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arxiv: 2509.20083 · v2 · pith:DWTOQRDQnew · submitted 2025-09-24 · 📊 stat.AP

Rethinking player evaluation in sports: Goals above expectation and beyond

Robert Bajons , Lucas Kook This is my paper

Pith reviewed 2026-05-22 11:59 UTC · model grok-4.3

classification 📊 stat.AP
keywords player evaluationdouble machine learninggoals above expectationsports analyticssemiparametric modelsfrequentist inferenceresidualization
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The pith

Residualized machine learning metrics enable valid frequentist inference for player performance in sports.

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

The paper presents a framework that uses flexible machine learning to estimate expected outcomes for events like shots in soccer, then derives player metrics such as goals above expectation from the difference between actual and expected results. Standard versions of these metrics suffer from bias and do not support reliable statistical inference when the underlying models are complex. By adding a residualization step drawn from double machine learning, the framework corrects for player involvement and restores valid frequentist properties. This matters for sports analysts because it turns point estimates into quantities that can be tested for significance, allowing identification of players who genuinely outperform expectations. The approach is also connected to semiparametric models that estimate directional player effects.

Core claim

Metrics based on differences between observed and model-predicted outcomes are equivalent to Rao's score tests in parametric regressions for the expected outcome; residualized versions of these metrics, obtained by additionally regressing on player involvement, inherit the Neyman orthogonality and rate conditions of double machine learning and therefore permit valid inference even when flexible nuisance estimators are used.

What carries the argument

Residualized outcome-difference metrics constructed via double machine learning, which adjust the original GAX-style quantities by an extra regression step that predicts player participation given the observed features.

If this is right

  • The residualized metrics support inference on whether individual players exert a positive directional effect on outcomes such as goals or shot success.
  • The same construction applies directly to goalkeeper save evaluation, basketball shooting skill, quarterback passing accuracy, and soccer player injury proneness.
  • Player-specific effect estimates become interpretable within semiparametric regression models that separate the contribution of each athlete from the baseline expectation.

Where Pith is reading between the lines

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

  • Analysts could use the framework to attach confidence intervals to player rankings, reducing the risk of overvaluing short-term luck in contract or lineup decisions.
  • The residualization idea may transfer to other observational domains where flexible models are used to benchmark individual performance against a population baseline.
  • Extensions could examine finite-sample coverage of the resulting confidence intervals under realistic sports-data sparsity.

Load-bearing premise

The chosen nuisance estimators for the expected outcome and for player involvement must be consistent at the rates required by double machine learning and must satisfy Neyman orthogonality after residualization.

What would settle it

In a large soccer dataset where the expected-goal model is known to be correctly specified, the residualized GAX statistic for a player with no true effect should be statistically indistinguishable from zero at conventional significance levels; systematic rejection would indicate that the regularity conditions fail in practice.

Figures

Figures reproduced from arXiv: 2509.20083 by Lucas Kook, Robert Bajons.

Figure 1
Figure 1. Figure 1: A snapshot of the data and the most important features used to compute xG models, GAX and rGAX. Premier League, and Serie A) provided by Hudl-Statsbomb and obtained via the StatsbombR R package [Yam, 2025]. These data comprise all soccer events captured during each game, where a soccer event is defined as any on-ball action performed by players, such as passes, dribbles, shots, crosses, etc. In particular,… view at source ↗
Figure 2
Figure 2. Figure 2: Goals and residualized goals above expectation plots for the 2015/16 season of the 5 big European soccer leagues. A: Scatterplot of empirical GAX and rGAX. The solid line indicates the identity. The correlation coefficient R is added to the plot. B: Player-wise empirical GAX and rGAX with one-sided 95% confidence intervals for rGAX for the top 10 players with respect to empirical rGAX and 5 selected well-k… view at source ↗
Figure 3
Figure 3. Figure 3: Player-wise empirical GAX and rGAX computed from three different xG models for the top 10 players with respect to empirical rGAX and 5 selected well-known players. The 95% confidence intervals for rGAX from the model using all data are shown. other aspects determining outstanding soccer players. On the other hand, there are a number of potential practical considerations which relate to the underlying assum… view at source ↗
Figure 4
Figure 4. Figure 4: Scatterplots of empirical GAX and rGAX values computed from different models. A: Scatterplot for empirical rGAX from an xG model computed with all data available against empirical rGAX from a model using only 2015/16 data. B: Scatterplot for empirical GAX from an xG model computed with all data available against empirical GAX from a model using only 2015/16 data C: Scatterplot for empirical rGAX from an xG… view at source ↗
Figure 5
Figure 5. Figure 5: Goals saved and residualized goals saved above expectation plots for the 2015/16 season of the 5 big European soccer leagues. A: Scatterplot of empirical GSAX and rGSAX. The solid line indicates the identity. The correlation coefficient R is added to the plot. B: Player-wise empirical GSAX and rGSAX with one-sided 95% confidence intervals for rGAX for the top 10 players with respect to empirical rGSAX. dif… view at source ↗
Figure 6
Figure 6. Figure 6: Goals and residualized goals above expectation plots for the 2015/16 season of the 5 big European soccer leagues using two different models for the regression of X on Z. A: Scatterplot of empirical GAX and rGAX using an untuned random forest. The solid line indicates the identity. The correlation coefficient R is added to the plot. B: Player-wise empirical GAX and rGAX from an untuned random forest with on… view at source ↗
Figure 7
Figure 7. Figure 7: Scatterplot of empirical rGAX values as obtained from different models for the regression of X on Z. A: Scatterplot of empirical rGAX from an untuned random forest and empirical rGAX from a tuned random forest. B: Scatterplot of empirical rGAX from an tuned xgboost model and empirical rGAX from a tuned random forest. R = 0.964 -100 0 100 200 300 -100 0 100 200 300 empricial rqSI empricial qSI p-value ≤ 0.0… view at source ↗
Figure 8
Figure 8. Figure 8: rqSI and qSI for the 2022/23 NBA seasons using a shot indicator as outcome (0 or 1). A: Scatterplot of empirical qSI and rqSI. The solid line indicates the identity. The correlation coefficient R is added to the plot. B: Player-wise empirical qSI and rqSI with one-sided 95% confidence interval for rqSI for the top 15 players with respect to empirical rqSI. where Y is again the outcome of a shot, X is a pla… view at source ↗
Figure 9
Figure 9. Figure 9: rqSI and qSI for the 2022/23 NBA seasons using the score value as outcome (0,2, or 3). A: Scatterplot of empirical qSI and rqSI. The solid line indicates the identity. The correlation coefficient R is added to the plot. B: Player-wise empirical qSI and rqSI with one-sided 95% confidence interval for rqSI for the top 15 players with respect to empirical rqSI. this season via the R package hoopR [Gilani, 202… view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of rqSI values and GCM test statistic when using different outcomes. A: Scatterplot of empirical rqSI when using score indicator outcome and score value outcome. B: Scatterplot of the empirical GCM test statistic when using score indicator outcome and score value outcome. The solid line in B indicates the identity. The correlation coefficient R is added to both plots. above expectation (CPAE, s… view at source ↗
Figure 11
Figure 11. Figure 11: rCPAE and CPAE for the 2022/23 NFL seasons. A: Scatterplot of empirical rCPAE and CPAE. The solid line indicates the identity. The correlation coefficient R is added to the plot. B: Player-wise empirical rCPAE and CPAE with one-sided 95% confidence interval for rCPAE for the top 15 players with respect to empirical rCPAE. Besides Y , we also observe the indicator random variable δ := 1(Y ∗ ≤ C), which ind… view at source ↗
Figure 12
Figure 12. Figure 12: Events and residualized events above expectation plots for time to first injury in the Liverpool F.C. data. A: Scatterplot of the empirical IAX and rIAX. The solid line indicates the identity. B: Player-wise empirical IAX and rIAX with 95% confidence intervals for rIAX. C: Feature-specific rIAX with 95% confidence intervals. For the survival regression, a random survival forest was used. For the feature r… view at source ↗
read the original abstract

A popular quantitative approach to evaluating player performance in sports involves comparing an observed outcome to the expected outcome ignoring player involvement, which is estimated using statistical or machine learning methods. In soccer, for instance, goals above expectation (GAX) of a player measure how often shots of this player led to a goal compared to the model-derived expected outcome of the shots. Typically, sports data analysts rely on flexible machine learning models, which are capable of handling complex nonlinear effects and feature interactions, but fail to provide valid statistical inference due to finite-sample bias and slow convergence rates. In this paper, we close this gap by presenting a framework for player evaluation with metrics derived from differences in actual and expected outcomes using flexible machine learning algorithms, which nonetheless allows for valid frequentist inference. We first show that the commonly used metrics are directly related to Rao's score test in parametric regression models for the expected outcome. Motivated by this finding and recent developments in double machine learning, we then propose the use of residualized versions of the original metrics. For GAX, the residualization step corresponds to an additional regression predicting whether a given player would take the shot under the circumstances described by the features. We further relate metrics in the proposed framework to player-specific effect estimates in interpretable semiparametric regression models, allowing us to infer directional effects, e.g., to determine players that have a positive impact on the outcome. Our primary use case are GAX in soccer. We further apply our framework to evaluate goal-stopping ability of goalkeepers, shooting skill in basketball, quarterback passing skill in American football, and injury-proneness of soccer players.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a framework for player evaluation in sports that compares observed outcomes to expected outcomes estimated via flexible machine learning models. It shows that standard metrics such as goals above expectation (GAX) are related to Rao's score test, then proposes residualized versions of these metrics motivated by double machine learning to restore valid frequentist inference. The residualization for GAX includes an auxiliary regression for shot-taking probability. The framework is further connected to player-specific effects in semiparametric models and is demonstrated on soccer GAX, goalkeeper performance, basketball shooting, American football passing, and soccer injury proneness.

Significance. If the inference claims hold after residualization, the work would offer a practical advance in sports analytics by permitting complex ML models for expected-outcome estimation while supplying valid standard errors and directional effect tests. The explicit links to Rao's score test and double ML provide theoretical grounding that is often missing in applied sports metrics, and the multi-sport applications illustrate generality.

major comments (2)
  1. [§3.2] §3.2 (residualized GAX construction): the validity of frequentist inference after residualization rests on the double ML nuisance estimators (shot probability model and outcome model) satisfying the n^{-1/4} rate and Neyman orthogonality conditions. The manuscript invokes these conditions but provides neither simulation verification under sports-data regimes (modest per-player samples, high-dimensional covariates, player heterogeneity) nor empirical rate diagnostics; without this, the reported standard errors for residualized GAX lack guaranteed coverage.
  2. [§4.1] §4.1 (semiparametric interpretation): the claim that residualized metrics correspond to player-specific coefficients in a partially linear model requires explicit derivation of the equivalence, including the precise form of the player indicator and the orthogonality condition after residualization. The current sketch leaves open whether the estimator remains consistent when the player-shot indicator is itself high-dimensional or sparse.
minor comments (2)
  1. The abstract and introduction would benefit from a concise statement of the exact regularity conditions invoked from the double ML literature (e.g., Chernozhukov et al.).
  2. [Figure 1] Figure 1 (GAX comparison): axis labels and legend should explicitly distinguish raw versus residualized versions to avoid reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have prompted us to strengthen the theoretical and empirical support in the manuscript. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (residualized GAX construction): the validity of frequentist inference after residualization rests on the double ML nuisance estimators (shot probability model and outcome model) satisfying the n^{-1/4} rate and Neyman orthogonality conditions. The manuscript invokes these conditions but provides neither simulation verification under sports-data regimes (modest per-player samples, high-dimensional covariates, player heterogeneity) nor empirical rate diagnostics; without this, the reported standard errors for residualized GAX lack guaranteed coverage.

    Authors: We agree that additional verification tailored to sports data regimes would enhance confidence in the finite-sample properties. While the double ML framework supplies asymptotic guarantees under the n^{-1/4} rate and Neyman orthogonality conditions, we acknowledge the referee's point that explicit checks are valuable when per-player samples are modest and covariates are high-dimensional. In the revised manuscript we will add a dedicated simulation study that generates data under realistic sports regimes (limited shots per player, high-dimensional features, and player heterogeneity) and reports empirical coverage of the residualized GAX standard errors together with diagnostics for nuisance estimator convergence rates. revision: yes

  2. Referee: [§4.1] §4.1 (semiparametric interpretation): the claim that residualized metrics correspond to player-specific coefficients in a partially linear model requires explicit derivation of the equivalence, including the precise form of the player indicator and the orthogonality condition after residualization. The current sketch leaves open whether the estimator remains consistent when the player-shot indicator is itself high-dimensional or sparse.

    Authors: We thank the referee for requesting a more explicit derivation. In the revised Section 4.1 we will supply the full equivalence proof: consider the partially linear model Y = m(X) + θ D + ε where D is the binary player (or shot-taking) indicator and m(X) is estimated by machine learning. The residualized metric is exactly the Neyman-orthogonal score for θ obtained by regressing the outcome residual on the player-indicator residual. We will show that the orthogonality condition holds after double residualization and that the resulting estimator for θ is consistent and asymptotically normal. For the high-dimensional or sparse case we will clarify that, provided the nuisance functions are estimated at the required rate (e.g., via lasso or other sparse methods) and the number of players grows appropriately with sample size, consistency is preserved; we will add a short discussion of these conditions. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external DML and score-test identity

full rationale

The paper first relates standard GAX-style metrics to Rao's score test via a direct algebraic connection in parametric models (a known statistical identity, not a self-definition). It then introduces residualized versions explicitly motivated by the external double machine learning literature (Chernozhukov et al. and follow-ups), with the auxiliary regression for shot probability presented as an application of Neyman orthogonality rather than a redefinition of the target metric. No self-citation chain, fitted parameter renamed as prediction, or ansatz smuggled via prior work by the same authors appears in the provided abstract or derivation outline. The central claim of valid frequentist inference therefore rests on independent external theory plus the paper's own residualization step, leaving the chain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the regularity conditions of double machine learning and on the validity of the score-test equivalence for the expected-outcome model; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Double machine learning regularity conditions (consistent nuisance estimation at appropriate rates and Neyman orthogonality after residualization) hold for the chosen ML estimators.
    Invoked when the authors propose residualized versions motivated by recent developments in double machine learning.

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