{"id":"b04bf43f-0865-43e2-a4d3-f3ab07c7cc62","arxiv_id":"2506.03445","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A stochastic-approximation EM algorithm with Metropolis-Hastings imputation is proposed for logistic regression with missing mixed-type covariates and is shown to beat MICE, MissForest, mean/mode, and complete-case baselines in most tested settings.","lead":"Researchers built a variant of the expectation-maximization algorithm that estimates logistic-regression coefficients when some covariates are missing and the covariates mix categorical and continuous values. They report lower bias and better predictions than several common imputation methods in simulations and four real datasets.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim depends on an untested independence assumption between continuous and discrete covariates (supplement Eq. 18); under realistic dependent hybrid data the method's imputation distribution is misspecified, so consistent superiority is not established.","rationale":"I agree with the reader that the independence assumption is the weakest link. The paper's own derivations and prediction formula depend on it, and the simulation design cannot detect violations because it generates covariates independently. I also note the paper's Table III undercuts 'consistently outperforms' in the beta-RMSE metric, which supports a conditional rather than unconditional reading. The algorithm is technically coherent under its stated model and the code release is a positive; the issue is an untested misspecification in the central claim, not an internal contradiction. A targeted correlated-covariate simulation would settle it. Thus I would keep the CONDITIONAL verdict.","tokens_in":14925,"tokens_out":8380,"duration_ms":94239,"concrete_test":"Simulate n=1000 with continuous covariates x_c ~ N(0,Sigma) and discrete covariates constructed to depend on x_c, e.g., xd1 = I(xc1 + epsilon1 > 0.5), xd2 multinomial with probabilities via softmax on xc2, plus an independent binary covariate. Generate y with a fixed beta. Impose 30% MCAR and 30% MAR missingness on both types. Run SAEM and the four benchmarks 100 times. If SAEM's mean absolute bias in beta and test AUC/Brier are not better than MissForest/MICE, or worsen relative to an independent-covariate control, the independence assumption is load-bearing. If SAEM remains best under dependence, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim—lower bias/RMSE and better prediction than MICE/MissForest over a range of settings—requires the complete-data model in Eq. (2)/(18) to be at least approximately correct. That model factorizes p(y_i,x_i;theta)=p(y_i|x_i;beta)p(xc_i;theta_c)p(xd_i;theta_d), i.e., the continuous and discrete covariates are independent, and the discrete covariates are mutually independent. This factorization is used in the E-step (4), the MH target (supplement Eq. 15), the discrete posterior weights, the SA-step sufficient statistics, and the prediction rule in Section III-B. Real hybrid data commonly contain dependencies between, e.g., age/sex and biomarkers. If such dependencies exist, the MH step samples p(xc_mis|xc_obs,y) rather than the correct p(xc_mis|xc_obs,xd_obs,y), and the weights for xd_mis ignore observed continuous covariates; both can bias beta under MCAR/MAR missingness. The simulations in Section IV generate continuous and discrete covariates independently, so they provide no evidence about this regime. Table III even shows SAEM is not uniformly best for beta RMSE under the favorable independent model (e.g., beta1 RMSE 0.4397 vs MissForest 0.3323), so 'consistently outperforms' is already overstated. Real-data results have no ground-truth beta and no sensitivity analysis. The concern is not that the algorithm is incoherent under its assumptions, but that the central practical claim rests on a model assumption the paper never tests.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15249,"tokens_out":5727,"duration_ms":65552,"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":[{"comment":"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.","section":"§IV and Supplementary Table III"},{"comment":"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.","section":"§II and Supplementary Eq. (18)"},{"comment":"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.","section":"§IV, smoothing schedule"}],"minor_comments":[{"comment":"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.","section":"Abstract and §V"},{"comment":"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.","section":"Eq. (2)"},{"comment":"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.","section":"Supplementary Table III caption"},{"comment":"The phrase 'almost 30 faster' should read 'almost 30 times faster'.","section":"§IV"},{"comment":"The French phrase 'dataset utilis´e dans R' is a leftover fragment; it should be translated or removed.","section":"Supplementary Materials, dataset descriptions"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the scope of a statistical methods journal, but the performance claims need to be brought in line with the evidence, and the independence assumption needs explicit testing. The supplied code and reproducible simulation framework are positive elements that should be retained in the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a workmanlike SAEM implementation for logistic regression with missing mixed-type covariates. The new part is not any single ingredient—SAEM, Metropolis-Hastings, Gaussian and multinomial covariate models are all established—but the specific assembly for hybrid covariates, with component-wise M-steps and a Monte Carlo prediction rule. That is a legitimate niche contribution, and the authors back it with a public GitHub release and simulations against MICE, MissForest, mean/mode, and complete-case analysis. The prediction-time comparison (much faster than MissForest at larger sample sizes) is a genuine practical point.\n\nWhere it wobbles is the headline claim. The abstract says the approach consistently outperforms traditional methods, but the paper's own Supplementary Table III shows SAEM's RMSE for β1 (0.4397) is worse than MissForest's (0.3323) and much worse than Full data (0.2618) under 50% MCAR. That is not \"consistent outperformance\" by the paper's own numbers. The real-data tables are more favorable, but there is no ground-truth β, so those only show prediction metrics.\n\nThe bigger soft spot is the independence assumption between continuous and discrete covariates, stated in Eq. (2) and supplement Eq. (18), and used throughout the E-step, the MH target, and the prediction rule. The paper generates continuous and discrete covariates independently in simulation, so it never exercises a regime where they are correlated—which is common in real hybrid data (e.g., age and a categorical biomarker). Under dependence, the imputation distribution is misspecified and β can be biased even under MCAR/MAR. The authors do not flag this limitation or provide a sensitivity analysis. That is a load-bearing gap, not a minor quibble.\n\nThere are also smaller issues: the closest likelihood-based competitor, Jiang et al. (ref [16]), is cited but not benchmarked, and the printed update equations have a notation slip (Eq. 8 lists G2 as μ,Σ and G3 as θd, swapped relative to their definitions). These are fixable.\n\nAll that said, the algorithm is coherent under its stated assumptions, the code is out, and the simulation design is sensible. It is not a broken paper; it is an overclaimed one. I would send it to peer review, but the referee should push for weaker claims, a test with dependent hybrid covariates, and a benchmark against a joint-modeling method. The authors know the machinery; they need to show it survives contact with realistic data.\n\nRecommendation: worth a serious referee, conditional on revision.","headline":"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.","tokens_in":15789,"tokens_out":2898,"would_cite":false,"duration_ms":28195,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F10","62J12"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["logistic regression","missing covariates","mixed-type data","SAEM algorithm","Metropolis-Hastings","multiple imputation","maximum likelihood","missing at random"],"falsifier":"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.","tokens_in":14711,"feed_emoji":"📊","tokens_out":4905,"duration_ms":50999,"temperature":0.7,"pith_summary":"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.","feed_headline":"SAEM beats MICE and MissForest for missing mixed-type covariates","feed_subtitle":"A stochastic-EM estimator jointly fits coefficients and covariate distributions, cutting bias and Brier scores up to 60% missingness.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the SAEM convergence theory that the algorithm's stochastic approximation step relies on.","marker":"[21]"},{"why":"Provides the MICE baseline that the paper's simulation and real-data comparisons must beat.","marker":"[26]"},{"why":"Provides the MissForest baseline, the strongest nonparametric imputation competitor in the comparisons.","marker":"[13]"},{"why":"Supplies the EM framework for missing covariates in generalized linear models that the paper extends to mixed types.","marker":"[24]"},{"why":"Justifies the complete-data factorization under independence, the structural assumption the whole algorithm builds on.","marker":"[23]"},{"why":"Provides the coupling of SAEM with MCMC and the decreasing smoothing sequence used in the S-step and SA-step.","marker":"[25]"}],"fun_headline_variants":["SAEM improves logistic regression with missing mixed covariates","New EM handles hybrid missing data better than imputation","Stochastic EM beats standard imputation for mixed-type missingness","Logistic regression with missing covariates: SAEM wins","SAEM yields accurate logistic regression with missing mixed covariates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["SAEM improves logistic regression with missing mixed covariates","New EM handles hybrid missing data better than imputation","Stochastic EM beats standard imputation for mixed-type missingness","Logistic regression with missing covariates: SAEM wins","SAEM yields accurate logistic regression with missing mixed covariates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000627,"raw_usage":{"total_tokens":2858,"prompt_tokens":861,"completion_tokens":1997,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":477,"completion_tokens_details":{"reasoning_tokens":1920}},"tokens_in":477,"tokens_out":1997,"duration_ms":16541,"temperature":1.0,"reasoning_tokens":1920,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:03:25.388389+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Convergence o f a stochastic approximation version of the em algorithm,","cited_arxiv_id":null,"evidence_quote":"Supplies the SAEM convergence theory that the algorithm's stochastic approximation step relies on."},{"cited_title":"mice: Multi variate imputation by chained equations in r,","cited_arxiv_id":null,"evidence_quote":"Provides the MICE baseline that the paper's simulation and real-data comparisons must beat."},{"cited_title":"Missforest—non-parametric missing value imputation for mixed- type data,","cited_arxiv_id":null,"evidence_quote":"Provides the MissForest baseline, the strongest nonparametric imputation competitor in the comparisons."},{"cited_title":"Missing co variates in generalized linear models when the missing data mechanism is non-ignorable,","cited_arxiv_id":null,"evidence_quote":"Supplies the EM framework for missing covariates in generalized linear models that the paper extends to mixed types."},{"cited_title":"Dynamic expectation-maximization algorithms for mixed-type data,","cited_arxiv_id":null,"evidence_quote":"Justifies the complete-data factorization under independence, the structural assumption the whole algorithm builds on."},{"cited_title":"Coupling a stochastic approxi mation version of emwith an mcmc procedure,","cited_arxiv_id":null,"evidence_quote":"Provides the coupling of SAEM with MCMC and the decreasing smoothing sequence used in the S-step and SA-step."}],"review_version":1}