Shapley Decomposition of R-Squared in Machine Learning Models

Nickalus Redell

REVIEW 3 major objections 5 minor 22 references

Shapley Decomposition of R-Squared in Machine Learning Models

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proposes a model-agnostic metric that decomposes any fitted model's $R^2$ into per-feature shares of explained variance, using precomputed Shapley values and requiring no model refitting.

desk verdict The proposed 'Shapley decomposition of R2' is not a Shapley decomposition: in the simplest two-feature uncorrelated linear model it produces shares (0.32, 0.48) instead of the true Shapley shares (0.20, 0.60), so the fairness claim is demonstrably false. read the letter →

arxiv 1908.09718 v1 pith:NRYEU3DM submitted 2019-08-26 stat.ME

Nickalus Redell This is my paper · ORCID

classification stat.ME

keywords ShapleyvaluesR-squaredvariancedecompositionfeatureimportancemodelinterpretabilitymodel-agnosticgradientboostedtrees

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 introduces $R^2_{\text{shap}}$, a global feature-importance metric that decomposes any fitted model's overall $R^2$ into per-feature shares using precomputed Shapley values. The shares are bounded between 0 and 1 and sum exactly to the model's $R^2$, so feature importance can be reported on the familiar scale of explained variance. The method subtracts each feature's instance-level Shapley values from the model's predictions, measures the resulting increase in residual variance, and normalizes these increases into additive shares. Because only one model fit and one Shapley-value pass are needed, the computational burden shifts from retraining models to the Shapley calculation itself. The paper also proposes a companion ratio, $\sigma_{\text{unique}}$, that quantifies how much of the explained variance is uniquely assignable to features when features are correlated.

What carries the argument

The central object is the Shapley-modified prediction $\hat{y}^{(f)}_{i,\text{shap}} = \hat{y}_i - \phi^{(f)}_i$, where Shapley values are the standard additive per-feature attributions of each prediction. The mechanism is a residual-variance ratio: the baseline residual variance is compared with the residual variance after removing each feature, the ratios are normalized into a simplex, and the result is multiplied by $R^2_{\text{baseline}}$. This converts the additive explanation property of Shapley values into an additive variance decomposition of $R^2$.

What would settle it

Fit a model to synthetic data generated as $y = x_1 + x_1 x_2 + \epsilon$ with correlated $x_1$ and $x_2$, compute the proposed shares, and compare them with the change in $R^2$ when each feature is dropped and the model is retrained. A mismatch in the direction or magnitude of the two importance rankings would show that subtracting Shapley values does not remove a feature's contribution, undercutting the metric's fairness claim.

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Extended reading notes

Core claim

For a fitted model with baseline $R^2_{\text{baseline}} = \operatorname{var}(\hat{y})/(\operatorname{var}(\hat{y})+\operatorname{var}(y-\hat{y}))$, the paper defines Shapley-modified predictions $\hat{y}^{(f)}_{i,\text{shap}} = \hat{y}_i - \phi^{(f)}_i$, where $\phi^{(f)}_i$ is feature $f$'s Shapley value for instance $i$. For each feature, the feature-level share $R^2_{\text{shap}}^{(f)}$ is the increase in residual variance after this subtraction, capped and normalized so that the shares sum to $R^2_{\text{baseline}}$. The central claim is that these shares fairly allocate the proportion of model-explained variability to each feature, and that this works for any class of prediction model without refitting.

Load-bearing premise

The load-bearing premise is that subtracting a feature's Shapley values from a prediction actually removes that feature's contribution, so the resulting increase in residual variance is uniquely attributable to that feature; this is not guaranteed for nonlinear or interacting models with correlated features.

Editorial extensions

If this is right

Practitioners can rank global feature importance for any fitted model using precomputed Shapley values, with no model refitting required.
Because the feature shares are bounded in $[0,1]$ and sum to the overall $R^2$, explanations can be reported on the same scale as classical regression fit.
For gradient boosted trees and neural networks, the metric's cost is dominated by the Shapley-value calculation, which recent algorithms make inexpensive.
The companion $\sigma_{\text{unique}}$ ratio gives a diagnostic for how much of the explained variance is uniquely attributable to individual features versus shared among correlated features.
The metric can summarize feature importance on either training or test data, though the paper leaves the choice as an open question.

Reading between the lines

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

If the removal interpretation behind Eq. (5) fails for a given nonlinear or interacting model, the shares become a normalized reallocation of residual-variance changes rather than a fair decomposition; this could be tested by comparing the metric's rankings with leave-one-feature-out retraining on a suite of models.
Because the normalization in Eq. (6) depends on the full set of features, adding an irrelevant but correlated feature could dilute the shares of the others; a model with nested feature sets would expose this.
Because the metric inherits whatever Shapley-value estimator produced the $\phi^{(f)}_i$, using different explainers or correlation-aware Shapley approximations could change the shares even for the same model, so part of the importance ranking may be an artifact of the explainer.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this paper's central claim doesn't survive contact with the simplest example. The idea is to take precomputed instance-level Shapley values, subtract each feature's values from the predictions, measure the increase in residual variance, and normalize so the shares sum to R2. That's a clean and computationally convenient construction, and the writing is honest about correlation issues and offers code. But it is not a Shapley decomposition of R2, and it does not fairly allocate explained variance.

I checked the stress-test counterexample and it holds. For y = x1 + sqrt(3)x2 + eps with all independent standard normal, R2 = 0.8. True Shapley/LMG shares are 0.2 and 0.6. The paper's Eq. (6) gives raw residual-variance terms 0.4 and 0.6, which normalize to 0.32 and 0.48. So in the cleanest possible setting—uncorrelated features, linear model, exact Shapley values—the metric misallocates variance. The fairness claim is false under the standard Shapley criterion.

The reason is structural. Shapley values are additive explanations of individual predictions, not removable additive components of the data-generating process. Subtracting phi_f from yhat changes residual variance by var(phi_f) plus interaction terms; normalizing the resulting quantities to sum to R2 imposes efficiency but does not impose Shapley fairness. The paper's sigma_unique discussion is a useful caveat but doesn't fix the attribution.

What's genuinely useful: the paper addresses a real pain point—global importance without refitting—and the empirical stability check is a reasonable start. The R package is a plus. But as written, the title and abstract overclaim. The metric is a heuristic, not a fair decomposition.

The practical fix is either to prove conditions under which Eq. (6) matches Shapley shares (I doubt it can, given the counterexample), or to reframe the paper as proposing a residual-variance-based importance heuristic and drop the fairness language. As is, it should not be accepted as a finished method. I'd still send it to peer review—the issue is timely and the counterexample is a valuable object lesson—but the revision burden is heavy.

Referee Report

3 major / 5 minor

Summary. This paper proposes a global feature-importance metric for machine learning models. The method first computes a baseline R2 as var(yhat)/(var(yhat)+varres), then for each feature f constructs a modified prediction yhat_i - phi_i^f using instance-level SHAP values, and finally normalizes the resulting residual-variance ratios in Eq. (6) so that the feature-level scores sum to the baseline R2. The authors claim that this metric 'fairly allocates' model-explained variability, is bounded between 0 and 1, requires no model refitting, and can be computed from precomputed Shapley values. The paper includes a correlation-adjustment index sigma_unique (Eq. (7)), a simulation of that index, and applied demonstrations on wine quality and forest-fire datasets using CatBoost.

Significance. The practical motivation is good: a model-agnostic, refit-free global importance measure that reuses exact TreeSHAP values would be useful, and the authors provide an open-source R package. The paper also correctly identifies that LMG-type Shapley decompositions require refitting and are computationally expensive. However, the central theoretical claim is not established: Eq. (6) is a normalization of residual-variance increases, not a Shapley decomposition of R2, and the paper's fairness claim is demonstrably false under the standard Shapley/LMG notion in a simple linear model. The sigma_unique formula also contains a definite error. As submitted, the contribution is a heuristic with a misleading name rather than a rigorous decomposition.

major comments (3)

[Proposed R2 Feature Importance Metric, Eq. (6), and Abstract] The claim that Eq. (6) is a 'Shapley-value variance decomposition' that 'fairly allocates' explained variance is not supported. The decomposition into shares that sum to R2_baseline is imposed by the normalization in the denominator of Eq. (6); no game over feature subsets is defined, and none of the Shapley axioms (efficiency, symmetry, dummy, additivity) is verified. The metric is therefore a normalized residual-variance-increase heuristic, not a Shapley decomposition. This is not a semantic quibble: in the linear model y = x1 + sqrt(3) x2 + eps with independent standard-normal x1, x2, eps, the classical Shapley/LMG shares of R2 = 0.8 are (0.2, 0.6), while Eq. (6) applied to exact SHAP values yields (0.32, 0.48). Thus the 'fair allocation' claim is false under the standard notion of Shapley fairness, even in the idealized uncorrelated case that the paper highlights.
[Proposed R2 Feature Importance Metric, Eq. (5)] The construction yhat_i - phi_i^f is interpreted as removing feature f's marginal contribution from the prediction. This is not generally valid for nonlinear or interacting models. SHAP values are local additive attributions that depend on all features; the vector y - (yhat_i - phi_i^f) is not the prediction error of a model trained or evaluated without feature f. The variance increase after this subtraction therefore reflects covariance and interaction terms involving other features, not a quantity that can be uniquely attributed to feature f. The discussion of 'unique variance' in the correlation section consequently overstates what the metric measures.
[Accounting for Correlation, Eq. (7)] The formula for sigma_unique is inconsistent with the definition in the text and with the claimed value of 1 for uncorrelated features. As written, the numerator is the sum of the raw residual variances after each feature is removed, not the sum of increases in residual variance. In the same two-feature example, the numerator is var(x1 + eps) + var(sqrt(3)x2 + eps) = 2 + 4 = 6, while the denominator is var(y) - var(eps) = 4, giving sigma_unique = 1.5 rather than 1. To obtain the intended quantity, each term would need to be the increase over the baseline residual variance, that is, sum_f [var(y - yhat_shap(f)) - var(y - yhat)]. As printed, Eq. (7), Figure 1, and the sigma_unique values in the applied section do not measure what is claimed.

minor comments (5)

[Related Work] There is a typo: 'Lundburg' should be 'Lundberg'.
[Proposed R2 Feature Importance Metric, Eq. (6)] The paper should state the edge case in which the denominator of Eq. (6) is zero, for example when R2_baseline = 0 or all raw numerators are zero.
[Applied Analysis] The applied analysis is purely descriptive: there is no comparison with LMG, permutation importance, or other global importance methods, and no uncertainty quantification for the reported rankings.
[Applied Analysis, Figure 2] Please define the axes in Figure 2 and explain how the 0-1 scaling affects the interpretation; as drawn, the figure makes it difficult to assess changes relative to the underlying model R2.
[Accounting for Correlation, Eq. (7)] The notation var(y - ybar) should be spelled out as var(y - mean(y)) to avoid confusion with the fitted values yhat.

Circularity Check

1 steps flagged · score 2.0 of 10

Sum-to-R2 property is enforced by normalization in Eq. (6), but the paper has no fitted-input or self-citation circularity.

self definitional [Eq. (6), Section 'Proposed R2 Feature Importance Metric']
"The denominator of (6) normalizes the R2 values to produce a simplex similar to that produced by the softmax function. Finally, multiplying this vector of feature-level variance explained attributions with R2baseline ensures that R2baseline := sum_f R2(f)shap."

The advertised property that the feature shares sum to the overall model R2 is not derived from Shapley axioms or from the data; it is inserted by the normalization in Eq. (6). The paper presents this summation as a desirable property of the method, but it holds by construction of the formula. Thus the 'decomposition summing to the overall R2' is definitional rather than an independent finding. This is a limited circularity because the metric is a proposed definition, not a fitted quantity that is recycled as a prediction.

full rationale

The main derivation chain is self-contained: Eq. (4) defines a baseline R2, Eq. (5) defines Shapley-modified predictions, Eq. (6) defines feature shares from residual-variance ratios, and Eq. (7) defines a uniqueness ratio. No parameter is fitted to a subset of the data and later renamed as a prediction, and no external benchmark is predicted from fitted constants. The wine and forest-fires analyses are illustrative demonstrations of stability rather than tests that could falsify the sum-to-R2 property. There are no load-bearing self-citations: the author's own shapFlex package appears only as an implementation link, while the Shapley additivity and LMG results are cited from independent literature. The one definitional circularity is the sum property, which Eq. (6) enforces by normalization; the broader 'fair allocation' claim is an interpretive choice about what the normalized residual-variance changes mean, not a circular derivation. Score 2 reflects this single self-definitional property while recognizing that the central construction is otherwise independent of its inputs.

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

The method introduces no new fitted parameters and no new physical or formal entities. It relies on three domain assumptions: the additive removal interpretation of Shapley values, the residual variance measure of unique attribution, and the correctness of the chosen Shapley value implementation. The free-parameter list is empty because the clipping and normalization choices are definitional, and the applied model sizes are tuning choices, not parameters of the metric itself.

assumptions (3)

domain assumption For a fitted model, Shapley values are additive: yhat_i = phi_0 + sum_f phi_i^f (Eq. 3), so subtracting phi_i^f removes feature f from the prediction.
This is the load-bearing premise connecting Shapley attribution to a counterfactual feature-removed prediction; it holds only for additive model structures and ignores interaction reweighting.
domain assumption An increase in residual variance after subtracting a feature's Shapley values measures that feature's unique explained variance (Eq. 7).
Requires approximate orthogonality between the feature's Shapley vector and the model residual and ignores covariance terms; the paper motivates this via sigma_unique but does not prove the decomposition.
domain assumption TreeSHAP values used in the applied analysis are treated as exact and reliable Shapley values.
The metric inherits any approximation error or feature-independence assumptions in the Shapley values, which the paper only partially acknowledges in the correlation section.

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Cite this review

Pith. "Pith review of Shapley Decomposition of R-Squared in Machine Learning Models." pith.science (2026). https://pith.science/paper/NRYEU3DM

@misc{pith2026190809718,
  author       = {Pith},
  title        = {Pith review of: Shapley Decomposition of R-Squared in Machine Learning Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NRYEU3DM}},
  note         = {Machine review of arXiv:1908.09718}
}

abstract

In this paper we introduce a metric aimed at helping machine learning practitioners quickly summarize and communicate the overall importance of each feature in any black-box machine learning prediction model. Our proposed metric, based on a Shapley-value variance decomposition of the familiar $R^2$ from classical statistics, is a model-agnostic approach for assessing feature importance that fairly allocates the proportion of model-explained variability in the data to each model feature. This metric has several desirable properties including boundedness at 0 and 1 and a feature-level variance decomposition summing to the overall model $R^2$. In contrast to related methods for computing feature-level $R^2$ variance decompositions with linear models, our method makes use of pre-computed Shapley values which effectively shifts the computational burden from iteratively fitting many models to the Shapley values themselves. And with recent advancements in Shapley value calculations for gradient boosted decision trees and neural networks, computing our proposed metric after model training can come with minimal computational overhead. Our implementation is available in the R package shapFlex.

Figures

Figures reproduced from arXiv: 1908.09718 by the authors.

**Figure 2.** This figure depicts the feature-level attribution of the total variance explained in (a) wine quality [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

J., Singh, C., Kumbier, K., Abbasi-Asl, R., & Yu, B

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arXiv 2017

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work page 2017

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work page 2018

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