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arxiv: 2606.24228 · v1 · pith:BE264UEInew · submitted 2026-06-23 · 📊 stat.ME

Uncertainty intervals for multilevel models with missing not at random data

Minna Genb\"ack This is my paper

Pith reviewed 2026-06-25 22:52 UTC · model grok-4.3

classification 📊 stat.ME
keywords sensitivity analysismissing not at randommultilevel modelspartial identificationuncertainty intervalslinear mixed-effectsdropout biasMNAR
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The pith

A sensitivity analysis method produces uncertainty intervals for parameters in multilevel models when data is missing not at random.

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 linear multilevel models that jointly models the outcome variable and the dropout process to derive a bias adjustment term. This adjustment is computed from observed data once values are supplied for one or more sensitivity parameters that capture the MNAR mechanism. When those parameters are restricted to a researcher-specified plausible range, the method yields bounds on the parameters of interest instead of point estimates. The resulting intervals support estimation and inference under assumptions weaker than the usual missing-at-random condition. The approach is checked through simulation and applied to longitudinal data on loneliness, physical activity, and memory trajectories.

Core claim

By modeling both the outcome and the dropout risk with multilevel structures, a bias adjustment can be derived that is identifiable from observed data conditional on fixed values of sensitivity parameters; restricting those parameters to a plausible interval then partially identifies the target parameters, producing uncertainty bounds that hold under MNAR mechanisms weaker than missing at random.

What carries the argument

The bias adjustment term obtained from the joint multilevel models for outcome and dropout, evaluated conditional on the sensitivity parameters.

If this is right

  • Parameters in multilevel models can be bounded rather than point-estimated when dropout is suspected to be MNAR.
  • Inference procedures can report intervals that reflect the range of possible bias adjustments consistent with the sensitivity assumptions.
  • The method applies directly to longitudinal studies with repeated measures and attrition.
  • Results remain valid under missingness mechanisms that violate the missing-at-random assumption but lie inside the specified sensitivity range.

Where Pith is reading between the lines

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

  • The same bounding strategy could be adapted to nonlinear or generalized linear multilevel models by deriving analogous bias terms.
  • Domain experts could use substantive knowledge to tighten or widen the sensitivity ranges in fields such as health or social science.
  • If the resulting bounds are narrow across a wide sensitivity range, that would indicate robustness even without assuming missing at random.

Load-bearing premise

The researcher can specify a plausible range for the sensitivity parameters that contains their true values.

What would settle it

A simulation in which the true MNAR mechanism is known and the computed uncertainty intervals do not contain the true parameter values for the chosen sensitivity ranges.

read the original abstract

We propose a sensitivity analysis method for missing not at random (MNAR) data in the context of linear multilevel (mixed-effects) models. The outcome and dropout risk are both modelled using multilevel models and a bias adjustment due to MNAR data is derived. This bias can be estimated from observed data conditional on specified values of sensitivity parameter(s). Under the assumption that these parameters lie within a plausible range, the method partially identify the parameters of interest, yielding bounds for estimation and inference under assumptions weaker than missing at random. The proposed analysis is investigated in a simulation study and illustrated with an analysis of the association between loneliness and physical activity with memory trajectories, adjusting for demographic, socioeconomic, and health covariates.

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 / 3 minor

Summary. The manuscript proposes a sensitivity analysis method for missing not at random (MNAR) data in linear multilevel (mixed-effects) models. Both the outcome and the dropout mechanism are modeled via multilevel structures; a bias adjustment is derived that can be estimated from the observed data conditional on researcher-specified values of one or more sensitivity parameters. Under the assumption that these parameters lie in a plausible range, the approach yields bounds (uncertainty intervals) on the parameters of interest, enabling estimation and inference under assumptions weaker than missing at random. The method is examined in a simulation study and illustrated with an analysis of the association between loneliness, physical activity, and memory trajectories, adjusting for demographic, socioeconomic, and health covariates.

Significance. If the bias-adjustment derivation and bounding procedure are valid, the work supplies a practical sensitivity-analysis framework for a common data structure in longitudinal research. Multilevel models are widely used for clustered or repeated-measures data, yet MNAR dropout is rarely addressed beyond MAR assumptions; the proposed partial-identification approach therefore fills a methodological gap. The combination of an explicit bias formula, simulation evidence, and a substantive application strengthens the contribution.

major comments (2)
  1. [§3] §3 (bias-adjustment derivation): the adjustment formula must be shown to correctly incorporate the random-effects covariance structure when both the outcome and selection models are multilevel; without an explicit step that integrates over the random effects in the conditional expectation, it is unclear whether the bias term remains consistent for the fixed-effect parameters of primary interest.
  2. [§4] §4 (simulation design): the reported coverage probabilities and interval widths are obtained under correctly specified sensitivity-parameter ranges; a supplementary set of simulations in which the true sensitivity parameters lie near the boundary of the assumed range is needed to verify that the uncertainty intervals retain their nominal coverage when the researcher-specified range is only approximately correct.
minor comments (3)
  1. [§2] The notation distinguishing the sensitivity parameters from the model parameters should be introduced once in §2 and used consistently thereafter.
  2. [Table 2] Table 2 (simulation results) would benefit from an additional column reporting the proportion of replications in which the bounds contain the true parameter under each MNAR scenario.
  3. [§5] The real-data application in §5 reports point estimates and bounds but does not state the exact range chosen for the sensitivity parameters; this information should be added for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: §3 (bias-adjustment derivation): the adjustment formula must be shown to correctly incorporate the random-effects covariance structure when both the outcome and selection models are multilevel; without an explicit step that integrates over the random effects in the conditional expectation, it is unclear whether the bias term remains consistent for the fixed-effect parameters of primary interest.

    Authors: We agree that an explicit integration step would clarify the derivation. The bias adjustment in Section 3 is obtained by conditioning on the observed data and selection indicators under the joint multilevel model for the outcome and dropout process; this implicitly integrates over the random effects via their joint covariance structure. In the revision we will insert an additional displayed equation that explicitly performs the integration over the random-effects distribution, confirming that the resulting bias term is consistent for the fixed-effect parameters of interest. revision: yes

  2. Referee: §4 (simulation design): the reported coverage probabilities and interval widths are obtained under correctly specified sensitivity-parameter ranges; a supplementary set of simulations in which the true sensitivity parameters lie near the boundary of the assumed range is needed to verify that the uncertainty intervals retain their nominal coverage when the researcher-specified range is only approximately correct.

    Authors: We concur that this additional check would strengthen the simulation evidence. We will add a supplementary simulation study in which the true sensitivity parameters are placed at the boundary (and slightly outside) of the researcher-specified ranges, and will report the resulting coverage probabilities and interval widths for the uncertainty intervals. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is conditional on external inputs

full rationale

The paper's central method derives a bias adjustment for MNAR data in multilevel models explicitly conditional on researcher-specified sensitivity parameters that are treated as external inputs (not fitted or derived from the target data). Partial identification then follows by varying those parameters over a plausible range. No load-bearing self-citations, self-definitional steps, or fitted inputs renamed as predictions are present in the abstract or described approach; the derivation remains self-contained against the stated conditioning.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method depends on user-specified sensitivity parameters whose plausible range is treated as an external modeling choice rather than derived from data.

free parameters (1)
  • sensitivity parameters
    User-specified values that control the magnitude of the MNAR bias adjustment.
axioms (1)
  • domain assumption The sensitivity parameters lie within a plausible range
    This assumption enables partial identification and the production of bounds.

pith-pipeline@v0.9.1-grok · 5636 in / 1213 out tokens · 29922 ms · 2026-06-25T22:52:54.420551+00:00 · methodology

discussion (0)

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Reference graph

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12 extracted references · 7 canonical work pages

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