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Time-dependent structural equation modeling of fans' football fever using activity tracking data during the 2025 DFB Cup final

This study utilizes time-dependent structural equation modeling on smartwatch data from Arminia Bielefeld fans during the 2025 DFB Cup final to characterize football fever as a latent variable with a V-shaped temporal trajectory and significant individual heterogeneity.

Original authors: Jonas Bauer, Christiane Fuchs, Tamara Schamberger

Published 2026-04-23

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CC BY 4.0

Original authors: Jonas Bauer, Christiane Fuchs, Tamara Schamberger

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 nail-biting football match. Your heart is pounding, you're sweating, and you're screaming at the TV. You feel a surge of "football fever." But what if we could measure that fever not just by asking you how you felt afterward (which is often unreliable because our memories are fuzzy), but by actually tracking your body's reactions second-by-second?

That is exactly what this paper does. The authors took a deep dive into the 2025 DFB Cup final (a massive German football match) to understand the "life cycle" of a fan's excitement.

Here is the story of their research, explained simply:

1. The Problem: Why Surveys Aren't Enough

Usually, scientists ask fans, "How stressed were you during the game?" a few days later. The problem is that our brains are terrible at remembering the exact intensity of emotions as they happened. We tend to rewrite history based on how we feel now.

To fix this, the researchers used smartwatches (like Garmin) worn by 137 fans of the underdog team, Arminia Bielefeld. These watches acted like "emotional thermometers," recording heart rates and stress levels every three minutes. This gave them a high-definition movie of the fans' emotions, rather than just a blurry snapshot.

2. The Mystery: What is "Football Fever"?

The researchers noticed that when the fans' heart rates went up, their stress levels went up, too. They were moving in perfect sync.

They decided to treat this synchronized reaction as a hidden force, which they named "Football Fever."

The Analogy: Think of "Football Fever" as the engine of a car. You can't see the engine itself, but you can see the wheels spinning (heart rate) and the exhaust fumes rising (stress level). The researchers built a mathematical model to reverse-engineer the engine based on what the wheels and exhaust were doing.

3. The Method: A Time-Traveling Map

They didn't just look at the average; they built a complex map (called a Structural Equation Model) that tracked how this "engine" changed over time. They accounted for:

The Trend: The general shape of the excitement curve.
The Hiccups: Short-term spikes (like when a goal was scored).
The Differences: How some fans are more emotional than others.

4. The Big Discovery: The "V-Shape" Rollercoaster

The most exciting finding was the shape of the "Football Fever" curve. It wasn't a straight line; it was a V-shape.

The Start (High Peak): The excitement was sky-high right at kickoff.
The Slide (The Valley): As the game went on, the opposing team scored early goals. The fans' fever slowly drained away, like a balloon losing air. The curve went down steadily.
The Rally (The Climb): In the second half, the underdog team (Arminia Bielefeld) started fighting back, scoring goals, and creating chances. The fans' fever shot back up, creating the right side of the "V."

The Metaphor: Imagine a rollercoaster that starts at the top, drops down a long, scary hill, and then suddenly shoots back up the other side. That was the emotional journey of the average fan.

5. The Twist: Not All Fans Are the Same

While the average fan followed that V-shape, the model showed that fans are very different people.

Some fans had a "flat" fever; they stayed calm regardless of the score.
Others had a "wild" fever; they were screaming at the start and screaming at the end.
The model revealed that the fans who started with the highest excitement were the ones who reacted the most intensely to the game's ups and downs. It's like the difference between a casual observer and a die-hard superfan.

6. Why This Matters

This study proves that we can use technology to scientifically measure emotions in real-time.

For Teams: They could know exactly when fans are most excited to send them a special message or offer a discount.
For Health: It shows how sports can impact our physical health (heart rate), which is good for understanding stress but also a warning for people with heart conditions.
For Science: It moves us away from asking "How did you feel?" to actually seeing how we feel.

In a Nutshell

The researchers used smartwatches to turn invisible emotions into visible data. They discovered that a fan's excitement during a big game is like a V-shaped rollercoaster: high at the start, dipping when things go wrong, and soaring again when hope returns. And just like on a rollercoaster, everyone rides it a little differently.

1. Problem Statement

Existing literature on sports fan behavior relies heavily on cross-sectional surveys and retrospective self-reports, which suffer from recall bias and fail to capture the temporal dynamics of emotional states. While previous studies have identified physiological reactions (e.g., heart rate spikes) during matches, they lack a rigorous statistical framework to model the underlying latent construct driving these reactions—termed "football fever." Furthermore, few studies utilize high-resolution longitudinal data to distinguish between stable individual traits, match-specific trends, and short-term autoregressive effects in real-time physiological responses.

2. Methodology

Data Collection and Pre-processing

Dataset: High-resolution data from 137 Arminia Bielefeld supporters wearing Garmin smartwatches during the 2025 DFB Cup final (vs. VfB Stuttgart).
Variables: Heart Rate (HR) and Stress Level (SL) recorded in 3-minute intervals over 97 match minutes (34 time points).
Missing Data Strategy: Due to sensor limitations during high physical activity, stress data had ~25% missingness. The authors employed Multivariate Multiple Imputation using Random Forests (via the mice package in R) to generate 10 imputed datasets, avoiding the biases of list-wise deletion or standard SEM imputation.

Statistical Model: Time-Dependent Structural Equation Modeling (SEM)

The authors constructed a complex SEM to model "football fever" as a common latent factor manifesting through HR and SL. The model integrates four key components:

Indicator-Construct Model:
- Assumes HR and SL are linear indicators of the latent football_fevert.
- Factor loadings ( $\lambda$ ) are fixed over time to ensure temporal invariance of the construct's definition.
- HR is set as the marker indicator ( $\lambda_{HR}=1$ ).
Measurement Error Variances:
- Distinguishes between time-invariant individual heterogeneity and time-specific variability.
- Specifically models higher variance at the start of halves ( $t=1, 17$ ) due to diverse fan activities (e.g., standing vs. sitting).
- Includes random intercepts for stress levels to capture between-half variability.
Latent Growth Curve (Trend Component):
- Models the trajectory of football fever using three random variables: an intercept ($IFF$) and two slopes ( $S_1, S_2$ ).
- Piecewise Linear Trend:
  - $t \le 7$ : Plateau (Kick-off).
  - $7 < t \le 25$ : Steady decline (following early opponent goals).
  - $t > 25$ : Steady incline (late rally).
- Allows for random effects (variances and covariances) to capture substantial between-fan heterogeneity in baseline arousal and reaction to events.
Autoregressive Process (AR(1)):
- Adds a stochastic process $R_t = \rho R_{t-1} + \zeta_t$ to account for short-term carry-over effects (e.g., residual excitement after a goal) that deviate from the long-term trend.

Estimation and Fit Evaluation

Software: R packages lsttheory, lavaan, and lavaan.mi.
Pooling: Parameters and standard errors were pooled across 10 imputed datasets using Rubin's rules.
Fit Indices: Standard indices (CFI, RMSEA) were adjusted for high-dimensional time-series data using corrections from Yuan et al. (2015) to prevent Type I errors.

3. Key Results

Model Fit

The time-dependent SEM demonstrated excellent fit (Corrected RMSEA = 0.015, CFI = 0.994, SRMR = 0.079).
It significantly outperformed a time-invariant baseline model ( $\Delta AIC = 10,920$ ), confirming that temporal dynamics are critical.

Parameter Estimates

Construct Validity: High standardized factor loadings (HR $\approx$ 0.98, SL $\approx$ 0.80) confirm that both indicators reliably reflect the latent "football fever."
Trajectory (V-Shape): The average fan's fever followed a V-shaped trajectory:
- Peak: Kick-off ( $\hat{\mu}_{IFF} \approx 89$ bpm equivalent).
- Decline: Linear decrease of ~10% between minutes 7–24 (driven by opponent goals).
- Recovery: Sharp increase from minute 25 onwards (driven by home team goals/chances).
Heterogeneity:
- Significant variance in the intercept ( $\hat{\phi}_{IFF} = 164.5$ ), indicating large differences in baseline arousal among fans.
- Significant variance in the first slope ( $\hat{\phi}_{S1}$ ), suggesting fans react differently to early match events.
- Autoregression: A strong carry-over effect ( $\hat{\rho} = 0.658$ ), meaning physiological states persist and influence the next 3-minute interval.

Residual Analysis

Residuals were near zero within halves, indicating the trend model captured intra-half dynamics well.
Systematic negative residuals appeared at the start of the second half and during specific goal events, suggesting potential structural breaks or non-invariance not fully captured by the linear trend.

4. Key Contributions

Theoretical Advancement: Successfully operationalized "football fever" as a latent variable, moving beyond descriptive statistics to a structural model of emotional dynamics.
Methodological Innovation: Integrated wearable technology data, multiple imputation, and time-dependent SEM with latent growth and autoregressive components. This framework addresses the specific challenges of high-dimensional, correlated, and missing longitudinal data.
Granular Insight: Revealed that while the average fan follows a V-shaped emotional curve, there is profound between-fan heterogeneity. Some fans maintain steady arousal, while others exhibit extreme volatility, likely linked to their level of fandom.

5. Significance and Implications

Sports Marketing & Management: The findings enable real-time, fan-specific communication strategies. Marketers can target "peak engagement" windows (e.g., the second-half rally) and segment audiences based on their emotional volatility (casual vs. die-hard fans).
Health & Safety: By quantifying the physiological load (heart rate/stress) of spectatorship, the study highlights potential cardiovascular risks for fans with high emotional reactivity, offering a basis for health interventions.
Scientific Paradigm: The study demonstrates that retrospective surveys are insufficient for studying dynamic emotional phenomena. It establishes a blueprint for using objective, high-frequency physiological data to study consumer behavior and group psychology in real-world settings.

Limitations Noted

Generalizability: The sample was restricted to one club's fans with specific smartwatches, potentially biasing toward tech-savvy or highly engaged demographics.
Measurement: The stress metric is proprietary to the manufacturer and requires external validation.
Model Structure: The linear trend could not fully capture complex non-linear structural breaks (e.g., halftime transitions), evidenced by residual patterns.