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REVIEW 5 major objections 7 minor 21 references

Cumulative Logit Ordinal Regression with Proportional Odds under Nonignorable Missing Response -- Application to Phase III Trial

T0 review · 5 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims an EM algorithm that jointly fits a cumulative-logit proportional-odds model and a logistic missingness model containing the outcome reduces bias and mean squared error, relative to complete-case analysis and MICE-based…

desk verdict The ordinal EM extension is a sensible piece of work, but the paper's own simulation tables contradict its headline claim, and the complete-case results look implausible, so the empirical core needs a hard recheck before the method can be trusted. read the letter →

arxiv 2505.04727 v1 pith:AP7BBQOC submitted 2025-05-07 stat.ME stat.AP

Arnab Kumar Maity , Huaming Tan , Vivek Pradhan , Soutir Bandyopadhyay This is my paper · ORCID
classification stat.MEstat.AP MSC 62F1062J1262P10
keywords cumulativelogitEMalgorithmnonignorablemissingordinalresponseproportionaloddsrandomizedtrialnotatrandom
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Clinical trials with ordinal endpoints lose information when outcomes are missing, and the bias is worst when missingness depends on the outcome itself, a nonignorable or missing-not-at-random mechanism. This paper proposes an expectation-maximization (EM) algorithm for cumulative-logit proportional-odds regression that jointly models the ordinal response and a logistic missingness model in which the outcome appears as a predictor. In simulations with about 10%, 25%, and 45% missing responses and sample sizes from 60 to 1000, the EM estimates show lower bias and mean squared error than complete-case analysis and MICE-based multiple imputation, with coverage probabilities approaching 0.95 as the sample grows. Applied to a Phase III psoriasis trial where 13.5% of five-point Physician's Global Assessment scores were missing, the method estimates a significant outcome-missingness association and yields treatment odds ratios that differ from both complete-case and MICE estimates.

What carries the argument

The carrying object is the joint likelihood $L(\boldsymbol{\gamma}) = \prod_i f_i(Y_i \mid X_i, \boldsymbol{\beta}) \, p_i(R_i \mid Z_i, \boldsymbol{\alpha})$, where $Z_i=(X_i, Y_i)$ and $p_i$ is a logistic missingness probability. The E-step assigns a missing response the weight $w_{iy} = f_i(Y_i\mid X_i,\boldsymbol{\beta}) p_i(R_i\mid Z_i,\boldsymbol{\alpha}) / \sum_{j=1}^J f_i(Y_i=j \mid X_i,\boldsymbol{\beta}) p_i(R_i\mid Z_i,\boldsymbol{\alpha})$, turning each missing observation into $J$ weighted pseudo-observations, one per ordinal category. The M-step then maximizes the weighted proportional-odds likelihood and the weighted logistic missingness likelihood separately, using standard software; iteration continues to convergence, and the final covariance matrix uses the Louis correction for the observed information. Because the missingness model includes $Y$ itself, the weights are the conditional predictive distribution of the missing outcome given covariates and missingness, which is the mechanism that removes the nonignorable bias.

What would settle it

Replicate the $n=60$, 45%-missingness scenario of the paper's Table 4 and compare EM with MICE on the treatment coefficient; the paper's own table shows MICE with bias 0.102 versus EM's 0.583, so this cell already questions the claim of uniform superiority, and repeating the protocol under a threshold or quadratic missingness mechanism would test robustness to model misspecification.

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Extended reading notes

Core claim

The central claim is that explicit modeling of the missingness mechanism removes most of the bias that complete-case analysis and MAR-based multiple imputation leave in proportional-odds ordinal regression when responses are missing not at random. The paper builds the joint likelihood as the product of a cumulative-logit model for the ordinal response $Y$ and a binary logistic model for the missing indicator $R$, with the unobserved $Y$ included as a covariate in the missingness model, so a nonzero coefficient on $Y$ signals nonignorable missingness. An EM algorithm treats each missing response as a weighted mixture over all ordinal categories, re-estimates the weights from the fitted models, and iterates between weighted proportional-odds and weighted logistic fits. Over 1000-replication simulations the EM estimator is reported to dominate complete-case and MICE in bias and mean squared error across missingness rates, with near-nominal coverage at larger sample sizes, and standard errors are computed from the Louis observed-information correction. In the psoriasis application the outcome coefficient in the missingness model is significant, and the EM-based odds ratios for both active doses are attenuated relative to complete-case estimates, while onset age becomes significant only under EM.

Load-bearing premise

The method works only if the assumed logistic missingness model, with the outcome entering as a linear predictor, is the true mechanism and the EM likelihood has a single peak; the paper does not prove the single peak, and a wrong missingness model could produce estimates as biased as the simpler approaches.

Editorial extensions

If this is right

  • Treatment-effect estimates in ordinal clinical-trial endpoints will move closer to the complete-data values when missingness depends on the outcome, making the method a concrete sensitivity check for regulatory submissions.
  • The coefficient on the outcome in the missingness model can be tested, giving a diagnostic for whether missingness is likely nonignorable.
  • Confidence intervals built from the Louis information matrix approach nominal coverage at moderate-to-large sample sizes, supporting valid inference rather than only point estimation.
  • Because the E-step re-weights all possible response categories, no ad hoc imputation model or rounding rule for ordinal values is needed.
  • The released SAS macro makes the procedure usable in the same software environment many clinical-trial statisticians already employ.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the true missingness mechanism is nonlinear in the outcome, such as dropout driven by an extreme-category threshold, the linear logistic model for $R$ is misspecified and could restore bias; testing the method under such misspecification would delimit its scope.
  • The same reweighting principle should transfer to other ordinal links such as probit or complementary log-log, and to ordinal mixed models where nonignorable dropout is common in longitudinal trials.
  • At the smallest sample size with heavy missingness, the paper's own tables show EM's intercept estimates still carry large absolute bias, so the method is best presented as a moderate-to-large-sample procedure rather than a universal fix.
  • A practical extension is to embed the outcome-coefficient test in routine sensitivity analyses: if the coefficient on $Y$ is insignificant, analysts could reasonably report the simpler MAR-based results.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 7 minor

Summary. The paper proposes an EM algorithm for fitting a cumulative logit proportional odds ordinal regression model when the response variable is missing not at random (MNAR). The missingness mechanism is modeled by a logistic regression that includes the outcome itself as a covariate (equations 3 and 8), and the E-step weights each possible value of the missing ordinal response by its posterior probability under the current parameter estimates. The M-step is performed by fitting a weighted proportional odds model and a weighted logistic model (Section 3.2). The authors evaluate the method in simulations with sample sizes 60 to 1000 and missingness rates 10%, 25%, and 45%, comparing complete-case analysis (CC), multiple imputation (MI) via MICE, and the proposed EM approach, and they apply the method to a Phase III psoriasis trial with a five-category PGA endpoint. The central claim in Section 4.2 is that the EM algorithm consistently outperforms CC and MI in reducing bias and MSE.

Significance. If the method delivered the claimed improvements, it would be a practically useful extension of Ibrahim and Lipsitz (1996) to ordinal outcomes in clinical trials, where nonignorable missingness is a common concern. The paper provides a concrete algorithmic recipe, an extensive simulation grid, a real-data application, and a SAS macro, which are valuable for reproducibility. However, the manuscript's own tables contradict the headline claim of consistent superiority at 25% and 45% missingness, and the theoretical convergence argument in Section 3.5 is not valid as stated. The method appears to perform well at 10% missingness, so the contribution is potentially salvageable, but the current claims and supporting evidence need substantial revision.

major comments (5)
  1. [§4.2, Tables 3 and 4] The central claim that EM 'consistently outperforms CC and MI in terms of reducing bias and MSE' is contradicted by the paper's own simulation results at higher missingness. At n=1000 with 25% missing (Table 3), EM absolute biases for y≤1, y≤2, x1, x3, and x4 are 1.609, 0.782, 0.554, 0.016, and 0.027, whereas CC biases are 0.001, 0.007, 0.001, 0.000, and 0.000, and MI biases are 0.963, 0.693, 0.020, 0.009, and 0.017. At 45% missing (Table 4, n=1000), EM biases are 2.989, 1.682, 0.850, 0.024, and 0.036, compared with CC biases of 0.000, 0.008, 0.001, 0.000, and 0.000 and MI biases of 1.465, 1.299, 0.214, 0.008, and 0.042. EM is the worst method for nearly every parameter in these settings, and the bias does not shrink from n=500 to n=1000, which is inconsistent with the claimed consistency. The authors should either correct the results, explain why EM behaves so poorly at higher missingness, or substantially revise the claim of consistent superiority.
  2. [§3.5] The convergence discussion is not a proof. The paper states that Fisher consistency of the cumulative logit surrogate loss (Pedregosa et al., Theorem 7) provides an 'indirect justification for the existence of a global maximum for Q(·|·)' and that minimizing a surrogate loss is equivalent to maximizing a likelihood. Fisher consistency is a property of the population risk minimizer for classification and does not imply unimodality of a finite-sample incomplete-data Q function, nor does it address the behavior of the EM fixed point. The authors should either provide a valid convergence argument (for example, standard EM ascent with explicit conditions on the observed-data likelihood) or explicitly state that only convergence to a stationary point is guaranteed.
  3. [Sections 2 and 3] Identifiability of the joint model (β, α) is never discussed. With Y missing not at random and p(R|X,Y,α) depending on the unobserved outcome, the observed-data likelihood can be non-identifiable without additional restrictions. Since the consistency claim in Section 4.2 rests on the likelihood being well-behaved, the paper should provide conditions under which the parameters are identified from the observed data, or at least acknowledge the issue and report a sensitivity analysis. This is load-bearing for the theoretical validity of the estimator.
  4. [§4.2] The simulation summaries are computed 'after excluding non-converged runs,' but the number and proportion of excluded runs are never reported. If EM fails to converge in a sizable fraction of replications, the reported means, MSEs, and coverage probabilities are conditional on convergence and may be systematically misleading. The authors should report convergence rates for every scenario, especially at n=60 and 45% missingness where the EM estimates show very large biases.
  5. [Supplementary Section 2.1 and Table 1] The supplementary five-category simulation also undercuts the paper's stated superiority. The text says EM is the second best method after 'whole' and outperforms CC, but Table 1 at n=1000 shows EM absolute bias of 0.923 for y≤1 compared with MI's 0.522 and CC's 0.000, and EM has the largest bias among the three for all five parameters. The discrepancy between the claims and the reported numbers needs to be addressed.
minor comments (7)
  1. [Keywords] The keyword 'Nonignorbale' is a typo and should be 'Nonignorable'.
  2. [§5] In the sentence 'In the above model (7, the variables...' the opening parenthesis after 'model' is missing its closing parenthesis; and 'condfidence' should be 'confidence'.
  3. [Equation (7)] Equation (7) uses j = 0, 1, 2, 3 for a five-category outcome, while the general model in equation (1) uses j = 1, ..., J-1. The indexing should be harmonized to avoid confusion.
  4. [Table 2, n=500, x4 row] The entry for the MI estimate in the x4 row appears misaligned or erroneous (the value 0.102 would imply a bias of 0.202 against the true -0.100, yet the reported absolute bias is 0.001). Please check the alignment of this row.
  5. [Figure labels and table headers] In the supplementary figures, the top-level labels 'β2 β3' are ambiguous because the panels display β01, β02, β1, β2, and β3. Clarify the panel labels and the mapping from β2/β3 to x3/x4 in Tables 2-4.
  6. [Supplementary Table 4] The citation 'Firth ( ?)' is unresolved; please provide the full reference for the Firth correction.
  7. [§4.2 and §6] The statement that the 95% CP of EM approaches 0.95 as n increases is not supported by Table 4 at n=1000, where CP for y≤1 is 0.932 and for x3 is 0.960. Please revise the summary to be consistent with the tables, and soften Section 6's claim that the convergence properties are 'supported by theoretical guarantees.'

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the EM likelihood fit and the simulations are self-contained, and the only self-citation is background rather than load-bearing.

full rationale

The proposed method maximizes the joint observed-data likelihood in Eqs. (4)-(5), built from the cumulative logit model (1) and the logistic missingness model (3) that includes Y as a predictor. All parameters are estimated from that likelihood, and no fitted quantity is relabeled as a prediction. The simulation evidence is generated from an external data-generating process: Y is drawn from model (6) with fixed true parameters, and the missingness indicator R is generated from a separate logistic model with fixed alpha; the CC, EM, and MI estimates are then compared on this independent simulated data. The single author-overlapping citation, Maity et al. (2019), is used only as background ('Subsequent extensions of this framework have demonstrated strong performance in simulation studies'), not as the justification for the paper's central claim, whose support is the paper's own simulation study. Section 3.5's appeal to Fisher consistency of cumulative logit surrogate losses is an imported external result and is not circular, although it is a correctness gap because Fisher consistency concerns classification risk rather than unimodality of the incomplete-data likelihood. The supplementary Firth section explicitly concedes that Firth-corrected estimates have lower bias than EM, and Tables 3-4 contradict the blanket claim that EM 'consistently outperforms' CC and MI; these are internal-evidence and correctness problems, not circular reductions of the derivation to its inputs.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The method assumes a parametric joint model for outcome and missingness. No new latent variables or fitted constants are introduced beyond standard model parameters. The main unproven structural inputs are correct specification and identifiability of the selection model and global convergence of EM.

assumptions (3)
  • domain assumption The ordinal response follows a cumulative logit proportional odds model (Eq. 1).
    The method and simulations assume this model; if incorrect, the estimates are biased.
  • domain assumption Missingness follows a logistic model that includes the outcome Y as a predictor (Eq. 3).
    Nonignorable MNAR inference relies on this selection model being correctly specified and identifiable.
  • ad hoc to paper The EM Q-function has a unique global maximum and standard EM convergence conditions hold (Section 3.5).
    No formal proof is given; Fisher consistency of a surrogate loss is not equivalent to likelihood unimodality.

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Cite this review

Pith. "Pith review of Cumulative Logit Ordinal Regression with Proportional Odds under Nonignorable Missing Response -- Application to Phase III Trial." pith.science (2026). https://pith.science/paper/AP7BBQOC

@misc{pith2026250504727,
  author       = {Pith},
  title        = {Pith review of: Cumulative Logit Ordinal Regression with Proportional Odds under Nonignorable Missing Response -- Application to Phase III Trial},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AP7BBQOC}},
  note         = {Machine review of arXiv:2505.04727}
}
read the original abstract

Missing data are inevitable in clinical trials, and trials that produce categorical ordinal responses are not exempted from this. Typically, missing values in the data occur due to different missing mechanisms, such as missing completely at random, missing at random, and missing not at random. Under a specific missing data regime, when the conditional distribution of the missing data is dependent on the ordinal response variable itself along with other predictor variables, then the missing data mechanism is called nonignorable. In this article we propose an expectation maximization based algorithm for fitting a proportional odds regression model when the missing responses are nonignorable. We report results from an extensive simulation study to illustrate the methodology and its finite sample properties. We also apply the proposed method to a recently completed Phase III psoriasis study using an investigational compound. The corresponding SAS program is provided.

Figures

Figures reproduced from arXiv: 2505.04727 by the authors.

Figure 1
Figure 1. The mean estimates of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The mean estimates of β from 1000 replications using different methods along with the true values when about 25% data are missing. We generate 100 simulated datasets keeping the true values of β = (0.6, 0.5, −0.2, −0.7, −1.3, 0.008, −0.02)T to summarize the different matrices when about 25% observations are missing. We present the simulations summaries in Tables 1 and 2. We note that, the results corroborate the evi… view at source ↗
Figure 3
Figure 3. The mean estimates of β from 1000 replications using different methods along with the true values when about 45% data are missing. example, in the following in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Reference graph

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