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REVIEW 3 major objections 5 minor 32 references

Maximum Likelihood for Logistic Regression Model with Incomplete and Hybrid-Type Covariates

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims a stochastic-approximation EM algorithm for logistic regression with missing mixed-type covariates estimates regression coefficients and covariate distributions jointly, and beats standard imputation pipelines on bias…

desk verdict A plausible SAEM recipe for logistic regression with missing mixed-type covariates, backed by code and simulations, but the headline claim of consistent superiority is contradicted by its own tables and the key independence assumption is untested. read the letter →

arxiv 2506.03445 v1 pith:AOBEWV5I submitted 2025-06-03 stat.ME eess.SP

Mohamed Cherifi , Xujia Zhu , Mohammed Nabil El Korso , Ammar Mesloub This is my paper
classification stat.MEeess.SP MSC 62F1062J12
keywords logisticregressionmissingcovariatesmixed-typedataSAEMalgorithmMetropolis-Hastingsmultipleimputationmaximumlikelihoodatrandom
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

The paper proposes that logistic regression with missing covariates of mixed type (discrete and continuous) is best handled by a stochastic-approximation version of the EM algorithm that jointly estimates the regression coefficients and the covariate distributions, rather than imputing first and fitting second. The authors try to establish that this SAEM procedure, with Metropolis-Hastings sampling for the missing continuous values and weighted sums for the missing discrete ones, gives lower bias and RMSE for the regression coefficients and better predictive accuracy, AUC, and Brier scores than MICE, MissForest, mean/mode imputation, and complete-case analysis. If true, practitioners with MCAR or MAR missingness in mixed covariate data would have a one-step alternative that dominates standard imputation pipelines in the tested regimes. The claim is supported by simulations at 30 to 60 percent missingness and by four real datasets.

What carries the argument

The machinery is SAEM (Stochastic Approximation Expectation-Maximization), driven by the complete-data factorization $p(y_i, x_i; \theta) = p(y_i \mid x_i; \beta) \, p(x_i^c; \theta_c) \, p(x_i^d; \theta_d)$. The S-step samples missing continuous covariates by Metropolis-Hastings from the conditional distribution that sums over missing discrete values; the SA-step updates a stochastic approximation of the expected complete log-likelihood component-wise with smoothing parameters $\delta_t$; and the M-step maximizes each component separately. The workhorse identities are the closed-form updates for the Gaussian sufficient statistics $T_1$ and $T_2$ and for the multinomial probabilities, which keep the stochastic approximation exact in those blocks.

What would settle it

Generate a dataset in which the discrete and continuous covariates are strongly correlated, delete values under a mechanism that depends on both, and compare SAEM against MICE on many replications; if SAEM's bias and RMSE are not lower than MICE's in that setting, the independence factorization is the point of failure. A simpler check is to fit the paper's model to a real mixed dataset and compare the implied joint distribution of the covariates with the empirical distribution.

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Extended reading notes

Core claim

The central claim is that the complete-data likelihood separates into a logistic term for the binary outcome, a multivariate Gaussian term for the continuous covariates, and a product of independent multinomial terms for the discrete covariates. Under this factorization, the E-step of EM can be replaced by a stochastic approximation in which missing continuous values are drawn from their conditional distribution by Metropolis-Hastings, missing discrete values are averaged with posterior weights, and the M-step splits into three separate updates: gradient ascent for the regression coefficients, closed-form sufficient-statistic updates for the Gaussian mean and covariance, and analytic updates for the multinomial probabilities. The paper asserts that this algorithm, run with a decreasing smoothing sequence, yields maximum-likelihood-type estimates under ignorable missingness and that in simulations it consistently has lower bias and RMSE for the coefficients and better test-set AUC, accuracy, and Brier scores than MICE, MissForest, mean/mode imputation, and complete-case analysis. Prediction with missing test covariates follows the same model by marginalizing over the missing parts, using Monte Carlo draws for continuous values and exact sums for discrete ones.

Load-bearing premise

The load-bearing premise is that, conditional on the regression, each sample's continuous and discrete covariates are independent of each other and the discrete covariates are mutually independent, so the joint covariate distribution factorizes into one Gaussian and a product of multinomials.

Editorial extensions

If this is right

  • For MCAR and MAR missingness in mixed covariates, the regression coefficient estimates from SAEM have lower bias and RMSE than MICE, MissForest, mean/mode imputation, and complete-case analysis in the simulated settings tested.
  • Prediction on test data with missing covariates can be made directly from the fitted joint model, without a separate imputation step, by marginalizing over the missing values.
  • The discrete-covariate parameters (multinomial probabilities) and continuous-covariate parameters (mean and covariance) are estimated in the same run at negligible extra cost, because their M-steps are closed form.
  • The method remains usable at high missingness rates: reported results include 50% and 60% MCAR and 30% to 40% MAR, with SAEM maintaining the highest AUC, accuracy, and lowest Brier score in those comparisons.
  • Testing-phase computation for SAEM is much faster than MissForest in the reported timings, for example about two seconds versus thirty-eight seconds in the largest configuration.

Reading between the lines

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

  • Editorial inference: because the covariate-distribution block does not depend on the form of the outcome model, the same SAEM skeleton should transfer to probit, Poisson, or other generalized linear models; only the update for the outcome coefficients would change.
  • Editorial inference: under the independence factorization, the method's advantage should shrink when discrete and continuous covariates are strongly dependent; a user with such data would need a coupled covariate model, which the paper does not provide.
  • Editorial inference: a practical benchmark implied by the paper is that gains over MissForest should be largest at high missingness rates and with informative covariates; users with weak covariates and low missingness may see near-tied performance.
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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

3 major / 5 minor

Summary. The paper proposes a Stochastic Approximation Expectation-Maximization (SAEM) algorithm for maximum-likelihood estimation in logistic regression with missing covariates of mixed continuous and discrete type. The complete-data model assumes multivariate Gaussian continuous covariates, independent multinomial discrete covariates, and full independence between the continuous and discrete blocks. The E-step uses Metropolis–Hastings to sample missing continuous covariates, the SA-step updates stochastic sufficient statistics, and the M-step updates the logistic coefficients, Gaussian parameters, and multinomial probabilities component-wise. The method is compared with MICE, MissForest, mean/mode imputation, and complete-case analysis on synthetic and four real datasets, with the abstract and conclusion claiming consistent superiority in accuracy and reliability.

Significance. If the method performed as claimed, it would provide a useful joint-modeling alternative to separate imputation plus logistic regression for hybrid missing covariates under ignorable missingness. The manuscript has strengths: the source code is provided, the simulation studies compare against ground truth and external baselines, and the likelihood objective is standard and clearly stated. However, the paper's central claim of consistent outperformance is not supported by its own numerical results, and the core independence assumption between continuous and discrete covariates is never tested in settings where that assumption fails. The contribution is therefore only partially validated and needs substantial revision before the performance claims can be accepted.

major comments (3)
  1. [§IV and Supplementary Table III] The abstract and conclusion claim that the proposed SAEM 'consistently outperforms traditional methods.' This is contradicted by the paper's own numbers: in Supplementary Table III, under 50% MCAR, SAEM has RMSE 0.4397 for β1, while MissForest achieves 0.3323 and the full-data reference 0.2618; SAEM also loses to MissForest on RMSE for β3 (0.1354 vs 0.0906), β4 (0.0860 vs 0.0689), and β7 (0.0961 vs 0.0909). SAEM has smaller bias for β1 but larger variance, so the broad claim is already overstated in the favorable independent-covariate setting. Please revise the superiority claims or provide a statistical comparison across the 100 runs that supports the stated conclusion.
  2. [§II and Supplementary Eq. (18)] The complete-data model factorizes as p(y_i, x_i; θ) = p(y_i | x_i; β) p(xc_i; θc) p(xd_i; θd), which imposes independence between continuous and discrete covariates and mutual independence among discrete covariates. This factorization drives the E-step in Eq. (4), the Metropolis–Hastings target in Supplementary Eq. (15), the discrete posterior weights, the stochastic-approximation sufficient statistics, and the prediction rule in Section III-B. The synthetic experiments in Section IV generate discrete and continuous covariates independently, so they provide no evidence about the realistic regime in which, for example, a discrete covariate such as sex or smoking status affects the mean of a continuous biomarker. Under such dependence, the MH sampler targets p(xc_mis | xc_obs, y) rather than p(xc_mis | xc_obs, xd_obs, y), and the discrete imputation weights ignore observed continuous covariates; both can bias β even under MCAR/MAR missingness. Please add simulation scenarios with dependence between the covariate blocks and a sensitivity analysis for the real data.
  3. [§IV, smoothing schedule] The description of the SAEM step-size sequence is internally inconsistent. The text states that δ_t = (t + t1)^{-τ} with t1 = 50, but also states that δ_t = 1 during the first 50 iterations. At t = 50, the formula gives (100)^{-1} = 0.01, not 1. Please specify the exact schedule, including how the transition from the warm-up phase to the decreasing phase is implemented, so that the experiments are reproducible.
minor comments (5)
  1. [Abstract and §V] The phrases 'consistently outperforms' and 'demonstrate that the SAEM approach outperforms classical techniques' should be tempered to reflect the scenarios actually tested, in light of the RMSE results in Supplementary Table III.
  2. [Eq. (2)] The displayed complete log-likelihood appears to have an extra closing parenthesis at the end of the Gaussian term; please check the formula and correct the typographical error.
  3. [Supplementary Table III caption] The caption states that the best values are highlighted in bold, but the table as printed contains no bold entries; this should be fixed or the caption amended.
  4. [§IV] The phrase 'almost 30 faster' should read 'almost 30 times faster'.
  5. [Supplementary Materials, dataset descriptions] The French phrase 'dataset utilis´e dans R' is a leftover fragment; it should be translated or removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation follows from an explicitly stated complete-data factorization and is validated against external baselines, not from self-citation or fitted inputs.

full rationale

The central derivation is self-contained. Equation (18) states the complete-data factorization p(y_i,x_i;theta)=p(y_i|x_i;beta)p(xc_i;theta_c)p(xd_i;theta_d) as an explicit modeling assumption, and the E-step (Eq. 4), MH target (Supplementary Eq. 15), component-wise stochastic approximation updates (Eqs. 6-11), and prediction rule (Section III-B) all follow algebraically from that factorization plus standard SAEM convergence results cited to external works [21,25]. The performance claim is tested against synthetic data with known beta and against MICE, MissForest, mean/mode, and complete-case baselines on real datasets; no fitted parameter from those experiments is renamed as a prediction, nor is any external benchmark imported from the authors' own prior work. The only self-citations ([17] in a literature survey and [28] for source code) are not load-bearing: the code link is for reproducibility, and the convergence of SAEM is attributed to Delyon et al. and Kuhn and Lavielle, not to the authors' own papers. The factorization in (18) is a substantive assumption that may fail on dependent hybrid data, but that is a misspecification and generalizability risk, not a circular step: the paper does not define its target in terms of its output. The simulations being generated under the method's own independence model is a standard validation design, and it does not make the estimator's output equal to its input. Accordingly, no circular step is identified.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a factorized parametric model, a standard SAEM framework, and several hand-tuned or unreported constants. No new physical or statistical entities are introduced. The most consequential assumptions are independence between continuous and discrete covariates and ignorable missingness, neither of which is checked on real data.

free parameters (3)
  • SAEM smoothing schedule (t1, tau, warm-up) = t1=50, tau=1, delta_t=1 for first 50 iterations
    Chosen by hand in Section IV for speed and stability. No sensitivity analysis is reported, and SAEM estimates can depend on this schedule.
  • Metropolis-Hastings chain length and burn-in S = not reported
    Algorithm 1 in supplement S.I uses S iterations per missing continuous block, but the paper never states S or the burn-in. This controls the quality of the imputed continuous covariates.
  • Prediction Monte Carlo sample size S = not reported
    Section III.B approximates the predictive integral with S samples from p(xc_mis | xc_obs; theta_c); S is not specified.
assumptions (5)
  • domain assumption Missingness mechanism is ignorable (MCAR/MAR)
    Invoked after Eq. (3): 'Under the assumption of independence and ignorable missing data mechanism'. Real datasets with naturally occurring missing values are treated as ignorable without verification.
  • domain assumption Continuous and discrete covariates are independent, and discrete covariates are mutually independent
    Appears in Eq. (2) and supplement Eq. (18): p(y_i,x_i;theta)=p(y_i|x_i;beta)p(xc_i;theta_c)p(xd_i;theta_d). The E-step, Metropolis-Hastings target, and prediction formula all rely on this factorization.
  • domain assumption Continuous covariates follow a multivariate Gaussian and each discrete covariate follows an independent multinomial distribution
    Section III.A sets theta_c={mu,Sigma} and theta_d_j as multinomial probabilities. These distributional forms are used in the complete-data likelihood and the MH proposal.
  • standard math SAEM convergence conditions from Delyon et al. and Kuhn-Lavielle hold for this model with the MH sampler
    The paper applies SAEM without proving that the cited regularity conditions are satisfied here. It relies on references [21] and [25] for convergence.
  • domain assumption Training and test covariates share the same distribution
    Section III.B states the prediction procedure 'assuming that training and test data come from the same distribution'. This is required for the Monte Carlo prediction rule.

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Cite this review

Pith. "Pith review of Maximum Likelihood for Logistic Regression Model with Incomplete and Hybrid-Type Covariates." pith.science (2026). https://pith.science/paper/AOBEWV5I

@misc{pith2026250603445,
  author       = {Pith},
  title        = {Pith review of: Maximum Likelihood for Logistic Regression Model with Incomplete and Hybrid-Type Covariates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AOBEWV5I}},
  note         = {Machine review of arXiv:2506.03445}
}
read the original abstract

Logistic regression is a fundamental and widely used statistical method for modeling binary outcomes based on covariates. However, the presence of missing data, particularly in settings involving hybrid covariates (a mix of discrete and continuous variables), poses significant challenges. In this paper, we propose a novel Expectation-Maximization based algorithm tailored for parameter estimation in logistic regression models with missing hybrid covariates. The proposed method is specifically designed to handle these complexities, delivering efficient parameter estimates. Through comprehensive simulations and real-world application, we demonstrate that our approach consistently outperforms traditional methods, achieving superior accuracy and reliability.

Figures

Figures reproduced from arXiv: 2506.03445 by the authors.

Figure 1
Figure 1. Boxplot of the difference between the estimate and the ground truth for β1 over 100 algorithm runs with 30% MAR missing data mechanisms. MICE approach (Multiple Imputation by Chained Equations), adapted to handle mixed data [26], and the Miss-Forest algorithm, which has also been tailored for mixed data [13]. Additionally, we include the MeanMode (MM) method [27], which imputes missing values for continuous variable… view at source ↗
Figure 2
Figure 2. Comparison of the Area Under the Curve (AUC), Accuracy, and Brier Score obtained on the test set for each method with 30% MCAR missing data over 100 algorithm runs. showcases the superior performance of the SAEM algorithm in terms of Area Under the Curve (AUC) and accuracy, as well as a significantly reduced Brier Score. The Brier Score, where lower values indicate better performance, reflects the quality of the pro… view at source ↗
Figure 3
Figure 3. Boxplot of the difference between the estimate and the ground truth for SAEM Parameters with 30% missing data under the MCAR mechanism over 100 algorithm runs. September 16, 2025 DRAFT [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Boxplot of the Euclidean norm of the βˆ estimates over 100 algorithm runs with 30% MCAR missing data mechanism. Metric SAEM MICE MM MissForest AUC 0.865 (0.022) 0.856 (0.021) 0.856 (0.022) 0.856 (0.021) Accuracy 0.813 (0.022) 0.801 (0.021) 0.805 (0.024) 0.808 (0.022) P…

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