispersion. The same auditor narrowed the claim that Poisson is too restrictive to cases with observed residual conditional overdispersion, and rejected the claim that hurdle and zero-inflated formulations require separate effect-modification structure, since those coefficients may be constrained, shared or omitted in one component; the literature auditor judged the entry supported, so the adjudicated verdict is unverifiable with both positions on the record.
Dissent. codex dissented (overstated): At the requested cpaic revision, cmlnmr() accepts only binomial, Gaussian, Poisson, and survival families. NAMESPACE and the Stan models contain no negative-binomial, Poisson-lognormal, hurdle, zero-inflated, recurrent-event, or frailty path. The core cpaic gap is therefore open, and C6-C7 correctly distinguish first recurrence from total burden. However, C1 needs observed residual conditional overdispersion; Poisson is not universally too restrictive. C3 is false as a general rule: when the conditional mean is correctly specified, the Poisson estimating score can remain unbiased under negative-binomial variance, leaving point effects consistent while model-based uncertainty is wrong. Transport bias requires misspecification of the conditional mean, zero mechanism, frailty, or exposure weighting. C5 also overstates necessity: hurdle and zero-inflated models have two predictors, but their effect-modifier coefficients may be constrained, shared, or omitted in one component. C8-C9 are universal prevalence or absence claims not established by cpaic source inspection.
- Literature and prior-art check
- open / supported. Verified from the multinma nma() reference that the supported count/rate likelihoods are binomial, bernoulli, poisson and normal plus the survival families, with no negative-binomial, Poisson-lognormal, hurdle or zero-inflated option, so ‘proposed but not yet available’ is accurate for the leading ML-NMR implementation. The general count-meta-analysis literature supports the Poisson-restriction claim: ‘Do alternative methods for analysing count data produce similar estimates? Implications for meta-analyses’ (doi 10.1186/s13643-015-0144-x) shows precision and pooled effects differ by whether rate ratio, Poisson, negative binomial, dichotomised risk ratio or time-to-first-event is used, which also supports the claim that time to first event and total burden are different estimands. I found no recurrent-event or joint-frailty model for population-adjusted network synthesis. One lead worth watching that I did not read: arXiv 2603.11019, ‘Don’t Disregard the Data for Lack of a Likelihood: Bayesian Synthetic Likelihood for Enhanced Multilevel Network Meta-Regression’, which if it does what its title says would remove the closed-form-likelihood barrier for non-standard count likelihoods in ML-NMR.
- GPT-5.6 Sol, technical and source-code lens
- open / overstated. At the requested cpaic revision, cmlnmr() accepts only binomial, Gaussian, Poisson, and survival families. NAMESPACE and the Stan models contain no negative-binomial, Poisson-lognormal, hurdle, zero-inflated, recurrent-event, or frailty path. The core cpaic gap is therefore open, and C6-C7 correctly distinguish first recurrence from total burden. However, C1 needs observed residual conditional overdispersion; Poisson is not universally too restrictive. C3 is false as a general rule: when the conditional mean is correctly specified, the Poisson estimating score can remain unbiased under negative-binomial variance, leaving point effects consistent while model-based uncertainty is wrong. Transport bias requires misspecification of the conditional mean, zero mechanism, frailty, or exposure weighting. C5 also overstates necessity: hurdle and zero-inflated models have two predictors, but their effect-modifier coefficients may be constrained, shared, or omitted in one component. C8-C9 are universal prevalence or absence claims not established by cpaic source inspection.
- Decision path
-
R5:default-open → R7:counter-and-support-both-present
How the evidence falls in time
2 papers in the reviewed corpus bear on this problem, published 2018 to 2021. Every paper found here asserts the problem is open; none reports progress.
| Year | Says | Paper |
|---|---|---|
| 2018 | confirms it is open | Different Methods for Modelling Severe Hypoglycaemic Events: Implications for Effectiveness, Costs and Health Utilities |
| 2021 | confirms it is open | Methods for the inclusion of real-world evidence in network meta-analysis |
Source
Derived from C:L214-215, C:L246, C:L1557-1561, C:L1557-1562, C:L1557-1565, C:L1608-1610 of the reviewed corpus.