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arxiv: 2606.25749 · v1 · pith:S6MXX3LAnew · submitted 2026-06-24 · 📊 stat.ME

A Bayesian Approach for Nonignorable Dropout in Bivariate Longitudinal Models

Andrea Gabrio , Michael J. Daniels , Gianluca Baio This is my paper

Pith reviewed 2026-06-25 20:15 UTC · model grok-4.3

classification 📊 stat.ME
keywords nonignorable dropoutbivariate longitudinal dataBayesian nonparametricsensitivity analysismissing datacost-effectivenessidentifying restrictions
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The pith

A Bayesian nonparametric model jointly handles nonignorable dropout for each of two longitudinal outcomes while allowing sensitivity analysis on the missingness assumptions.

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

The paper develops a method for analyzing incomplete bivariate longitudinal data, such as paired measures of effectiveness and costs in clinical trials, where participants drop out at different times for each outcome. It models the observed data flexibly with a nonparametric Bayesian approach and then uses restrictions tied to the dropout indicators plus sensitivity parameters to partially identify the distribution of the missing values. Different priors on those sensitivity parameters let the analyst explore a range of nonignorable missingness scenarios without assuming the dropout is unrelated to the unseen outcomes. This setup directly addresses the identification problems that arise when the outcome is multivariate and the data show skewness or point masses.

Core claim

The central claim is that a flexible nonparametric Bayesian specification for the observed bivariate longitudinal responses, combined with identifying restrictions that condition on the observed dropout indicators and on chosen sensitivity parameters, permits joint modeling of the two dropout processes and supports exploration of nonignorable missingness through alternative priors on the sensitivity parameters.

What carries the argument

Joint Bayesian nonparametric model for the observed data together with identifying restrictions conditional on dropout indicators and sensitivity parameters.

If this is right

  • Different dropout times for the two response types can be accommodated without forcing a common dropout process.
  • Skewness and point masses in the data are captured without parametric assumptions on the outcome distributions.
  • Multiple nonignorable missingness scenarios can be examined by varying the priors placed on the sensitivity parameters.
  • The approach supplies cost-effectiveness estimates that incorporate uncertainty arising from the dropout mechanism.

Where Pith is reading between the lines

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

  • The same structure could be used to handle trivariate or higher-dimensional longitudinal outcomes if the identifying restrictions are extended accordingly.
  • Policy conclusions drawn from cost-effectiveness trials may shift when the sensitivity parameters are allowed to differ across treatment arms.
  • The method suggests a route for sensitivity analysis in other settings where dropout times differ across multiple correlated endpoints.

Load-bearing premise

The distribution of the missing data can be partially identified through restrictions that depend on the dropout indicators and on the chosen values of the sensitivity parameters.

What would settle it

Collect a new trial dataset in which all participants complete every scheduled assessment and then compare the model's posterior predictions for the would-be missing values (under each prior on the sensitivity parameters) against the actually observed values.

Figures

Figures reproduced from arXiv: 2606.25749 by Andrea Gabrio, Gianluca Baio, Michael J. Daniels.

Figure 1
Figure 1. Figure 1: Empirical mean profiles for different utility and cost dropout times (1 = baseline, 2 = 6-months, 3 = 12-months) in the control (panel a) and intervention (panel b) group of the PBS study. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Posterior predictive distributions from DPM for the pairwise rank correlations between utilities and costs variables in the control (panel a) and intervention (panel b) arm for the replicated data (light blue bars) compared with the estimates based on the observed data in the real dataset (vertical blue lines). 5.3 Results This section summarises the results from the posterior distribution of the main quan… view at source ↗
Figure 3
Figure 3. Figure 3: shows the posterior means and 95% highest posterior density (HPD) credible intervals for the marginal utility and cost means µjt = (µ u jt, µc jt), obtained from fitting the model to all cases under six alternative scenarios: nonignorable MAR without using any identifying restrictions (NOIG), ignorable based on MAR identifying restrictions (IG), benchmark nonignorable based on identifying restrictions cond… view at source ↗
Figure 4
Figure 4. Figure 4: CEPs (panel a) and CEACs (panel b) associated with alternative missingness scenarios. In the CEPs, the ICERs based on the results from NOIG, BENCH and MNAR M are indicated with corresponding darker coloured dots, while the portion of the plane on the right-hand side of the straight line passing through the plot (evaluated at k = £25, 000) denotes the sustainability area. For the CEACs, in addition to the r… view at source ↗
read the original abstract

Longitudinal data collected in clinical trials are almost always incomplete due to some of the participants dropping out from the study during the planned follow-up. A common strategy to handle nonresponse expresses missingness in terms of a dropout process, which is jointly analysed with the outcome process to facilitate the formulation of the missingness assumptions. However, when the outcome is multivariate, the identification of the dropout process becomes problematic, especially when individuals have different dropout times for each type of response, and sensitivity analysis is difficult. The modelling task may be also be complicated by data complexities (e.g. skewness and spikes) which are difficult to capture through standard parametric methods. An example of this analysis framework occurs in trial-based economic evaluations, where a longitudinal bivariate response, formed by suitably-defined measures of effectiveness and costs, is analysed to inform policymakers about the cost-effectiveness of alternative interventions. We present a novel Bayesian nonparametric approach to handle a missing bivariate longitudinal outcome by jointly modelling the dropout process associated with each type of response while also taking into account the complexities of the data. We specify a flexible nonparametric model for the observed data and partially identify the distribution of the missing data with identifying restrictions conditional on the dropout indicators and sensitivity parameters. We explore alternative nonignorable scenarios through different priors for the sensitivity parameters. Our approach is motivated by, and applied to, data from a trial assessing the cost-effectiveness of a new treatment for intellectual disability and challenging behaviour.

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

2 major / 2 minor

Summary. The paper proposes a Bayesian nonparametric approach to handle nonignorable dropout in bivariate longitudinal data (e.g., effectiveness and cost measures in clinical trials). It jointly models the two dropout processes, specifies a flexible nonparametric model for the observed data, and partially identifies the missing data distribution using identifying restrictions that condition on the dropout indicators and sensitivity parameters, with alternative nonignorable scenarios explored via priors on those parameters. The method is motivated by and applied to trial data on treatment for intellectual disability and challenging behaviour.

Significance. If the partial identification strategy is valid, the work provides a flexible, nonparametric framework for sensitivity analysis under nonignorable missingness in bivariate longitudinal settings with possibly differing dropout times per response. This addresses a practical gap in trial-based economic evaluations where standard parametric models struggle with skewness, spikes, and multivariate dropout, potentially leading to more robust cost-effectiveness inferences.

major comments (2)
  1. [Abstract / identifying restrictions] Abstract and modeling section on identifying restrictions: the claim that the distribution of the missing bivariate outcomes is partially identified via restrictions conditional on the two dropout indicators and sensitivity parameters does not address the case (explicitly noted in the abstract) where individuals drop out at different times for each response. When dropout times differ, the required conditional distribution of the unobserved outcomes given observed history and the pair of dropout indicators may involve cross-response dependence not captured by the nonparametric model fitted only to observed data; varying priors on the sensitivity parameters would then fail to correctly bound bias in the cost-effectiveness parameters.
  2. [Joint dropout model] Section describing the joint dropout model: no derivation or explicit factorization is provided showing how the bivariate dropout process is specified when the two responses have misaligned observation times, which is load-bearing for the claim that the approach jointly models both dropout processes while remaining computationally tractable.
minor comments (2)
  1. [Abstract] Abstract contains a repeated word: 'The modelling task may be also be complicated'.
  2. [Notation] Notation for the sensitivity parameters and the nonparametric components should be introduced with explicit definitions and distinguished from standard MNAR parameters in the literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which highlight important aspects of our approach to nonignorable dropout in bivariate longitudinal settings. We address each major comment below and indicate the revisions planned for the manuscript.

read point-by-point responses

  1. Referee: [Abstract / identifying restrictions] Abstract and modeling section on identifying restrictions: the claim that the distribution of the missing bivariate outcomes is partially identified via restrictions conditional on the two dropout indicators and sensitivity parameters does not address the case (explicitly noted in the abstract) where individuals drop out at different times for each response. When dropout times differ, the required conditional distribution of the unobserved outcomes given observed history and the pair of dropout indicators may involve cross-response dependence not captured by the nonparametric model fitted only to observed data; varying priors on the sensitivity parameters would then fail to correctly bound bias in the cost-effectiveness parameters.

    Authors: We appreciate the referee's point on the challenges posed by misaligned dropout times. Our nonparametric model for the observed data is specified jointly across both responses, capturing dependence up to the last observed time for each. The identifying restrictions are formulated conditionally on the pair of dropout indicators to permit the sensitivity parameters to encode cross-response associations in the unobserved data. That said, we acknowledge that the original manuscript did not provide a sufficiently explicit derivation showing how this conditional distribution preserves or bounds the relevant dependence when observation times differ. We will revise the modeling section to include this derivation and clarify how the sensitivity analysis correctly informs bounds on bias for the cost-effectiveness parameters. revision: yes

  2. Referee: [Joint dropout model] Section describing the joint dropout model: no derivation or explicit factorization is provided showing how the bivariate dropout process is specified when the two responses have misaligned observation times, which is load-bearing for the claim that the approach jointly models both dropout processes while remaining computationally tractable.

    Authors: We agree that an explicit factorization of the bivariate dropout process under misaligned observation times is necessary to substantiate the claim of joint modeling and computational tractability. Although the model construction relies on a joint specification that factors through the observed data likelihood and appropriate priors for the dropout indicators, the manuscript did not include the full derivation. In the revision we will add this factorization, along with details on how the joint process is implemented in the MCMC algorithm to maintain tractability. revision: yes

Circularity Check

0 steps flagged

No circularity: nonparametric observed-data model and sensitivity-parameter restrictions are introduced as independent modeling choices

full rationale

The paper defines a joint nonparametric model on the observed bivariate longitudinal data and then applies standard partial-identification restrictions that condition on the observed dropout indicators plus user-specified sensitivity parameters. No equation reduces a target quantity to a fitted parameter by construction, no prediction is obtained by refitting a subset of the same data, and no load-bearing premise rests on a self-citation whose content is itself unverified. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on domain assumptions about identifying restrictions and the ability of the nonparametric model to capture data complexities; sensitivity parameters function as free parameters whose priors are varied to explore scenarios.

free parameters (1)
  • sensitivity parameters
    Priors placed on sensitivity parameters to explore alternative nonignorable dropout scenarios; values are not fixed by data.
axioms (1)
  • domain assumption Identifying restrictions conditional on the dropout indicators and sensitivity parameters suffice to partially identify the missing data distribution
    Invoked to enable inference on missing outcomes when dropout is nonignorable.

pith-pipeline@v0.9.1-grok · 5788 in / 1165 out tokens · 22039 ms · 2026-06-25T20:15:43.822364+00:00 · methodology

discussion (0)

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