Chapter 29 Quiz: Comprehensive Case Studies

Multiple Choice Questions

Question 1

When building an xG model, which calibration method is inherently built into the model structure and does not require post-hoc adjustment?

A) Isotonic regression calibration B) Platt scaling C) Logistic regression D) Temperature scaling

Answer: C

Explanation: Logistic regression models are inherently calibrated because the logistic function directly outputs probabilities that minimize the log-loss. The model parameters are estimated via maximum likelihood, which ensures that predicted probabilities match observed frequencies in the training data. Other methods like isotonic regression and Platt scaling are post-hoc calibration techniques applied to models (such as gradient boosting or random forests) whose raw outputs may not be well-calibrated.

Question 2

In a scouting campaign, per-90 normalization is applied to raw counting statistics. What is the primary risk of using per-90 metrics without a minimum minutes threshold?

A) It makes players from different leagues incomparable B) It inflates the metrics of players with very limited playing time C) It penalizes players who play full 90-minute matches D) It removes the effect of playing position

Answer: B

Explanation: Per-90 normalization divides total counts by minutes played and multiplies by 90. For a player with very few minutes (e.g., 100 minutes), a single goal would yield a per-90 rate of 0.9 goals per 90, which far exceeds what most elite strikers achieve over a full season. A minimum minutes threshold (typically 900-1,500 minutes) ensures statistical stability and prevents misleading inflation of metrics.

Question 3

The formula $\theta = \arctan\left(\frac{9.32 \cdot (120 - x)}{(120 - x)^2 + (y - 40)^2 - 3.66^2}\right)$ computes which xG feature?

A) Distance to the goal line B) Distance to the nearest defender C) The angle subtended by the goal from the shot location D) The angle between the shot direction and the goal line

Answer: C

Explanation: This formula computes the angle subtended by the two goal posts as seen from the shot location $(x, y)$. The constant 9.32 is the goal width in meters, 3.66 is half the goal width, and (120, 40) represents the goal center on a standardized 120x80 pitch. A larger subtended angle generally corresponds to a higher scoring probability.

Question 4

In the Acute:Chronic Workload Ratio (ACWR), what does the "chronic" component typically represent?

A) The maximum single-day load in the past 7 days B) The exponentially weighted moving average of load over 28 days C) The total accumulated load since the start of pre-season D) The player's career average training load

Answer: B

Explanation: The chronic workload is typically computed as an exponentially weighted moving average (EWMA) over 28 days. This provides a smoothed representation of the player's "fitness" or habitual workload level. The acute component (7-day rolling mean) represents recent workload, and their ratio indicates whether the player is experiencing a spike relative to what they are conditioned for.

Question 5

An ACWR value of 1.8 for a player indicates:

A) The player's acute load is 80% higher than their chronic load B) The player is in the optimal training zone C) The player's chronic load exceeds their acute load D) The player has been undertraining

Answer: A

Explanation: An ACWR of 1.8 means the acute (recent 7-day) workload is 1.8 times the chronic (28-day) baseline, representing an 80% increase. This falls well above the generally accepted "safe zone" of 0.8-1.3, indicating a significant workload spike that is associated with elevated injury risk. The medical team would likely flag this player for load reduction.

Question 6

PPDA (Passes Per Defensive Action) is a metric for measuring pressing intensity. A team with a PPDA of 8.5 compared to one with a PPDA of 14.3 is:

A) Pressing less intensely B) Pressing more intensely C) Allowing fewer passes in general D) Making fewer tackles per match

Answer: B

Explanation: PPDA measures the number of opponent passes allowed per defensive action in the opponent's half. A lower PPDA means the team engages in defensive actions more frequently relative to opponent passes, indicating more intense pressing. A PPDA of 8.5 means the team makes a defensive action for roughly every 8.5 opponent passes, while 14.3 indicates much less frequent engagement.

Question 7

When computing expected points using a Poisson model, which assumption is most commonly violated in practice?

A) Goals are scored in whole numbers B) The number of goals follows a discrete distribution C) Goals scored by each team are independent of each other D) The match lasts exactly 90 minutes

Answer: C

Explanation: The standard expected points model assumes that the number of goals scored by the home and away teams are independent Poisson random variables. In practice, this assumption is violated because of game-state effects: a team that falls behind may push forward more aggressively (increasing both their xG and the opponent's xG on counterattacks). Bivariate Poisson models or copula-based approaches can partially address this dependence.

Question 8

In a passing network, a player with high betweenness centrality but low degree centrality is best described as:

A) A hub who connects to many teammates directly B) A bridge who links otherwise disconnected groups of players C) An isolated player with few passing connections D) The most prolific passer on the team

Answer: B

Explanation: Betweenness centrality measures how often a node lies on the shortest path between other nodes. High betweenness with low degree means the player does not pass to many teammates directly, but the passes they do make are critical for connecting different parts of the team's passing structure. This profile is common for deep-lying playmakers who link defense to attack through a small number of high-value connections.

Question 9

When building a scouting similarity engine, what advantage does Mahalanobis distance have over Euclidean distance?

A) It is faster to compute B) It accounts for correlations between features C) It always produces smaller distance values D) It works only with categorical variables

Answer: B

Explanation: Mahalanobis distance accounts for the covariance structure of the data, effectively normalizing by the variance of each feature and accounting for correlations between features. This is important in scouting because many metrics are correlated (e.g., shots and goals). Euclidean distance treats all dimensions equally and independently, which can lead to misleading similarity scores when features have different scales or are correlated.

Question 10

In the Expected Calibration Error (ECE) formula $\frac{1}{B} \sum_{b=1}^{B} |p_b - \hat{p}_b|$, what do $p_b$ and $\hat{p}_b$ represent?

A) $p_b$ is the prior probability; $\hat{p}_b$ is the posterior probability B) $p_b$ is the observed goal rate in bin $b$; $\hat{p}_b$ is the mean predicted probability in bin $b$ C) $p_b$ is the prediction for player $b$; $\hat{p}_b$ is the actual outcome for player $b$ D) $p_b$ is the probability of a win; $\hat{p}_b$ is the probability of a loss

Answer: B

Explanation: In calibration analysis, predictions are sorted into bins. For each bin $b$, $\hat{p}_b$ is the mean predicted probability (e.g., mean xG), and $p_b$ is the observed frequency of the positive outcome (e.g., the proportion of shots that were actually goals). A perfectly calibrated model has $p_b = \hat{p}_b$ for all bins, yielding ECE = 0.

Question 11

A player's growth curve is modeled as $y(t) = \beta_0 + \beta_1 t + \beta_2 t^2$. If $\beta_2 < 0$, what does this imply about the player's development trajectory?

A) The player is improving at an accelerating rate B) The player's improvement rate is decelerating and will eventually peak C) The player's metric is constant over time D) The model is a poor fit for the data

Answer: B

Explanation: A negative $\beta_2$ coefficient creates an inverted parabola (concave down), meaning the rate of improvement decreases over time and eventually the metric reaches a peak before declining. This is consistent with the typical development trajectory of most physical and some technical metrics, where players improve through their early career, peak, and then gradually decline.

Question 12

In an automated match preparation report, what is the primary reason for analyzing the opponent's build-up patterns by side (left/center/right)?

A) To compute the opponent's total passing volume B) To identify which side the opponent preferentially attacks, informing defensive positioning C) To determine the opponent's formation D) To calculate the opponent's possession percentage

Answer: B

Explanation: Understanding whether an opponent preferentially builds up through the left, center, or right channel directly informs defensive preparation. If the opponent overloads the left side (e.g., 45% of build-up passes), the coaching staff can prepare to shift defensive resources to that side, press the left-back more aggressively, or set specific marking instructions for the opposing left winger.

Question 13

In the context of player development tracking, why is a minimum observation period of 3 months recommended before classifying a development trajectory as "declining"?

A) Because data collection only happens quarterly B) Because short-term fluctuations are normal and do not indicate genuine decline C) Because 3 months is the standard contract period D) Because younger players cannot decline over shorter periods

Answer: B

Explanation: Player development is inherently non-linear. Temporary dips in performance metrics can result from growth spurts (especially in youth players), tactical adjustments, positional changes, minor niggles, or simply normal statistical variance. Requiring a sustained 3+ month declining trend before flagging a concern reduces false positives and avoids unnecessary interventions based on short-term noise.

Question 14

When using $k$-means clustering for player archetype discovery, the silhouette score measures:

A) The proportion of variance explained by the clusters B) How similar each object is to its own cluster compared to other clusters C) The number of iterations until convergence D) The distance between cluster centroids

Answer: B

Explanation: The silhouette score for a data point measures how similar it is to its own cluster (cohesion) compared to the nearest neighboring cluster (separation). Values range from -1 to 1, where higher values indicate that the point is well-matched to its own cluster and poorly matched to neighboring clusters. The mean silhouette score across all points is used to evaluate the overall quality of the clustering.

Question 15

In the injury risk model, why is class_weight='balanced' used in the logistic regression?

A) To ensure equal numbers of injured and non-injured observations B) To up-weight the minority class (injuries) in the loss function, counteracting class imbalance C) To balance the number of features with the number of observations D) To ensure the model predicts exactly 50% probability for each class

Answer: B

Explanation: Injuries are rare events (typically 2-5% of player-days result in injury), creating a severely imbalanced classification problem. Without adjustment, the model would learn to predict "no injury" for everything and achieve high accuracy but miss virtually all injuries. class_weight='balanced' adjusts the loss function to penalize misclassification of the minority class (injuries) proportionally more, encouraging the model to learn the injury signal despite its rarity.

Short Answer Questions

Question 16

Explain why isotonic regression calibration might overfit when applied to a small dataset of shots (e.g., fewer than 1,000 shots). What would you recommend instead?

Answer: Isotonic regression is a non-parametric method that fits a step function to the calibration data without any assumptions about the functional form. With a small dataset, the step function can overfit to noise in the calibration data, producing a calibration curve that fits the training data well but generalizes poorly. The limited number of shots means that the observed goal rates in each probability bin have high variance, and isotonic regression will faithfully reproduce these noisy observations. For small datasets, Platt scaling (logistic calibration) is recommended because it constrains the calibration to a sigmoid shape (only 2 parameters), providing a strong regularization effect that prevents overfitting while still improving calibration.

Question 17

A scouting composite score assigns weights of 0.25 to npxG/90 and 0.15 to pressing actions per 90. Explain why simply summing weighted raw values (without normalization) would produce a flawed ranking.

Answer: Raw metrics exist on completely different scales. npxG/90 typically ranges from 0 to 0.8, while pressing actions per 90 might range from 5 to 30. Without normalization, the pressing actions metric would dominate the composite score due to its much larger absolute values, regardless of the assigned weights. A player with 25 pressing actions and 0.2 npxG/90 would receive a pressing contribution of $0.15 \times 25 = 3.75$ versus an npxG contribution of $0.25 \times 0.2 = 0.05$, making the composite score almost entirely determined by pressing. Min-max normalization or z-score standardization maps all metrics to a common scale, ensuring that the weights reflect the intended relative importance of each criterion.

Question 18

Describe two limitations of using PPDA as the sole measure of pressing intensity.

Answer: (1) PPDA does not account for the location or effectiveness of defensive actions. A team could achieve a low PPDA by committing tactical fouls in the opponent's half without genuinely pressing the ball, or by making many unsuccessful tackle attempts. True pressing intensity should also consider whether the ball is actually recovered. (2) PPDA does not capture off-ball pressing activity. A team might execute a disciplined high press that restricts the opponent to short, lateral passes without requiring many defensive actions because the pressing shape itself denies progressive passing options. Complementary metrics like counterpressing recovery rate, high turnovers, and distance covered in the pressing phase provide a more complete picture of pressing behavior.

Question 19

Explain the concept of "field tilt" and how it differs from possession percentage as a measure of territorial dominance.

Answer: Field tilt measures the proportion of total passes that occur in the attacking third. For example, if Team A completes 80 passes in the final third and Team B completes 120, Team A's field tilt is $80/200 = 40\%$. Unlike possession percentage, which measures ball control regardless of location, field tilt captures where on the pitch the action is taking place. A team can have high possession but low field tilt if they dominate the ball in their own half (often seen with teams that play out from the back against a high press). Conversely, a team with lower possession but high field tilt is spending their time on the ball in dangerous areas, which is generally more indicative of attacking dominance.

Question 20

A gradient boosting xG model achieves log-loss of 0.321, while logistic regression achieves 0.338. Under what circumstances might you recommend deploying the logistic regression model instead?

Answer: Logistic regression would be preferred when: (1) Interpretability is paramount---coaching staff need to understand why a shot was rated highly, and logistic regression coefficients provide direct, interpretable effect sizes. (2) Calibration is critical without post-hoc adjustment---logistic regression is inherently calibrated, while gradient boosting may require additional calibration that could degrade with distribution shift. (3) The model needs to be robust to small dataset shifts---logistic regression's simplicity makes it less prone to overfitting specific data patterns. (4) Computational constraints exist---logistic regression is much faster for inference, which matters in real-time applications. (5) The improvement from 0.338 to 0.321 may not be practically significant at the individual match level, given the inherent randomness in goal-scoring.

Question 21

In survival analysis for return-to-play, explain what "censoring" means and why it must be handled correctly.

Answer: Censoring occurs when the exact time of the event of interest (return to play) is not observed for all individuals. Right censoring is most common: a player may still be injured when the observation period ends, so we know they were out for at least $t$ days but not the actual total duration. If censored observations are simply excluded, the analysis will underestimate the true average recovery time because only the shorter (completed) recoveries are included. If they are treated as complete recoveries at the censoring time, the analysis will underestimate recovery times by truncating long recoveries. Kaplan-Meier and Cox regression properly handle censoring by incorporating the partial information---that the player survived (remained injured) up to the censoring point---without making assumptions about the actual recovery time.

Question 22

Describe the difference between a player's percentile rank and their z-score. When would you use each in a development report?

Answer: A percentile rank indicates what percentage of the comparison group a player's value exceeds (e.g., 75th percentile means the player's value is higher than 75% of peers). A z-score indicates how many standard deviations a player's value is from the group mean (e.g., z = 1.5 means 1.5 standard deviations above average). Percentile ranks are preferred in development reports for coaching audiences because they are intuitive (everyone understands "top 25%") and are robust to outliers and non-normal distributions. Z-scores are preferred for analytical work because they preserve the distance between values and are needed for statistical modeling (e.g., composite score construction, growth curve fitting). A development report might display percentile ranks on radar charts for communication while using z-scores internally for computing development indices and trajectory models.

Question 23

A match preparation report identifies that the opponent delivers 60% of their corners as inswingers to the near post. Describe two specific tactical recommendations the coaching staff could implement based on this finding.

Answer: (1) Station a dedicated near-post defender---the tallest and most aerially dominant outfield player should be positioned at the near post with instructions to attack any delivery into the near-post zone. This player should be the first to any ball in the near-post area and should prioritize clearing over trying to retain possession. (2) Position a pressing player on the corner taker to force a change in delivery. By having a player close down the corner taker from the inswinger side, the team can either block the inswinger delivery angle entirely or force the opponent to use an outswinger instead, which the defending team can prepare for with different zonal assignments. Additionally, the coaching staff might rehearse a specific zonal marking scheme that overloads the near-post zone with 2-3 defenders rather than the usual 1, anticipating the opponent's preferred delivery pattern.

Question 24

Explain why the expected points calculation uses a Poisson distribution for goals rather than a normal distribution.

Answer: Goals are discrete count events that can only take non-negative integer values (0, 1, 2, 3, ...), which is exactly what the Poisson distribution models. The normal (Gaussian) distribution is continuous and extends to negative infinity, making it unsuitable for modeling counts. Additionally, goals in soccer are relatively rare events with a typical rate of 1-2 per team per match, and the Poisson distribution is specifically designed for modeling the number of events occurring in a fixed interval when events happen at a known average rate and independently of each other. While the independence assumption is imperfect (game-state effects), the Poisson distribution provides a much better fit to observed goal distributions than the normal distribution, particularly at the tails (e.g., 0-goal and 4+-goal matches).

Question 25

In a player development context, explain the concept of "development velocity" and propose a formula for computing it.

Answer: Development velocity measures the rate at which a player's metrics are improving relative to the expected trajectory for their age group. It captures not just the current level but the speed of improvement. A proposed formula is:

$$v_d = \frac{\Delta P_{player}}{\Delta P_{expected}}$$

where $\Delta P_{player}$ is the change in the player's percentile rank over a specified period (e.g., 6 months), and $\Delta P_{expected}$ is the expected change for a typical player at their developmental stage (often close to zero for an average developer). Alternatively, development velocity can be computed as the derivative of the fitted growth curve at the current time point, normalized by the average derivative across the peer group. A development velocity greater than 1 indicates the player is improving faster than average, suggesting accelerated development. Values consistently above 1.5 might indicate a player ready for promotion to a higher age group. Values below 0.5 sustained over 6+ months might trigger a review of the player's training program, playing time, or personal circumstances.