Chapter 9 Quiz: Regression Analysis for Sports Modeling

Instructions: Answer all 25 questions. This quiz is worth 100 points. You have 60 minutes. A calculator is permitted; no notes or internet access. For multiple choice, select the single best answer.

Section 1: Multiple Choice (10 questions, 3 points each = 30 points)

Question 1. In a linear regression model predicting NFL point differentials, the coefficient on "home team offensive EPA per play" is 0.15. This means:

(A) The home team scores 0.15 more points per game on average

(B) For every one-unit increase in offensive EPA per play, the predicted point differential increases by 0.15, holding other variables constant

(C) Offensive EPA explains 15% of the variance in point differentials

(D) The correlation between offensive EPA and point differentials is 0.15

Question 2. Which of the following is the primary reason to use logistic regression instead of linear regression when modeling the probability of a team winning?

(A) Logistic regression is always more accurate than linear regression

(B) Linear regression can produce predicted probabilities outside the [0, 1] range

(C) Logistic regression requires fewer data points to fit

(D) Linear regression cannot handle categorical predictor variables

Question 3. A regression model has an $R^2$ of 0.35 on the training set and 0.12 on the test set. This pattern most likely indicates:

(A) The model is underfitting

(B) The model is overfitting

(C) The data has high measurement error

(D) The training and test sets come from different populations

Question 4. The Variance Inflation Factor (VIF) for a feature is 10.5. This indicates:

(A) The feature explains 10.5% of the variance in the outcome

(B) The feature's coefficient variance is inflated by a factor of 10.5 due to multicollinearity

(C) Removing the feature would improve model accuracy by 10.5%

(D) The feature has 10.5 times the predictive power of the average feature

Question 5. In the context of sports regression models, "feature engineering" refers to:

(A) Removing outliers from the training data

(B) Creating new predictor variables by transforming or combining raw data

(C) Selecting the best algorithm for the prediction task

(D) Tuning the hyperparameters of the regression model

Question 6. A logistic regression model outputs a predicted probability of 0.65 for the home team winning. The sportsbook's implied probability (after removing vig) is 0.58. The bettor should:

(A) Bet on the home team because the model probability exceeds the market probability

(B) Bet on the away team because the market is always more accurate

(C) Not bet because the difference is within the margin of error

(D) Bet on the home team only if the model's Brier score is below 0.20

Question 7. Which regularization method would you choose if you suspect that many of your features are irrelevant and you want the model to automatically set their coefficients to exactly zero?

(A) Ridge regression (L2)

(B) Lasso regression (L1)

(C) Elastic Net

(D) Ordinary least squares with backward elimination

Question 8. The Durbin-Watson statistic for a regression model is 0.85. This suggests:

(A) Strong positive autocorrelation in the residuals

(B) No autocorrelation in the residuals

(C) Strong negative autocorrelation in the residuals

(D) Heteroscedasticity in the residuals

Question 9. Cross-validation is used in sports modeling primarily to:

(A) Increase the training set size

(B) Estimate out-of-sample performance and guard against overfitting

(C) Automatically select the best features

(D) Ensure the model meets the normality assumption

Question 10. When building a regression model for betting, a "look-ahead bias" occurs when:

(A) You use too many predictor variables

(B) You include information in training features that would not have been available at prediction time

(C) You test on the same data used for training

(D) You fail to account for home-field advantage

Section 2: True/False (5 questions, 3 points each = 15 points)

Write "True" or "False." Full credit requires correct identification only.

Question 11. True or False: Adding more features to a linear regression model will always increase (or leave unchanged) the $R^2$ on the training data.

Question 12. True or False: If a regression coefficient has a p-value of 0.03, it means there is a 3% probability that the coefficient is actually zero.

Question 13. True or False: In sports modeling, time-series cross-validation (walk-forward validation) is preferred over standard k-fold cross-validation because it prevents future information from leaking into the training set.

Question 14. True or False: A model with an RMSE of 10 points for predicting NBA game totals is too inaccurate to be profitable for betting.

Question 15. True or False: Ridge regression (L2 regularization) is preferred over Lasso (L1 regularization) when you believe all features contribute some predictive value and none should be entirely excluded.

Section 3: Fill in the Blank (3 questions, 4 points each = 12 points)

Question 16. In logistic regression, the function that maps the linear predictor to a probability between 0 and 1 is called the __________ function, defined as $\sigma(z) = \frac{1}{1 + e^{-z}}$.

Question 17. The diagnostic statistic that measures the influence of a single observation on the fitted regression by comparing the full model to the model with that observation removed is called __________ distance.

Question 18. When the variance of regression residuals is not constant across fitted values (e.g., prediction errors are larger for high-scoring games than low-scoring games), the condition is called __________.

Section 4: Short Answer (3 questions, 5 points each = 15 points)

Answer each question in 3-5 sentences.

Question 19. Explain the difference between "training error" and "generalization error" in the context of a sports betting regression model. Why is the distinction critical for bettors?

Question 20. Describe what a "calibration plot" shows for a logistic regression model predicting game outcomes, and explain why good calibration is essential for betting applications.

Question 21. Explain why using a team's current season statistics (e.g., season-to-date points per game) as features early in the season is problematic, and describe one approach to address this issue.

Section 5: Code Analysis (2 questions, 6 points each = 12 points)

Question 22. Examine the following code that builds a logistic regression model:

import pandas as pd
from sklearn.linear_model import LogisticRegression
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler

df = pd.read_csv("nba_games.csv")
X = df[["net_rating", "pace", "rest_days", "travel_miles"]]
y = df["home_win"]

scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)

X_train, X_test, y_train, y_test = train_test_split(
    X_scaled, y, test_size=0.2, random_state=42
)

model = LogisticRegression()
model.fit(X_train, y_train)
print(f"Test accuracy: {model.score(X_test, y_test):.3f}")

(a) This code contains a data leakage error. Identify it and explain why it matters.

(b) Write the corrected version of the problematic lines.

X_train, X_test, y_train, y_test = train_test_split(
    X, y, test_size=0.2, random_state=42
)

scaler = StandardScaler()
X_train_scaled = scaler.fit_transform(X_train)  # Fit only on training data
X_test_scaled = scaler.transform(X_test)         # Transform test using training stats

model = LogisticRegression()
model.fit(X_train_scaled, y_train)
print(f"Test accuracy: {model.score(X_test_scaled, y_test):.3f}")

Question 23. Examine the following feature engineering code:

def compute_features(games_df, team, game_date):
    """Compute features for a team as of a given game date."""
    team_games = games_df[games_df["team"] == team]
    season_games = team_games[team_games["season"] == game_date.year]

    features = {
        "avg_points": season_games["points_scored"].mean(),
        "avg_allowed": season_games["points_allowed"].mean(),
        "win_pct": season_games["win"].mean(),
        "last_5_avg_points": season_games.tail(5)["points_scored"].mean(),
    }
    return features

(a) Identify at least two bugs or methodological problems in this function.

(b) For each problem, explain the potential impact on model performance.

(c) Write corrections for the issues you identified.

def compute_features(games_df, team, game_date, season_id):
    """Compute features for a team using only games before game_date."""
    team_games = games_df[
        (games_df["team"] == team) &
        (games_df["season"] == season_id) &
        (games_df["game_date"] < game_date)  # Only past games
    ].sort_values("game_date")

    if len(team_games) < 3:
        return None  # Not enough data for stable features

    features = {
        "avg_points": team_games["points_scored"].mean(),
        "avg_allowed": team_games["points_allowed"].mean(),
        "win_pct": team_games["win"].mean(),
        "last_5_avg_points": team_games.tail(5)["points_scored"].mean(),
        "games_played": len(team_games),
    }
    return features

Section 6: Applied Problems (2 questions, 8 points each = 16 points)

Question 24. You build a linear regression model predicting NFL game totals (combined score). The model produces the following predictions and the corresponding sportsbook lines for five upcoming games:

(a) (2 points) For each game, state whether the model favors the Over or the Under, and calculate the difference between the model's prediction and the sportsbook total.

(b) (2 points) Using a normal distribution assumption with the model's RMSE of 12.0 points, calculate the model's implied probability of the Over hitting for Game A (model prediction 48.2, line 44.5).

(c) (2 points) Convert the Game A sportsbook vig-adjusted line (-110/-110) into implied probabilities for Over and Under. Does the model see value on the Over?

(d) (2 points) If you were deploying this model with a minimum edge threshold of 3 percentage points, which games would you bet, and on which side?

Question 25. A colleague builds a logistic regression model to predict NBA moneyline outcomes and reports the following evaluation metrics:

• Training accuracy: 68.5%
• Test accuracy: 62.1%
• Training log-loss: 0.58
• Test log-loss: 0.67
• Training Brier score: 0.195
• Test Brier score: 0.232
• AUC-ROC: 0.67

They also report that when they bet on all games where the model's predicted probability exceeds the market's implied probability by at least 5 percentage points, they achieved a 54.2% win rate over 240 bets at average odds of -108.

(a) (2 points) Assess whether the model is overfitting based on the training vs. test metrics.

(b) (2 points) Calculate the approximate profit or loss from 240 bets at -108 average odds with a 54.2% win rate, assuming a flat $100 stake per bet.

(c) (2 points) Is the 54.2% win rate over 240 bets statistically significant at the 5% level against a null of 52.4% (the break-even rate at -108)? Use the normal approximation.

(d) (2 points) What three improvements would you recommend for this model before deploying it with real money?

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