Tompkins, Dubin and Wallace (2024; published 2025) showed that flexible inverse probability of treatment and intensity weighting (FIPTIW) — which multiplies inverse-probability-of-treatment weights and inverse-intensity weights to adjust simultaneously for treatment confounding and informative observation times in irregular longitudinal data — is sensitive to violations of its noninformative-censoring assumption, and that simply multiplying by inverse-probability-of-censoring weights does not restore validity, because outcome-dependent censoring corrupts the visit-intensity weights at their source. They identified as open problems: (i) valid estimation of intensity weights under informative censoring, jointly with informative monitoring and nonrandomized treatment; and (ii) extension to clustered samples with within-household correlation. A follow-up by Stefan and Pullenayegum (2025 preprint, arXiv:2510.19154; published in Statistics in Medicine, 2026) resolved a substantial part of the non-treatment-weighted analogue (inverse-intensity-weighted GEEs under observable informative dropout), but the joint treatment-confounded FIPTIW setting, the multiple-imputation route for intensity estimation, and the clustered extension remain open.

Background

Many longitudinal studies — observational cohorts and electronic-health-record analyses in particular — observe subjects at irregular, subject-specific times, and the timing of visits may itself be related to the outcome being measured (for example, patients visit more often during disease flares). Generalized estimating equations (GEE), the standard regression tool for repeated measures, presuppose that observation times carry no information about the outcome; when that fails, naive GEE estimates of the outcome trajectory are biased. Inverse intensity weighting (IIW) corrects this: the visit process is modeled as a counting process with an intensity (an instantaneous visit rate, typically fit by Cox-type regression), and each observed outcome contribution is weighted by the reciprocal of the modeled cumulative intensity, so that heavily and lightly observed subjects contribute appropriately; a weighted GEE with an independence working correlation then yields consistent estimates of the marginal outcome-model parameters. In observational data this bias source coexists with two others: non-randomized treatment assignment (handled by inverse probability of treatment weighting, IPTW, which weights observations by the reciprocal of the probability of receiving the treatment actually received, given covariates) and loss to follow-up. Tompkins, Dubin and Wallace survey and study the combined FIPTIW estimator, which multiplies IPTW and IIW weights. A key requirement of the IIW weights is a noninformative-censoring assumption (Assumption O2 in that paper): withdrawal from the study must be unrelated to the unobserved outcome history given the observed data. They show by simulation that FIPTIW is biased when O2 fails, and — importantly — that the standard remedy of further multiplying by inverse probability of censoring weights (IPCW, the device that reweights observed records to represent a study population free of dropout) does not reliably restore validity: the visit-intensity model is itself estimated from visit data observed only while subjects remain under follow-up, so outcome-dependent censoring corrupts the IIW weights at their source. Their data application (malaria diagnoses in the PRISM cohort, the Program for Resistance, Immunology, Surveillance, and Modeling of Malaria in Uganda) also featured household clustering, which they handled by analyzing only one child per household, thereby discarding data and leaving the clustered extension open. On variable choice, they showed that omitting covariates related to both the visit process and the outcome biases the estimates, while including covariates related only to the visit process costs little variance — so conservative inclusion is advisable — yet no formal variable-selection or diagnostic tool for the intensity model exists; the paper also recommends trimming combined weights at the 95th percentile when the treatment assignment process is highly informative and drives extreme weights. Two constructions left unexplored for the open problem are extending the multiple-imputation-and-bootstrap approach of Jackson et al. for Cox models under dependent censoring to intensity-model estimation, and providing full theory and validation for the multiply robust framework sketched by Coulombe and Yang. Prior work on clustered irregularly observed longitudinal data includes the clustered irregular-observation methodology of Pullenayegum, Birken and Maguire (2021).

Problem Statement

The problem, as identified by Tompkins, Dubin and Wallace, is to develop and validate estimation methodology for combined treatment-and-intensity weighting that remains valid when its noninformative-censoring assumption fails. The scientific objective is a combined intensity-and-treatment weighting methodology that is valid and usable in the settings real cohort studies present — outcome-dependent withdrawal, informative monitoring, and non-randomized treatment assignment — so that studies such as the source's motivating PRISM cohort can be analyzed without assuming censoring away. Concretely, an answer must deliver:

  1. A coherent estimation strategy for inverse-intensity weights — and for the combined FIPTIW estimator built on them — that yields consistent estimates of the outcome-model (causal) parameters under informative, outcome-dependent censoring, jointly with informative monitoring and non-randomized treatment assignment. The strategy must state its identification assumptions explicitly, justify that they genuinely permit outcome-dependent censoring, and demonstrate by theory and simulation that bias is removed in settings where the source showed that naive multiplication by censoring weights fails.

Separately, the source identifies as a distinct area of future research — not a component required for completion of the core problem above — an extension incorporating within-cluster correlation when several individuals share a cluster (for example, households), so that clustered samples such as the full PRISM household cohort can be analyzed in full rather than discarding all but one child per household.

References

  1. Grace Tompkins, Joel A. Dubin, Michael Wallace. On flexible inverse probability of treatment and intensity weighting: Informative censoring, variable selection, and weight trimming. Statistical Methods in Medical Research 34(5):915-937 (2025); arXiv:2405.15740 (2024). DOI: 10.1177/09622802241313289; https://doi.org/10.1177/09622802241313289; https://arxiv.org/abs/2405.15740
  2. George Stefan, Eleanor Pullenayegum. Inverse-intensity-weighted generalized estimating equations with irregularly measured longitudinal data and informative dropout. Statistics in Medicine 45(18-19) (2026; online 21 July 2026); arXiv:2510.19154 (2025). DOI: 10.1002/sim.70674; https://doi.org/10.1002/sim.70674; https://arxiv.org/abs/2510.19154
  3. Janie Coulombe, Shu Yang. Multiply robust estimation of marginal structural models in observational studies subject to covariate-driven observations. Biometrics 80(3) (2024). DOI: 10.1093/biomtc/ujae065; https://doi.org/10.1093/biomtc/ujae065
  4. Xiawen Zhang, Anna Heath, Wei Xu, Eleanor Pullenayegum. Omitting patients with no follow-up leads to bias when using inverse-intensity weighted GEEs to handle irregular and informative assessment times. BMC Medical Research Methodology 26 (2025). DOI: 10.1186/s12874-025-02721-z; https://doi.org/10.1186/s12874-025-02721-z
  5. Eleanor M. Pullenayegum, Catherine Birken, Jonathon Maguire, TARGet Kids! Collaboration. Clustered longitudinal data subject to irregular observation. Statistical Methods in Medical Research 30(4):1081-1100 (2021). DOI: 10.1177/0962280220986193; https://doi.org/10.1177/0962280220986193
  6. Dan Jackson, Ian R. White, Shaun Seaman, Hannah Evans, Kathy Baisley, James Carpenter. Relaxing the independent censoring assumption in the Cox proportional hazards model using multiple imputation. Statistics in Medicine 33(27):4681-4694 (2014). DOI: 10.1002/sim.6274; https://doi.org/10.1002/sim.6274