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arxiv: 2606.27952 · v1 · pith:JVHLCT7Pnew · submitted 2026-06-26 · 📊 stat.ME

Robust estimation of occupation probabilities for coarsened multistate processes

Niklas Nyboe Maltzahn , Gergely D\'aniel Luk\'ats , Kjetil R{\o}ysland This is my paper

Pith reviewed 2026-06-29 03:05 UTC · model grok-4.3

classification 📊 stat.ME
keywords multistate modelsoccupation probabilitiesinverse probability weightingcoarsening at randomright-censoringaugmented estimatorscausal inferencemissing data
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The pith

Augmented inverse probability weighted estimators yield robust occupation probability estimates for coarsened multistate processes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to estimate the probability that a multistate process is in a particular state at a given time when observations are coarsened by right-censoring and by baseline exposure. The estimators are constructed using augmented inverse probability weighting, which draws on causal inference techniques to achieve double robustness. They remain consistent if either the model for the coarsening process or the model for the state transitions is correctly specified. The method allows for time-varying confounders and does not assume the process follows Markov dynamics. Theoretical results and simulations establish that the estimators are consistent and can achieve optimal efficiency when both models are correct.

Core claim

The authors derive augmented inverse probability weighted estimators for occupation probabilities under coarsening at random. These estimators are doubly robust to misspecification of either the coarsening mechanism or the conditional expectations of the state indicators, and they are efficient when both are correct. The approach identifies the target parameters from observed data without requiring the multistate process to satisfy the Markov property.

What carries the argument

Augmented inverse probability weighted estimators, which weight observed state indicators by the inverse probability of coarsening and augment with predictions from an outcome model.

If this is right

  • Consistency holds if at least one of the coarsening or outcome models is correctly specified.
  • Efficiency is achieved when both models are correct.
  • The estimators apply to processes with time-varying confounders.
  • No Markov assumption is needed for identification or estimation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These estimators could be adapted for use in longitudinal studies with similar coarsening patterns.
  • Violations of coarsening at random due to unmeasured confounding would require additional methods like instrumental variables for correction.
  • Combining the approach with flexible machine learning models for the nuisance parameters may enhance applicability in complex data settings.

Load-bearing premise

Coarsening at random must hold, so that whether data is observed depends only on what has been observed so far.

What would settle it

Generate data from a multistate process where the coarsening probability depends on an unobserved state; the proposed estimators will then exhibit bias, whereas they will be unbiased when coarsening depends only on observed data.

Figures

Figures reproduced from arXiv: 2606.27952 by Gergely D\'aniel Luk\'ats, Kjetil R{\o}ysland, Niklas Nyboe Maltzahn.

Figure 1
Figure 1. Figure 1: The illness-death model with recurrence and cen [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Latent occupation probability curves 𝜓0 (𝑡|𝑎), given treatment 𝐴 = 𝑎, for the population average and bands be￾tween the minimum, the quantiles 10%, 25%, 75%, 90%, and the maximum values of 𝜓 𝑖 0 (𝑡|𝑎). In order to benchmark our estimators’ performances to the true occupation probability curves 𝜓0 (𝑡) of the model, as defined in (1), we must ensure that we are able to first approximate the latter with minim… view at source ↗
Figure 3
Figure 3. Figure 3: Example run of the estimators 𝜓ˆ 0 𝑛 , 𝜓ˆ 𝜈 𝑛 , 𝜓ˆ 𝜇 𝑛 , 𝜓ˆ 𝜈,𝑚𝑜𝑑 𝑛 and 𝜓ˆ 𝜇,𝑚𝑜𝑑 𝑛 , all using the consistent nuisance parameter estimators ˆ𝜋0, 𝜆ˆ𝐺 0 , 𝑄ˆ 0 and ˆ𝜂0. For 𝜓ˆ 𝜇 𝑛 , we include the 95 % pointwise confidence intervals. Population size is 500, on a single example dataset. 𝑓1, 𝑓2 and the numeric values of the vectors 𝛽, 𝜉 and the CTMC coefficient matrix are also detailed in Appendix E.1. 4.2. Te… view at source ↗
Figure 4
Figure 4. Figure 4: MBE and RMSE metrics of the estimators 𝜓ˆ 0 𝑛 , 𝜓ˆ 𝜈 𝑛 , 𝜓ˆ 𝜇 𝑛 , 𝜓ˆ 𝜈,𝑚𝑜𝑑 𝑛 and 𝜓ˆ 𝜇,𝑚𝑜𝑑 𝑛 , all using the consistent nuisance parameter estimators ˆ𝜋0, 𝜆ˆ𝐺 0 , 𝑄ˆ 0 and ˆ𝜂0. Averages are over 100 independent datasets with a population size of 9000 in each. The resulting outputs are compared to the true occupation probability curves 𝜓0 (𝑡|𝑎) from [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The mean SD plots agree with the theoretical hierarchy outlined in Section 3.3, and the [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Robustness to 𝜋: MBE and RMSE metrics of the estimators 𝜓ˆ 0 𝑛 , 𝜓ˆ 𝜈 𝑛 , 𝜓ˆ 𝜇 𝑛 , 𝜓ˆ 𝜈,𝑚𝑜𝑑 𝑛 and 𝜓ˆ 𝜇,𝑚𝑜𝑑 𝑛 , all using the biased nuisance estimator ˆ𝜋1. Averages are over 100 inde￾pendent datasets with a population size of 9000 in each. The IPW estimator 𝜓ˆ 0 𝑛 is significantly biased, due to the biased nuisance estimator ˆ𝜋1 and the IPW estimator’s lack of robustness properties. The AIPW estimators, ho… view at source ↗
Figure 7
Figure 7. Figure 7: Robustness to 𝐺: MBE and RMSE metrics of the estimators 𝜓ˆ 0 𝑛 , 𝜓ˆ 𝜈 𝑛 , 𝜓ˆ 𝜇 𝑛 , 𝜓ˆ 𝜈,𝑚𝑜𝑑 𝑛 and 𝜓ˆ 𝜇,𝑚𝑜𝑑 𝑛 , all using the biased nuisance estimator 𝜆ˆ𝐺 1𝑎 . Averages are over 100 inde￾pendent datasets with a population size of 9000 in each. The IPW estimator 𝜓ˆ 0 𝑛 is significantly biased, due to the biased nuisance estimator 𝜆ˆ𝐺 1𝑎 and the IPW estimator’s lack of robustness properties. The AIPW estimat… view at source ↗
Figure 8
Figure 8. Figure 8: Robustness to 𝜂: RMSE for the estimators 𝜓ˆ 0 𝑛 , 𝜓ˆ 𝜇 𝑛 , and 𝜓ˆ 𝜇,𝑚𝑜𝑑 𝑛 , all using the biased nuisance estimator ˆ𝜂1. Av￾erages are over 100 independent datasets with a population size of 9000 in each. We present the Figures for robustness to 𝜂 and 𝑄 in Appendix E.3. When using the biased nuisance estimator ˆ𝜂1, we see a slight improvement in the RMSE for 𝜓ˆ 𝜇,𝑚𝑜𝑑 𝑛 compared to 𝜓ˆ 𝜇 𝑛 in some plots, suc… view at source ↗
Figure 9
Figure 9. Figure 9: 𝑓1 and 𝑓2 risk transformations of Age and BMI. r The state transition hazards 𝛼𝑖 𝑗 in the CTMC are linear sums of A, Sex, 𝑓1 (Age) and 𝑓2 (BMI), with interaction terms between A and the other variables. Overall, this means that there are 32 coefficients chosen:         𝛼12 𝛼13 𝛼21 𝛼23         =         −3.950 −1.20 0.60 0.80 0.80 −0.30 −0.30 −0.30 −5.550 −1.20 0.60 0.80 0.80 −0.30 −… view at source ↗
Figure 10
Figure 10. Figure 10: Propensity score distribution in our example sce [PITH_FULL_IMAGE:figures/full_fig_p038_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The observed state occupation process, demon [PITH_FULL_IMAGE:figures/full_fig_p038_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: MBE and RMSE metrics of the estimators 𝜓ˆ 0 𝑛 , 𝜓ˆ 𝜈 𝑛 , 𝜓ˆ 𝜇 𝑛 , 𝜓ˆ 𝜈,𝑚𝑜𝑑 𝑛 and 𝜓ˆ 𝜇,𝑚𝑜𝑑 𝑛 , all using the biased nuisance esti￾mator 𝜆ˆ𝐺 1𝑏 (E1). Averages are over 100 independent datasets with a population size of 9000 in each [PITH_FULL_IMAGE:figures/full_fig_p039_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Robustness to 𝑄: Mean SD and RMSE metrics of the estimators 𝜓ˆ 0 𝑛 , 𝜓ˆ 𝜇 𝑛 and 𝜓ˆ 𝜇,𝑚𝑜𝑑 𝑛 , all using the biased nui￾sance estimator 𝑄ˆ 1 (𝑡). Averages are over 100 independent datasets with a population size of 9000 in each [PITH_FULL_IMAGE:figures/full_fig_p040_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Robustness to 𝜂: Mean SD of the influence func￾tions for estimators 𝜓ˆ 0 𝑛 , 𝜓ˆ 𝜇 𝑛 and 𝜓ˆ 𝜇,𝑚𝑜𝑑 𝑛 , all using the biased nuisance estimator ˆ𝜂1 (𝑡, 𝑢). Averages are over 100 indepen￾dent datasets with a population size of 9000 in each [PITH_FULL_IMAGE:figures/full_fig_p041_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Modifying regression coefficients of 𝜓ˆ 𝜈,𝑚𝑜𝑑 𝑛 and 𝜓ˆ 𝜇,𝑚𝑜𝑑 𝑛 when using the consistent nuisance parameter esti￾mators ˆ𝜋0, 𝜆ˆ𝐺 0 , 𝑄ˆ 0 and ˆ𝜂0. Averages are over 100 indepen￾dent datasets with a population size of 9000 in each. Healthy Illness Death A: 0 A: 1 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 2 4 6 0 2 4 6 Regression Coefficients Estimator βπ *, ν βπ *, µ βG *, µ [PITH_FULL_IMAGE:fi… view at source ↗
Figure 16
Figure 16. Figure 16: Robustness to 𝜂: Modifying regression coefficients of 𝜓ˆ 𝜈,𝑚𝑜𝑑 𝑛 and 𝜓ˆ 𝜇,𝑚𝑜𝑑 𝑛 when using the biased nuisance esti￾mator ˆ𝜂1 (𝑡, 𝑢). Averages are over 100 independent datasets with a population size of 9000 in each [PITH_FULL_IMAGE:figures/full_fig_p042_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Robustness to 𝑄: Modifying regression coeffi￾cients of 𝜓ˆ 𝜈,𝑚𝑜𝑑 𝑛 and 𝜓ˆ 𝜇,𝑚𝑜𝑑 𝑛 when using the biased nui￾sance estimator 𝑄ˆ 1 (𝑡). Averages are over 100 independent datasets with a population size of 9000 in each [PITH_FULL_IMAGE:figures/full_fig_p043_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: RMSE of the five estimators and the modifying [PITH_FULL_IMAGE:figures/full_fig_p044_18.png] view at source ↗
read the original abstract

We derive augmented inverse probability weighted estimators for occupation probabilities of multistate models under two levels of coarsening; right-censoring and baseline exposure. The key exchangeability assumption for identification is coarsening at random, while allowing for time-varying confounders, but not requiring Markov properties. Using existing techniques from causal inference and missing data literature, the derived estimators have highly desirable robustness and efficiency properties. These properties are demonstrated through both theoretical results, and a simulation study.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper derives augmented inverse probability weighted estimators for occupation probabilities of multistate models under two levels of coarsening (right-censoring and baseline exposure). Identification uses the coarsening-at-random assumption, permitting time-varying confounders but not requiring Markov properties. The estimators are claimed to achieve double robustness and efficiency by applying standard techniques from causal inference and missing-data literature. These properties are supported by theoretical results and a simulation study.

Significance. If the derivations are correct, the work offers a useful extension of robust estimation methods to multistate processes with coarsened data, avoiding Markov assumptions while handling time-varying confounders. Credit is due for explicitly leveraging existing AIPW techniques to obtain the stated robustness and efficiency properties rather than deriving ad-hoc estimators. This is relevant for applications in epidemiology and survival analysis where occupation probabilities are of interest under incomplete observation.

minor comments (2)
  1. Abstract: the phrase 'highly desirable robustness and efficiency properties' is vague; replace with a precise statement of the double-robustness and efficiency results (e.g., consistency under correct specification of either the coarsening or outcome model).
  2. The simulation study section should report the exact data-generating processes, sample sizes, and performance metrics (bias, variance, coverage) for each estimator to allow direct comparison with the theoretical claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation of minor revision. The referee summary correctly captures the paper's focus on augmented IPW estimators for occupation probabilities under coarsening at random in multistate processes, allowing time-varying confounders without Markov assumptions. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives augmented IPW estimators for occupation probabilities by invoking standard identification results and double-robustness techniques from the existing causal inference and missing-data literature (coarsening at random, time-varying confounders, no Markov assumption required). No load-bearing step reduces by construction to a quantity defined only inside the paper, to a fitted parameter renamed as a prediction, or to a self-citation chain whose content is itself unverified. The central claims remain externally grounded and are additionally checked by theory and simulation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the coarsening at random assumption for identification and on standard results from causal inference and missing data methods; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Coarsening at random assumption holds and permits identification
    Stated as the key exchangeability assumption for identification in the abstract.

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