arxiv: 2512.07578 · v3 · submitted 2025-12-08 · 📊 stat.ML · cs.LG· stat.ME

φ-Table: A Statistical Explanation for Global SHAP

Dongseok Kim , Hyoungsun Choi , Mohamed Jismy Aashik Rasool , Gisung Oh This is my paper

Pith reviewed 2026-05-17 00:25 UTC · model grok-4.3

classification 📊 stat.ML cs.LGstat.ME

keywords SHAPglobal explanationblack-box modellinear surrogatefeature importancemodel interpretabilityregressionstatistical table

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The pith

A standardized linear surrogate on SHAP-selected features turns global rankings into a table of directional coefficients, uncertainties, fidelity, and stability.

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

The paper introduces the φ-table to give statistical substance to global SHAP explanations for tabular black-box regression models. It first ranks features by SHAP importance, then fits a standardized linear surrogate directly to the black-box response values on those features. The table reports the original SHAP ranks alongside the surrogate coefficients, their uncertainty, a fidelity score showing how closely the linear fit matches the original model outputs, and bootstrap stability checks. A reader would care because pure importance rankings leave open whether a feature reliably raises or lowers predictions and whether that pattern holds across the data. If the method works as described, it supplies concrete directional and reliability information while staying anchored in the original model's behavior.

Core claim

The φ-table selects features by SHAP importance and fits a standardized linear surrogate to the fitted model response f(X), reporting SHAP importance together with model-response coefficients, uncertainty summaries, surrogate fidelity, and bootstrap coefficient stability. The resulting coefficients are interpreted as projections of the fitted model response onto the SHAP-selected feature set. Across synthetic, semi-synthetic, and real-data experiments, the φ-table extends ranking-only SHAP into a statistical global explanation by exposing direction, uncertainty, fidelity, and stability as distinct components of fitted model behavior.

What carries the argument

The φ-table, which selects features via SHAP importance and then fits a standardized linear surrogate to the black-box response f(X) to produce a multi-column statistical summary.

If this is right

The table supplies explicit directional summaries for how each selected feature affects the model response.
Uncertainty summaries quantify the precision of those directional effects.
Surrogate fidelity scores indicate how much of the original fitted response is captured by the linear projection.
Bootstrap stability measures test whether the reported coefficients remain consistent under resampling.

Where Pith is reading between the lines

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

The fidelity and stability columns could serve as built-in diagnostics for deciding when a simple linear summary is sufficient versus when a more flexible global explanation is needed.
The same selection-plus-surrogate pattern might be tested on classification models by replacing the linear fit with an appropriate probabilistic surrogate.
Comparing φ-table coefficients against coefficients from other global surrogate techniques on the same data would clarify whether the SHAP-first selection step adds unique value.

Load-bearing premise

A standardized linear surrogate fitted to the black-box response on the SHAP-selected features can still recover directional effects and uncertainties even when the underlying model contains strong nonlinearity or interactions.

What would settle it

A controlled case where the black-box model is constructed with known strong interactions among the top SHAP features, yet the linear surrogate shows low R-squared or unstable bootstrap coefficients, would show that the directional summaries do not reliably describe the model's behavior.

read the original abstract

Global SHAP explanations are typically presented as feature-importance rankings, which identify variables that matter to a black-box model but do not indicate whether their effects admit clear directional summaries, how uncertain those summaries are, or how faithfully they represent the fitted response. This paper proposes the $\phi$-table, a SHAP-based statistical explanation table for tabular black-box regression models. The procedure selects features by SHAP importance and fits a standardized linear surrogate to the fitted model response $f(X)$, reporting SHAP importance together with model-response coefficients, uncertainty summaries, surrogate fidelity, and bootstrap coefficient stability. The resulting coefficients are interpreted as projections of the fitted model response onto the SHAP-selected feature set. Across synthetic, semi-synthetic, and real-data experiments, the $\phi$-table extends ranking-only SHAP into a statistical global explanation by exposing direction, uncertainty, fidelity, and stability as distinct components of fitted model behavior.

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

The φ-table adds a clean reporting format on top of SHAP rankings but rests on a linear surrogate whose reliability is not yet shown in the numbers.

read the letter

The paper's core move is straightforward: pick features by global SHAP, fit a standardized linear surrogate to the black-box output on those features, and bundle the resulting coefficients with bootstrap stability, surrogate fidelity, and the original SHAP values into one table. That specific combination of elements is not in the SHAP literature the abstract cites, so the reporting format itself counts as new. For someone already running SHAP on tabular regression problems, the table gives a compact way to see direction and rough uncertainty alongside importance, which is a practical step forward from rankings alone. The procedure is clearly described and the intent to treat the coefficients as projections of the fitted response is stated plainly. Credit for that. The main weakness is that the linear surrogate is asked to carry the directional claim. When the black-box has interactions among the selected features, the coefficients average over those effects instead of isolating marginal directions, so sign or magnitude can shift from what a conditional view would show. The abstract says experiments on synthetic, semi-synthetic, and real data demonstrate the extension, yet supplies no fidelity numbers, no ablation checks on interaction strength, and no comparison of surrogate signs against the true model behavior. Without those quantities it is hard to know how often the separation of direction from fidelity and stability actually holds. The work is aimed at applied users who want more statistical context from their SHAP output without new theory. A reader who already trusts linear approximations on SHAP-selected subsets will find the table useful as a reporting device. If the full experiments include quantitative checks on surrogate quality and interaction robustness, the paper deserves a serious referee to sort out whether the added components are reliable in practice. I would send it for review rather than desk reject, mainly to get the missing numbers and to test the interaction concern directly.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes the φ-table, a SHAP-based statistical explanation table for tabular black-box regression models. Features are selected by SHAP importance; a standardized linear surrogate is then fitted to the model response f(X) on this subset. The table reports SHAP importances together with the surrogate coefficients (interpreted as directional projections of the fitted response), uncertainty summaries, surrogate fidelity, and bootstrap coefficient stability. The authors claim that experiments on synthetic, semi-synthetic, and real-data sets demonstrate that the φ-table extends ranking-only SHAP by exposing direction, uncertainty, fidelity, and stability as distinct components of model behavior.

Significance. If the linear surrogate faithfully recovers directional effects without being confounded by interactions or nonlinearities among the selected features, the φ-table would supply a practical statistical layer on top of global SHAP rankings. This could help practitioners assess not only which features matter but also the reliability and directional consistency of those effects, addressing a recognized limitation of pure ranking explanations.

major comments (2)

[§3 (Method)] §3 (Method): The central construction fits a standardized linear surrogate to f(X) on the SHAP-selected feature subset and treats its coefficients as directional projections. When the black-box contains interactions or nonlinearities among those features, the linear projection averages over interaction effects rather than isolating marginal directions; reported coefficient signs and magnitudes can therefore reverse or attenuate relative to the true conditional behavior of f. The manuscript should supply explicit checks (e.g., interaction-strength diagnostics or ablation studies) to confirm that direction remains separable from fidelity and stability.
[§4 (Experiments)] §4 (Experiments): The abstract states that experiments across synthetic, semi-synthetic, and real data demonstrate the claimed extension, yet no quantitative fidelity metrics (R², MSE, or surrogate-vs-black-box agreement), stability intervals, or comparisons against nonlinear surrogates are reported. Without these numbers it is impossible to judge whether the linear projection actually isolates directional effects or merely reproduces post-hoc summaries of f(X).

minor comments (2)

[Introduction] The notation distinguishing φ-table coefficients from standard SHAP values should be introduced earlier and used consistently throughout.
[§3.3] Bootstrap stability procedure should state the number of replicates and the exact resampling scheme employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. The comments highlight important nuances in interpreting the linear surrogate and the presentation of experimental evidence. We respond to each major comment below and will revise the manuscript accordingly to strengthen the exposition.

read point-by-point responses

Referee: [§3 (Method)] The central construction fits a standardized linear surrogate to f(X) on the SHAP-selected feature subset and treats its coefficients as directional projections. When the black-box contains interactions or nonlinearities among those features, the linear projection averages over interaction effects rather than isolating marginal directions; reported coefficient signs and magnitudes can therefore reverse or attenuate relative to the true conditional behavior of f. The manuscript should supply explicit checks (e.g., interaction-strength diagnostics or ablation studies) to confirm that direction remains separable from fidelity and stability.

Authors: We thank the referee for this observation. The manuscript explicitly frames the surrogate coefficients as projections of the fitted response f(X) onto the SHAP-selected feature set (see abstract and §3), rather than as isolated marginal effects. Any linear projection necessarily averages over interactions present in the selected subset; this is an inherent property of the construction. To address the request for explicit checks, we will add to the revised §3 an interaction diagnostic that contrasts the linear surrogate R² with that of a nonlinear surrogate (e.g., random forest) on the same features. We will also include a controlled ablation study on synthetic data with tunable interaction strength, showing how fidelity and stability metrics flag cases where directional projections may be attenuated. revision: yes
Referee: [§4 (Experiments)] The abstract states that experiments across synthetic, semi-synthetic, and real data demonstrate the claimed extension, yet no quantitative fidelity metrics (R², MSE, or surrogate-vs-black-box agreement), stability intervals, or comparisons against nonlinear surrogates are reported. Without these numbers it is impossible to judge whether the linear projection actually isolates directional effects or merely reproduces post-hoc summaries of f(X).

Authors: We agree that consolidated quantitative metrics strengthen the claims. While the manuscript reports per-experiment fidelity and bootstrap stability, we acknowledge these could be more prominently summarized and augmented with nonlinear comparisons. In the revision we will add a dedicated summary subsection (and table) in §4 that aggregates average R², MSE, and stability intervals across all three data regimes, together with a direct comparison against a nonlinear surrogate fitted on the identical SHAP-selected features. This will provide the numerical evidence needed to evaluate the linear projection's behavior. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the φ-table procedure

full rationale

The paper defines the φ-table explicitly as a post-hoc procedure: select features by SHAP importance, then fit a standardized linear surrogate directly to the black-box response f(X) on those features, and report the resulting coefficients (interpreted as projections), uncertainty, fidelity, and stability. These quantities are defined as outputs of the fitting and bootstrapping steps rather than independent predictions or first-principles derivations. No load-bearing self-citation, uniqueness theorem, or reduction of a claimed result to its own inputs by construction appears in the described chain. The method is intentionally a summary of the fitted model, so the reported statistics are tautological with the surrogate construction by design but do not constitute circularity under the specified patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on standard assumptions of linear regression (linearity in the surrogate, independence of residuals) and on the premise that SHAP importance ranks are a sufficient basis for feature selection; no new entities are postulated and no free parameters beyond ordinary regression coefficients are introduced in the abstract.

axioms (1)

domain assumption A linear surrogate can meaningfully approximate directional effects of a black-box model on selected features
Invoked when the paper interprets the fitted coefficients as projections of the model response

pith-pipeline@v0.9.0 · 5471 in / 1303 out tokens · 87501 ms · 2026-05-17T00:25:09.546387+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · 1 internal anchor

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