A statistical view on xG and GAX: Rethinking goals above expectation (GAX)
Vienna University of Economics and Business
Jun 5, 2025
Expected Goals (xG) and Goals Above Expectation (GAX)
Motivation
How can we measure team or player performance?
- Classical approaches: Count-based statistics such as goals, assists, shots, etc.
Problems:
- Goals are rare (only 10% of shots end up in a goal)
- Chances/shots are not created equally
- Assessing shots by binary outcome (goal/no goal) is not adequate (loss of information)!
Solution: expected goals (xG) models
- Use statistical models to assign a probability of scoring to each shot
- Take into account shot-specific features
- Evaluate players and teams based on aggregation of xG values
xG for player evaluation
How can we identify outstanding shooters?
Treat xG as performance measure for average player
Compare actual outcome of shot to expected outcome
xG for player evaluation
How can we identify outstanding shooters?
Treat xG as performance measure for average player
Compare actual outcome of shot to expected outcome
Goals above expectation (GAX)1: over time frame (e.g. season), compute difference between goals and xG for all shots of player \(i\)
\[ \operatorname{GAX}_i = \sum_{j=1}^{N_i} (Y_j - \hat h(Z_j))\] \(Y_j\) … actual outcome for shot \(j\)
\(\hat h(Z_j)\) … estimator for \(h(Z) = \mathbb{E}[Y|Z]\) (xG for shot \(j\))
GAX criticism
GAX not optimal for evaluating shooting skills:
Low stability (Baron et al. 2024):
- a player’s GAX in one season is poorly predictive of their GAX in the next season
High variance and no (direct) uncertainty quantification (Davis and Robberechts 2024)
Biases in traditional xG models due to overrepresented players and team strengths (Davis and Robberechts 2024)
A semiparametric approach toward GAX
A parametric model
How can we identify outstanding shooters?
- \(Y\) … binary outcome of a shot (goal/no goal)
- \(Z\) … shot-specific variables
- \(X\) … binary indicator of a players’ involvement (shooter/not shooter)
Logistic regression model:
\(Y \mid X,Z \sim \operatorname{Ber}(\pi(X,Z)), \quad \pi(X,Z) = P(Y=1 \mid X,Z)\) and
\[ \begin{aligned} \log\left(\frac{\pi(X,Z)}{1-\pi(X,Z)}\right) = X\beta + Z^{\top}\gamma. \end{aligned} \]
Goal: Inference on \(\beta\).
- Wald test, LR test, score test
Score test and GAX
- Score (target of score tests):
Score test on \(\beta\) uses score under \(H_0: \beta = 0\):
\[\sum_{j=1}^{N} (Y_j - \hat h(Z_j))X_j\]
- \(\hat h(Z_j) = \text{expit} (Z_j^{\top}\hat \gamma)\) … \(\hat \gamma\) is the MLE of \(\gamma\) under \(H_0\).
Since \(X_j\) binary \(\Rightarrow\) score is exactly GAX for a player
- High GAX (in absolute terms) correctly standardized indicates significant players
Given i.i.d data \((Y_i,X_i,Z_i)_{i = 1}^N\) from the logistic regression model
\[\sum_{i = 1}^N\frac{\partial\log L(\beta,\gamma \mid Y_i,X_i,Z_i)}{\partial \beta}\]
GAX for player evaluation
Conclusion: GAX relates to a classical score test in logistic regression model
- Uncertainty quantification (and significance testing) via score test
- Interpretation of player quality as effect on log-odds (probability) of scoring
Problems:
- Linear model assumptions unrealistic
- Biases arising from taking into account only shot-specific variables
- High-dimensionality if accounting for team, goalkeeper or position effects
traditional GAX via machine learning model:
- \(\sum_{j=1}^{N} (Y_j - \hat{h}(Z_j))X_j, \quad \hat h\) estimated via arbitrary ML algorithm
- No (valid) uncertainty quantification
A semiparametric reformulation
How can we identify outstanding shooters?
Problem reformulation (\(Y\), \(X\), and \(Z\) as before): partially linear logistic regression model (PLLM), where
\[ \log\left(\frac{\pi(X,Z)}{1-\pi(X,Z)}\right) = X\beta + g(Z) \]
Under PLLM: Test for \(Y\) conditionally independent of \(X\) given \(Z\) (\(Y \perp\!\!\!\perp X \mid Z\)) \(\Leftrightarrow\) Test for \(H_0 : \beta = 0\)
- Under \(H_0\): No modelling assumptions on relationship between \(Y\) and \(Z\), and \(X\) and \(Z\)
- Use Generalised Covariance Measure (GCM) test (Shah and Peters 2020) for inference
GCM test
Generalised Covariance Measure:
\[ \operatorname{GCM} = \mathbb{E}[\operatorname{Cov}(Y,X \mid Z)] =\mathbb{E}[(Y - \mathbb{E}[Y | Z])(X - \mathbb{E}[X | Z])]\]
Basis for GCM test:
\[Y \perp\!\!\!\perp X \mid Z \Rightarrow \mathbb{E}[\operatorname{Cov}(Y,X \mid Z)] = 0\]
GCM test in practice:
- Given i.i.d. data, use arbitrary machine learning models and
- Regress \(Y\) on \(Z\) and obtain estimate \(\hat h\) for \(\mathbb{E}[Y | Z]\)
- Regress \(X\) on \(Z\) and obtain estimate \(\hat f\) for \(\mathbb{E}[X | Z]\)
- Under mild rate conditions (similar to DML Chernozhukov et al. (2018)):
- Sample version of GCM: \(\operatorname{\widehat{GCM}} : = \sum_{i = 1}^N (Y_i-\hat h(Z_i))(X_i - \hat f(Z_i)) \leadsto \mathcal{N}(0,\frac{\sigma^2}{N})\)
- Given i.i.d. data, use arbitrary machine learning models and
GCM test and PLLM
- Interested in testing \(H_0 : \beta = 0\) in PLLM
Takeaways:
- GCM test allows directional testing: alternatives of the form \(H_1 : \beta > 0\)
- Interpretation: identify players with significant (positive) impact on probability of success from shot
GCM test and PLLM
- Interested in testing \(H_0 : \beta = 0\) in PLLM
Takeaways:
- GCM test allows directional testing: alternatives of the form \(H_1 : \beta > 0\)
- Interpretation: identify players with significant (positive) impact on probability of success from shot
Proposition
Consider a PLLM and let \(X\) be a binary variable with \(P(X = 1 | Z) > 0\). Then \[\beta = 0 \Leftrightarrow \mathbb{E}[(\operatorname{Cov}(Y,X \mid Z)] = 0\]
GCM test and PLLM
- Interested in testing \(H_0 : \beta = 0\) in PLLM
Takeaways:
- GCM test allows directional testing: alternatives of the form \(H_1 : \beta > 0\)
- Interpretation: identify players with significant (positive) impact on probability of success from shot
Proposition
Consider a PLLM and let \(X\) be a binary variable with \(P(X = 1 | Z) > 0\). Then \[\beta = 0 \Leftrightarrow \mathbb{E}[(\operatorname{Cov}(Y,X \mid Z)] = 0\]
Proposition
Consider a PLLM and let \(X\) be a binary variable with \(P(X = 1 | Z) > 0\). Then \[\operatorname{sign}(\beta) = \operatorname{sign}(\mathbb{E}[\operatorname{Cov}(Y,X \mid Z)])\]
GCM test and PLLM
- Interested in testing \(H_0 : \beta = 0\) in PLLM
Takeaways:
- GCM test allows directional testing: alternatives of the form \(H_1 : \beta > 0\)
- Interpretation: identify players with significant (positive) impact on probability of success from shot
Proposition
Consider a PLLM and let \(X\) be a binary variable with \(P(X = 1 | Z) > 0\). Then \[\beta = 0 \Leftrightarrow \mathbb{E}[(\operatorname{Cov}(Y,X \mid Z)] = 0\]
Proposition
Consider a PLLM and let \(X\) be a binary variable with \(P(X = 1 | Z) > 0\). Then \[\operatorname{sign}(\beta) = \operatorname{sign}(\mathbb{E}[\operatorname{Cov}(Y,X \mid Z)])\]
GAX vs. RGAX
GAX in parametric model:
\[\sum_{j=1}^{N} (Y_j - \hat h(Z_j))X_j\]
- Only valid inference if linear model assumption hold
GAX via machine learning:
\[\sum_{j=1}^{N} (Y_j - \hat{h}(Z_i))X_j\]
- \(\hat h\) learned via arbitrary ML algorithm
- No valid inference
RGAX: Use sample GCM as score
\[\sum_{j=1}^{N} (Y_i-\hat h(Z_i))(X_i - \hat f(Z_i))\]
- Doubly robust score (rate conditions fulfilled) \(\Rightarrow\) valid inference
- Interpretation RGAX: additional regression accounts for whether a player would take the shot under the circumstances described by \(Z_i\)
GAX vs. RGAX
GAX in parametric model:
\[\sum_{j=1}^{N} (Y_j - \hat h(Z_j))X_j\]
- Only valid inference if linear model assumption hold
GAX via machine learning:
\[\sum_{j=1}^{N} (Y_j - \hat{h}(Z_i))X_j\]
- \(\hat h\) learned via arbitrary ML algorithm
- No valid inference
RGAX: Use sample GCM as score
\[\sum_{j=1}^{N} (Y_i-\hat h(Z_i))(X_i - \hat f(Z_i))\]
- Doubly robust score (rate conditions fulfilled) \(\Rightarrow\) valid inference
- Interpretation RGAX: additional regression accounts for whether a player would take the shot under the circumstances described by \(Z_i\)
GAX vs. RGAX
GAX in parametric model:
\[\sum_{j=1}^{N} (Y_j - \hat h(Z_j))X_j\]
- Only valid inference if linear model assumption hold
GAX via machine learning:
\[\sum_{j=1}^{N} (Y_j - \hat{h}(Z_i))X_j\]
- \(\hat h\) learned via arbitrary ML algorithm
- No valid inference
RGAX: Use sample GCM as score
\[\sum_{j=1}^{N} (Y_i-\hat h(Z_i))(X_i - \hat f(Z_i))\]
- Doubly robust score (rate conditions fulfilled) \(\Rightarrow\) valid inference
- Interpretation RGAX: additional regression accounts for whether a player would take the shot under the circumstances described by \(Z_i\)
Application
Quick data overview
Freely available event stream data from Statsbomb.
- 2015/16 season of the big 5 European leagues
- 45197 shots (4308 goals)
- 728 relevant players
Shot specific features \(Z\):
- Preprocessed features (22 Variables)
- Team information via Poisson strength model (defensive strength)
xG model:
- XGBoost model (properly tuned)
Shooting skills evaluation: GAX vs. RGAX
RGAX and GCM test results conveniently obtained via comets R package (Kook 2025)
Conclusion
In a logistic regression model: GAX is directly related to a score test on a players effect on the probability of goal.
If you don’t believe the GLM setup: GAX using ML models does not allow valid inference! \(\Rightarrow\) Residualize \(X\) as well, i.e. use RGAX.
- Valid p-values, confidence intervals and so on
If you want interpretation: RGAX directly related to parameter in popular semi-parametric model!
Outlook:
General framework usable beyond player evaluation via GAX:
- Enhancing player evaluation metrics such as EGA (soccer), EPA (American football), SI (basketball), …
- Coverage type in the NFL
- Identifying drivers of injuries in survival setup
Conclusion
In a logistic regression model: GAX is directly related to a score test on a players effect on the probability of goal.
If you don’t believe the GLM setup: GAX using ML models does not allow valid inference! \(\Rightarrow\) Residualize \(X\) as well, i.e. use RGAX.
- Valid p-values, confidence intervals and so on
If you want interpretation: RGAX directly related to parameter in popular semi-parametric model!
General framework usable beyond player evaluation via GAX:
- Enhancing player evaluation metrics: EGA (soccer), EPA (American football), SI (basketball), …
- Goalkeeper evaluation
- Identifying relevant variables for xG
- Coverage type in the NFL
- Identifying drivers of injuries in survival setup
Thank you for your attention!
References
Appendix
The essentials of xG models
Earliest version of xG dates back to Pollard and Reep (1997):
Logistic regression model on binary shot outcome
Most important features: shot location and goal angle
Distinction between kicked and headed shots
The essentials of xG models
Earliest version of xG dates back to Pollard and Reep (1997):
Logistic regression model on binary shot outcome
Most important features: shot location and goal angle
Distinction between kicked and headed shots
The essentials of xG models
Modern xG Models (Robberechts and Davis 2020; Anzer and Bauer 2021; Hewitt and Karakuş 2023):
- Flexible machine learning methods \(\Rightarrow\) account for non-linearities and interactions:
The essentials of xG models
Modern xG Models (Robberechts and Davis 2020; Anzer and Bauer 2021; Hewitt and Karakuş 2023):
Flexible machine learning methods \(\Rightarrow\) account for non-linearities and interactions:
Extreme gradient boosting machines (XGBoost)
Random forests
Neural networks
The essentials of xG models
Modern xG Models (Robberechts and Davis 2020; Anzer and Bauer 2021; Hewitt and Karakuş 2023):
Flexible machine learning methods \(\Rightarrow\) account for non-linearities and interactions:
Extreme gradient boosting machines (XGBoost)
Random forests
Neural networks
Broad set of shot-specific features:
Classical features: Distance to goal, angle, body part
Extended features from event and tracking data: distances to defenders and goalkeeper, shot type and technique, speed of and space for shooter
- Trained on large amount of data and properly tuned
Shot-specific features
Advanced xG models
Goalkeeper evaluation: GSAX vs RGSAX
