Robert Bajons - Rethinking Goals Above Expectation (GAX): A semiparametric approach

Expected Goals (xG) are the output of a statistical model assigning a probability of success to shot using shot-specific covariates and are one of the most popular metrics in modern football (soccer) analytics. Popular xG models are based on flexible machine learning algorithms, such as extreme gradient boosting machines, that account for non-linear and interaction effects of the shot-specific covariates. As a measure of a shot’s value, it is commonly used to evaluate the shooting skills of players by considering goals over expectation (GAX), i.e., the difference between actual and expected goals for each shot. However, GAX is often criticized for being unstable over seasons and for not providing (direct) means of uncertainty quantification. In this work, we address both issues by showing how the player-specific GAX relates to a score test when the xG model is a logistic regression and using a nonparametric extension which can be based on any xG model derived from sufficiently powerful machine learning algorithms. Thus, we are able to leverage commonly used black-box xG models, while still obtaining valid statistical inferences on the player-specific odds (or probability) of scoring a goal. Moreover, in order to make the results more interpretable, we show how the proposed procedure relates to player-specific effect estimates in a partially linear logistic regression model of additive effects on the log-odds of scoring a goal from a shot. Finally, we apply our framework to the 2015/16 season of the top five European leagues, determine the best shooters, and compare results across state-of-the-art xG models.