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Simultaneous Estimation of Ballpark Effects and Team Defense Using Total Bases Residuals

This paper introduces a unified regression framework using Total Bases Residuals derived from Statcast data to simultaneously estimate and distinguish between ballpark effects and team defense in baseball, offering a standardized index that aligns with recent league-wide trends and provides empirical support for its validity against official metrics.

Original authors: Jhe-Jia Wu, Tian-Li Yan, Ting-Li Chen

Published 2026-03-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jhe-Jia Wu, Tian-Li Yan, Ting-Li Chen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a baseball game, and a batter hits a massive home run. The crowd goes wild. But here's the tricky question: Who deserves the credit?

Was it the batter's incredible swing? Was it the wind blowing out from the stadium? Was it the fact that the opposing team's outfielders were standing in the wrong spot? Or was it just a lucky bounce?

For years, baseball statisticians have struggled to untangle these knots. This paper, written by researchers from Taiwan and the US, proposes a clever new way to solve this puzzle. They call it the "Total Bases Residual" method.

Here is the breakdown of their idea, using some everyday analogies.

1. The Problem: The "Noise" in the Signal

Imagine you are trying to measure how good a chef is. You taste a soup.

If the soup is salty, is it because the chef added too much salt?
Or is it because the water they used was already salty?
Or maybe the bowl they served it in was pre-salted?

In baseball, the "soup" is the result of a hit. The "salt" is the ballpark (some stadiums are huge and hard to hit over; others are tiny and easy). The "chef" is the defense (some teams are amazing at catching balls; others are sloppy).

Old methods tried to guess the "saltiness" of the stadium by just looking at the final score. But that's like blaming the water for the soup being salty without tasting the ingredients first. It often confused the stadium's effect with the team's defense.

2. The Solution: The "Expected Outcome" Baseline

The authors decided to start with a simple, fair rule: If you hit a ball with the same speed and angle, it should go the same distance, right?

They used high-tech data (called Statcast) to measure exactly how hard the ball was hit (Exit Velocity) and at what angle (Launch Angle).

The Analogy: Imagine a giant video game. You input the speed and angle of a hit, and the computer tells you, "Based on physics and history, this ball should travel 300 feet and result in a single."

This is their Baseline. It's the "expected" result if everything were perfectly average.

3. The "Residual": The Surprise Factor

Now, they look at what actually happened.

Scenario A: The computer said 300 feet (a single), but the ball rolled all the way to the wall for a double.
- The Residual: +1 Base. Something good happened.
Scenario B: The computer said 300 feet, but the ball was caught right in front of the outfielder.
- The Residual: -1 Base. Something bad happened.

This difference between "what should have happened" and "what actually happened" is called the Residual. It strips away the batter's skill (since we already accounted for the speed/angle) and leaves only the environment and the defense.

4. The Magic Trick: Separating the Stadium from the Team

Here is where the math gets clever. The researchers looked at thousands of these "surprises" across the whole league.

The Stadium Effect: If every team visiting a specific stadium (like Coors Field in Colorado) consistently gets "bonus bases" (positive residuals), that stadium is a Hitter-Friendly Park. It's like a trampoline that helps the ball bounce higher.
The Defense Effect: If a specific team (like the Atlanta Braves) consistently forces balls to result in fewer bases than expected (negative residuals), no matter who they are playing against, that team has Great Defense. It's like a team of vacuum cleaners that sucks the ball out of the air.

By running a giant regression model (a fancy way of saying "let's solve for X and Y at the same time"), they could separate the Trampoline (Stadium) from the Vacuum Cleaners (Defense).

5. The Results: What Did They Find?

They tested this from 2015 to 2024 and found some interesting things:

Stadiums are stable, but not perfect: Some parks are consistently great for hitters (like Coors Field), and some are great for pitchers (like Oracle Park in San Francisco). Their method confirmed what fans knew, but with more precision.
Defense is tricky: Their new defense numbers sometimes disagreed with the official MLB numbers.
- Example: In 2015, the Houston Astros' home field looked like a "Hitter's Paradise" in their model (Index 154), while official stats said it was neutral (Index 92). Why? Because the Astros' defense was actually so bad that it made the stadium look like it was helping hitters, even though the stadium itself wasn't that special. Their model caught this; the old stats missed it.
The "Juiced Ball" Era: They noticed a trend. From 2017–2019, the league-wide "surprise" was positive (balls went further). Then, in the early 2020s, it dropped. This perfectly matches the story of the "Juiced Ball" (a livelier baseball) and the subsequent changes MLB made to the ball to make it less lively.
The Shift Ban: In 2023, MLB banned extreme defensive shifts. The data showed a slight uptick in offensive outcomes right after the ban, proving their model was sensitive enough to catch rule changes.

6. The New Scorecard: The "Standardized Index"

Finally, they created a new way to report these numbers. Instead of saying "This park adds 0.05 runs," they use a Z-Score Index (similar to a test score).

100 = Average.
120 = One standard deviation better than average (Very good).
80 = One standard deviation worse than average (Not so good).

This makes it easy to compare a pitcher's park in 2015 with a defense in 2024, even though the game has changed so much.

The Bottom Line

This paper is like putting on 3D glasses for baseball statistics. Before, we were looking at a flat picture where the stadium, the defense, and the batter were all blended together. This new method separates the layers, letting us see exactly how much the stadium helped, how much the defense hurt, and how much the batter actually did.

It's a simpler, cleaner way to understand the beautiful chaos of a baseball game.

1. Problem Statement

In baseball analytics, accurately evaluating offensive performance and defensive capability is complicated by the fact that batted-ball outcomes are influenced by multiple interacting factors:

Contact Quality: The physical properties of the hit (Exit Velocity and Launch Angle).
Ballpark Environment: Dimensions, altitude, and atmospheric conditions specific to a stadium.
Team Defense: The ability of the fielding team to convert batted balls into outs.
Random Variation: Unpredictable elements in play.

Existing methods face significant limitations:

Park Effects: Traditional "Home/Road" ratios suffer from "inflationary bias" due to mathematical dependencies and assumptions of balanced schedules. They often fail to separate park effects from the specific defensive quality of the home team.
Defense Metrics: Modern metrics like Defensive Runs Saved (DRS), Ultimate Zone Rating (UZR), and Outs Above Average (OAA) rely on complex tracking data or specific modeling frameworks. They are often developed independently of park effect estimation, making it difficult to isolate the pure environmental impact of a stadium from the defensive performance of the team playing there.

The core challenge is confounding: without separating these components, observed differences in outcomes cannot be uniquely attributed to the ballpark or the defense.

2. Methodology

The authors propose a unified regression framework based on Total Bases Residuals (TBR) to simultaneously estimate ballpark effects and team defense.

A. Construction of Outcome Residuals (TBR)

Instead of using binary outcomes (out vs. hit) or simple hit frequency, the authors use Total Bases (TB) as the outcome metric, as it correlates more strongly with run production.

Control for Contact Quality: Using Statcast data (2015–2024), the authors discretize the Exit Velocity (EV) and Launch Angle (LA) space into a fine grid (3 mph $\times$ 3 degrees).
Expected Baseline: For each grid cell $g$ , they calculate the empirical mean total bases ( $\mu_g$ ) for all batted balls falling within that EV/LA range across the league.
Residual Definition: For an individual batted ball $i$ $i$ , the Total Bases Residual ( $R_i$ $R_{i}$ ) is defined as:
$R_i = TB_i - \mu_{g(i)}$
Where $g(i)$ $g (i)$ is the grid cell containing the ball's EV and LA.
- $R_i > 0$ : The ball resulted in more bases than expected given its contact quality.
- $R_i < 0$ : The ball resulted in fewer bases than expected.
- This residual isolates the combined influence of the ballpark environment and the defensive team, removing the primary effect of contact quality.

B. Simultaneous Estimation Model

The authors model the residuals using a weighted least squares regression with an additive structure:
$R_i = \beta_0 + \beta_{park}^{p(i)} - \beta_{def}^{d(i)} + \varepsilon_i$

$\beta_{park}^{p}$ : The environmental effect of ballpark $p$ .
$\beta_{def}^{d}$ : The defensive effect of team $d$ (subtracted because better defense reduces total bases).
Aggregation: To improve efficiency, observations are aggregated by the $(Park, Defensive Team)$ pair, reducing the sample size from $\sim 10^5$ batted balls to $\le 900$ cells per season.
Identifiability: One park and one team (Atlanta Braves/Truist Park) are set as reference categories (zero effect).

C. Standardization and Indexing

To facilitate comparison across seasons and metrics:

Centering: Estimates are re-centered so the league average is zero.
Z-Score Transformation: Effects are converted to z-scores based on the distribution of estimates for that season.
Standardized Index: A 100-based index is created: $Index = 100 + 20 \times z$ $I n d e x = 100 + 20 \times z$ .
- 100 = League Average.
- 120 = 1 Standard Deviation above average.
- This allows for intuitive comparison of magnitude (e.g., how "extreme" a park or defense is).

3. Key Contributions

Unified Framework: The first model to simultaneously estimate ballpark effects and team defense within a single regression, explicitly controlling for contact quality (EV/LA) via residuals.
Decoupling Confounding Factors: By using TBR residuals, the method isolates environmental and defensive effects from the quality of contact, addressing the "inflationary bias" found in traditional home/road ratios.
Standardized Metric: Introduction of a z-score-based index that provides a clearer interpretation of effect magnitude (in units of standard deviation) compared to traditional ratio-based metrics (like Park Factors or Def).
Empirical Validation: The study provides a rigorous comparison against official MLB metrics (Park Factors, Def, OAA) using home/away TBR patterns to validate the estimates.

4. Key Results

A. Ballpark Effects

Stability: The estimated park effects show high temporal stability (lower standard deviation over 10 years than official MLB metrics), consistent with the physical nature of stadiums.
Discrepancies: The model identifies significant differences from official MLB park factors in specific cases (e.g., Comerica Park in 2017, Minute Maid Park in 2015).
- Example: For Minute Maid Park (HOU, 2015), the official metric suggested a neutral/pitcher-friendly park (92), while the authors' model estimated a hitter-friendly park (154). Analysis of home/away TBR showed both Houston and opponents had significantly higher TBR at home, supporting the authors' "hitter-friendly" conclusion.
Asymmetry: The model struggles slightly with parks having extreme asymmetries (e.g., Fenway, PNC Park) because it does not explicitly model batted-ball direction, only EV/LA.

B. Team Defensive Effects

Comparison with Def and OAA: The authors' estimates show moderate correlation with existing metrics (highest correlation with OAA at ~0.67, lower with Def at ~0.44).
Validation via TBR: In cases of large discrepancies, the authors' estimates often align better with observed home/away TBR patterns.
- Example: When a team's opponents consistently have lower TBR at home (indicating strong defense), the authors' model correctly assigns a high defensive rating, whereas some official metrics (like Def) may assign a lower rating due to roster composition or other factors not captured in the batted-ball residual.
Scope: The metric specifically measures defense on batted-ball events, excluding catcher framing, blocking, or running game control, which differentiates it from comprehensive metrics like Def.

C. League-Wide Trends (Intercept Analysis)

The "Juiced Ball" Era: The model's intercept (representing the league-wide baseline) shows a clear downward trend from 2015 to 2024.
- 2017–2019: Positive intercepts align with the "juiced ball" era (reduced drag, increased carry).
- 2020–2022: Negative intercepts align with MLB efforts to reduce ball liveliness and the introduction of humidors.
- 2023: A slight upward shift aligns with the restriction on extreme defensive shifts.
This suggests the model successfully captures macro-level changes in the game's physical and strategic environment.

5. Significance and Future Work

Interpretability: The framework offers a transparent, data-driven way to separate environmental and defensive influences without relying on opaque "black box" models.
Strategic Insight: The results highlight how defensive positioning and ball physics have evolved, providing a baseline for evaluating team performance relative to the league environment.
Limitations & Future Directions:
- Directionality: The current model ignores batted-ball direction, which limits accuracy in asymmetric parks. Future work could incorporate direction but risks data sparsity.
- Player-Level Decomposition: The model estimates team-level defense, not individual player value. Linking team estimates to individual contributions is a future goal.
- Non-Linearity: The use of Total Bases is a linear proxy for run value. Future iterations could explore non-linear transformations based on run expectancy.
- Interactions: The current additive model assumes parks and defenses are independent. Future research could explore interactions (e.g., teams building rosters specifically for their home park's dimensions).

In conclusion, this paper presents a robust statistical approach to disentangling the complex factors influencing baseball outcomes, offering a refined perspective on both stadium effects and team defense that complements and, in some cases, corrects existing industry standards.