arxiv: 2605.08897 · v1 · submitted 2026-05-09 · 💻 cs.LG · cs.AI

Recognition: 2 theorem links

· Lean Theorem

Shapley Regression for Rare Disease Diagnosis Support: a case study on APDS

Safa Alsaidi , Tom\'as Brogueira , Nizar Mahlaoui , Marc Vincent , Guilherme Pelegrina , Nicolas Garcelon , Adrien Coulet , Miguel Couceiro

Authors on Pith no claims yet

Pith reviewed 2026-05-12 00:54 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords Shapley regressionrare disease diagnosisAPDSelectronic health recordsinterpretable machine learningsymptom interactionslogistic regressiongame theory

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The pith

Shapley regression replaces linear predictors with k-additive games to model symptom co-occurrences for APDS diagnosis.

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

The paper develops Shapley regression to detect rare APDS from routine health records by treating symptoms as players in a cooperative game that captures their joint effects. This keeps the model convex and interpretable like logistic regression while handling interactions that linear scores miss. Tests across eight public datasets identify a 2-additive version with l2 regularization as the best balance of accuracy and robustness to noise. On a matched cohort of 222 patients the method separates APDS cases from controls and surfaces both established and new pairwise symptom links that clinical experts confirm.

Core claim

Shapley regression is a game-theoretic model that replaces the linear predictor with a k-additive cooperative game, explicitly modeling co-occurrence of symptoms while retaining the transparency and convexity of logistic regression. On eight biomedical datasets a 2-additive model with l2 regularization achieves the optimal trade-off between predictive power and noise robustness. Applied to a real-world cohort of 222 patients, the approach accurately distinguishes APDS cases from matched controls, confirms known associated phenotypes, and enables exploration of pairwise symptom interactions validated by clinical experts.

What carries the argument

Shapley regression: a k-additive cooperative game that substitutes for the linear term inside logistic regression to encode symptom co-occurrences.

If this is right

The model can flag potential APDS patients earlier from standard electronic records without requiring specialist input.
Pairwise symptom interactions become directly inspectable and can be checked against clinical knowledge.
The same lightweight approach applies to other rare diseases whose symptoms overlap with common conditions.
Interpretability is preserved so clinicians can trace which symptom combinations drive each prediction.

Where Pith is reading between the lines

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

Similar game-based replacements could be tested in other medical domains where interactions among binary features matter but full deep models are too opaque.
Deploying the method inside hospital record systems might shorten the typical multi-year diagnostic delay for APDS and comparable disorders.
Larger or multi-center cohorts would reveal whether the 2-additive limit remains sufficient or whether certain patients require higher-order terms.
The convexity property may allow efficient integration with existing clinical decision-support tools that already use logistic scores.

Load-bearing premise

Truncating the game to pairs of symptoms plus l2 regularization is enough to capture the clinically important interactions without missing higher-order effects or inheriting bias from the control-matching process.

What would settle it

An independent APDS cohort where adding triple-wise symptom terms raises predictive accuracy by more than 5 percent or where the 2-additive model fails to separate cases from controls at the reported level.

Figures

Figures reproduced from arXiv: 2605.08897 by Adrien Coulet, Guilherme Pelegrina, Marc Vincent, Miguel Couceiro, Nicolas Garcelon, Nizar Mahlaoui, Safa Alsaidi, Tom\'as Brogueira.

**Figure 1.** Figure 1: Empirical Validation of Algorithmic Stability. We measure sensitivity (Euclidean shift ∥θˆS − θˆ S′∥2 after flipping a single label) across varying regularisation strengths C. The dashed line represents the theoretical linear bound (β ∝ C) derived in Regime 3. Crucially, the empirical points follow this linear trend in the high-regularisation regime (small C) but naturally deviate and saturate around C ≈ 1… view at source ↗

**Figure 2.** Figure 2: Effective Dimension Validation. Comparison of dimensions and generalization gaps (Settings: N = 1000, n = 8, averaged over 10 iterations). The plot demonstrates the divergence between the combinatorial dimension Dk (black dotted line), which grows exponentially, and the effective dimension deff (blue line), which saturates. Crucially, the generalization gap of the ℓ2-regularized model (blue dots) tracks th… view at source ↗

**Figure 3.** Figure 3: Marginal contribution (main effects) among phenotypes in APDS dataset. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Interaction effects among phenotypes in APDS dataset. The heatmap illustrates the mean pairwise interaction strength between the top interacting phenotypes. Red indicates positive interaction (synergy), while blue indicates negative (redundancy). while providing a key advantage: explanatory global Shapley Interaction Indices that offer insights into phenotype interactions. For APDS, this framework allows u… view at source ↗

read the original abstract

Activated PI3K8 Syndrome (APDS) is a rare genetic immune disorder caused by variants in PIK3CD or PIK3R1, with highly heterogeneous symptoms that often delay diagnosis. Early recognition is hampered by overlapping clinical presentations and limited clinician awareness, motivating systematic, data-driven approaches to detect APDS-associated phenotypic patterns in routine electronic health records. Traditional linear scoring systems cannot capture complex symptom interactions, while deep learning models, though expressive, often lack interpretability. To bridge this gap, we propose Shapley regression, a novel game-theoretic model replacing the linear predictor with a k-additive cooperative game, explicitly modeling co-occurrence of symptoms while maintaining the transparency and convexity of logistic regression. We carry out an empirical study of our lightweight method on eight public biomedical datasets, showing that a 2-additive model with $l_{2}$ regularization achieves an optimal trade-off between predictive power and noise robustness. We also apply it to a real-world cohort of 222 patients, on which Shapley regression accurately distinguished APDS cases from matched controls, confirming and validating phenotypes known to be associated with APDS, and facilitating the exploration of pairwise interactions between symptoms, validated by clinical experts.

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.

Desk Editor's Note private letter to a colleague

Shapley regression swaps the linear term in logistic regression for a k-additive game to capture symptom interactions, which is a clean idea for rare-disease work, but the APDS results rest on limited metrics and an untested matching step.

read the letter

The main thing to know is that the paper replaces the usual linear predictor in logistic regression with a k-additive cooperative game. This keeps convexity and interpretability while letting the model pick up pairwise symptom co-occurrences directly. They show on eight public biomedical datasets that the 2-additive version plus l2 regularization gives a reasonable accuracy-robustness trade-off, then apply it to a 222-patient APDS cohort where it separates cases from matched controls, recovers known phenotypes, and flags some pairwise interactions that experts signed off on. The expert validation step is a practical plus for clinical use. The soft spots are straightforward. The abstract and available description give no AUCs, confidence intervals, cross-validation folds, or direct comparisons to plain logistic regression or other interaction baselines, so the performance lift is hard to judge. The matched-control setup also leaves room for selection bias—if matching criteria overlap with the symptom features, the reported separation and interactions could partly reflect the matching rather than the model itself. No sensitivity checks on unmatched controls or alternative schemes are mentioned, which is the load-bearing assumption the stress-test note flags. This paper is for clinical informaticians and rare-disease groups who want something more expressive than linear scoring but still transparent enough for EHR screening. A reader working on similar diagnostic tools could borrow the model formulation even if the specific numbers need more support. It deserves a serious referee because the modeling choice is grounded and the application is concrete, though the evidence will need bolstering on metrics and robustness. I would send it to peer review with requests for full results tables, ablations, and matching sensitivity analyses.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes Shapley regression, a k-additive cooperative game extension to logistic regression that explicitly models symptom co-occurrences while preserving convexity and interpretability. It reports that a 2-additive model with L2 regularization achieves an optimal trade-off on eight public biomedical datasets. The method is then applied to a 222-patient APDS cohort, where it distinguishes cases from matched controls, validates known phenotypes, and identifies expert-confirmed pairwise symptom interactions.

Significance. If the quantitative claims hold, the work provides a transparent, convex alternative to linear models or black-box approaches for phenotyping rare diseases from EHR data, with explicit handling of pairwise interactions. The expert validation step adds clinical utility, and the focus on a real-world rare-disease cohort demonstrates practical applicability. No machine-checked proofs or fully reproducible code artifacts are described, but the game-theoretic framing offers a clear path for falsifiable follow-up studies.

major comments (2)

[Abstract and Empirical Evaluation] Abstract and Empirical Evaluation sections: the claim that a 2-additive model with L2 regularization achieves an optimal trade-off between predictive power and noise robustness is unsupported by any reported metrics (AUC, accuracy, F1), cross-validation details, confidence intervals, or ablation tables comparing k values and regularization strengths. This is load-bearing for the central empirical claim and must be addressed with specific numbers and baselines.
[APDS Cohort Application] APDS Cohort Application section: the matching procedure used to select controls in the 222-patient cohort is unspecified (variables, criteria, or algorithm). This is load-bearing for the claim that Shapley regression 'accurately distinguished APDS cases from matched controls' and validated phenotypes, because matching on age, sex, or correlated factors can alter symptom co-occurrence statistics and introduce spurious signals, as highlighted by the study design critique. Sensitivity checks (unmatched controls or alternative schemes) are required.

minor comments (1)

[Abstract] The abstract lists 'eight public biomedical datasets' without naming them or providing accession details, which hinders immediate reproducibility assessment.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback. We address each major comment point by point below. Where the comments identify gaps in reporting or description, we have revised the manuscript to incorporate the requested details and analyses.

read point-by-point responses

Referee: [Abstract and Empirical Evaluation] Abstract and Empirical Evaluation sections: the claim that a 2-additive model with L2 regularization achieves an optimal trade-off between predictive power and noise robustness is unsupported by any reported metrics (AUC, accuracy, F1), cross-validation details, confidence intervals, or ablation tables comparing k values and regularization strengths. This is load-bearing for the central empirical claim and must be addressed with specific numbers and baselines.

Authors: We agree that the central empirical claim requires explicit quantitative support. The manuscript reports results from an empirical study across eight public biomedical datasets but does not include the specific performance metrics, cross-validation details, confidence intervals, or ablation tables in the main text. In the revised manuscript we will add a table in the Empirical Evaluation section reporting AUC, accuracy, and F1 scores for k=1, 2, and 3 under no regularization, L1, and L2 regularization, together with 5-fold cross-validation results and 95% confidence intervals. Standard logistic regression will be included as an explicit baseline to demonstrate the claimed trade-off. revision: yes
Referee: [APDS Cohort Application] APDS Cohort Application section: the matching procedure used to select controls in the 222-patient cohort is unspecified (variables, criteria, or algorithm). This is load-bearing for the claim that Shapley regression 'accurately distinguished APDS cases from matched controls' and validated phenotypes, because matching on age, sex, or correlated factors can alter symptom co-occurrence statistics and introduce spurious signals, as highlighted by the study design critique. Sensitivity checks (unmatched controls or alternative schemes) are required.

Authors: We agree that the matching procedure must be fully specified and that sensitivity checks are necessary to support the claims. In the revised manuscript we will expand the APDS Cohort Application section to describe the matching variables, criteria, and algorithm in detail. We will also add sensitivity analyses using unmatched controls and at least one alternative matching scheme, reporting the resulting performance and phenotype validation outcomes to assess robustness against potential confounding. revision: yes

Circularity Check

0 steps flagged

No circularity: model definition and empirical results are independent

full rationale

The paper defines Shapley regression by replacing the linear term in logistic regression with a k-additive cooperative game (standard axioms plus convexity), then reports empirical performance on public datasets and a 222-patient cohort. No equation reduces a reported prediction or validation result to a fitted parameter by construction, no self-citation chain is load-bearing for the central claims, and the expert-validated phenotype interactions are external to the model's equations. The matching-procedure concern is a potential validity issue, not a circularity reduction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The approach rests on the assumption that symptom co-occurrence can be faithfully represented by a low-order additive game and that l2 regularization suffices to control noise; no new physical entities are postulated.

free parameters (2)

additivity order k = 2
Set to 2 after empirical comparison on eight public datasets; the choice directly controls model capacity.
l2 regularization coefficient
Applied to the 2-additive model but its specific value is not reported in the abstract.

axioms (2)

domain assumption Symptom interactions are adequately captured by coalitions of size at most k.
Invoked when the linear predictor is replaced by the k-additive game value function.
standard math The resulting game value function remains convex and therefore compatible with logistic regression optimization.
Required to preserve the transparency and convexity properties claimed for the model.

pith-pipeline@v0.9.0 · 5542 in / 1580 out tokens · 27643 ms · 2026-05-12T00:54:30.112771+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose Shapley regression, a novel game-theoretic model replacing the linear predictor with a k-additive cooperative game... 2-additive model with ℓ2 regularization achieves an optimal trade-off between predictive power and noise robustness.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and orbit embedding unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The learnable coefficients I(A) are exactly the Shapley Interaction Indices... ϕ_A(x) = sum ... min_i∈C x_i

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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