arxiv: 2604.20285 · v1 · submitted 2026-04-22 · 📊 stat.AP · stat.OT

Recognition: unknown

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

Jonas Bauer , Christiane Fuchs , Tamara Schamberger

Authors on Pith no claims yet

Pith reviewed 2026-05-09 23:13 UTC · model grok-4.3

classification 📊 stat.AP stat.OT

keywords football feverstructural equation modelinglatent growthautoregressive effectswearable datatemporal dynamicssports spectatorshipintensive longitudinal data

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The pith

Smartwatch data and time-dependent modeling show fans' football fever follows a V-shaped trajectory during matches.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies time-dependent structural equation modeling to intensive longitudinal smartwatch data collected from football supporters during a cup final. It constructs a latent variable for football fever from heart rate, stress, and related physiological indicators, then models its evolution using latent growth components for overall trends and autoregressive terms for short-term carry-over effects. Results aggregated across multiple imputations reveal a consistent V-shaped pattern: high arousal at kick-off, steady decline through the first half, and renewed increase in the second half, alongside large differences between fans in baseline levels and change rates. The work establishes that wearable devices can track such dynamic emotional processes when paired with structural equation models that properly handle temporal dependence and high-dimensional data.

Core claim

Football fever is adequately represented as a latent variable reflected in wearable technology data and unfolds according to a V-shaped trajectory: high at kick-off, followed by a steady decline until the renewed arousal in the second half, with substantial between-fan heterogeneity in both baseline level and temporal dynamics. This pattern is captured by a time-dependent structural equation model that includes latent growth components and autoregressive effects, with model fit evaluated after adjustments for missing measurements and high data dimensionality.

What carries the argument

Time-dependent structural equation model with latent growth components for overall trends and autoregressive effects for short-term carry-over in the latent football fever process derived from heart rate and stress indicators.

If this is right

Dynamic emotional phenomena in sports spectatorship require explicit modeling of temporal dependence to avoid biased conclusions about arousal levels.
Wearable technology combined with structural equation modeling can adequately represent latent constructs like football fever from physiological signals.
Substantial heterogeneity between fans in both starting levels and change patterns implies that average trajectories alone miss important individual differences.
The V-shaped pattern ties emotional responses to specific phases of the match, such as initial excitement and second-half renewal.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same modeling strategy could apply to emotional dynamics during other spectator events like concerts or public gatherings where physiological data are available.
If the V-shape generalizes, event organizers might use it to time interventions that counteract mid-event dips in audience engagement.
Individual differences in fever trajectories could be linked to measurable traits such as team loyalty or prior match experience in follow-up studies.
Extending the approach to non-sports contexts might reveal whether V-shaped emotional responses are specific to competitive matches or common to other high-arousal situations.

Load-bearing premise

Heart rate and stress readings from smartwatches are valid and sufficient indicators of the latent football fever construct, and the time-dependent SEM captures the true dynamics without substantial bias from missing data or high dimensionality.

What would settle it

Replicating the smartwatch data collection during another high-stakes football match and finding that the estimated trajectory lacks the V-shape or that the time-dependent model with autoregressive terms shows poor fit compared to simpler alternatives would challenge the central claim.

Figures

Figures reproduced from arXiv: 2604.20285 by Christiane Fuchs, Jonas Bauer, Tamara Schamberger.

**Figure 2.** Figure 2: Our structural equation model for the dynamics of football fever. Co [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Heatmap of variances and covariances of standardized residuals, av [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: Baseline model. B Results for different imputation methods 21 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗

read the original abstract

Football fans frequently exhibit pronounced emotional and physiological reactions during high-stakes matches. However, the temporal dynamics of this football fever are rarely modeled as a latent process. Using intensive longitudinal data from Arminia Bielefeld supporters who wore smartwatches during the 2025 German Football Association (DFB) Cup final, we investigate how football fever unfolds. The devices recorded heart rate, stress level, and related indicators in short intervals, allowing us to construct a latent variable for football fever and model its dynamics. We specify a time-dependent structural equation model with latent growth components and autoregressive effects to capture both overall trends and short-term carry-over effects in fans' physiological responses. Results are aggregated across multiple imputations of missing measurements. Model fit is evaluated using adjustments for the high data dimensionality. The results show that football fever follows a V-shaped trajectory: high at kick-off, followed by a steady decline until the renewed arousal in the second half, with substantial between-fan heterogeneity in both baseline level and temporal dynamics. Our findings demonstrate that football fever can be adequately represented as a latent variable using structural equation modeling and reflected by wearable technology data. This highlights the importance of accounting for temporal dependence when studying dynamic emotional phenomena, e. g., in sports spectatorship.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper fits a time-dependent SEM with latent growth and autoregression to smartwatch heart rate and stress data from one football final and reports a V-shaped latent trajectory plus individual differences, but the measurement model rests on unvalidated physiological proxies.

read the letter

This paper takes smartwatch readings from Arminia Bielefeld fans during the 2025 DFB Cup final and models a latent construct called football fever with a time-dependent SEM that includes latent growth terms and autoregressive carry-over. The main result is a V-shaped pattern: elevated at kick-off, declining through the first half, then rising again in the second half, with noticeable between-person variation in starting level and slope. They handle missing data via multiple imputation and adjust fit measures for the high dimensionality of the intensive longitudinal records. That is the core contribution on offer. The application itself is straightforward once the data exist, but collecting synchronized physiological time series from spectators at a live match is not routine, so the dataset and the modeling choice together give the work its edge. The authors are clear that they treat the wearable signals as indicators of an unobserved emotional process rather than direct measures. The soft spots sit mainly in the measurement step. Heart rate and stress scores from consumer devices are treated as sufficient for the latent variable without any reported check against self-reported excitement, behavioral observations, or established physiological benchmarks. If those signals contain substantial noise or systematic mismatch with the intended construct, the estimated growth factors and autoregressive coefficients will inherit the bias, and the V-shape interpretation becomes harder to trust. The abstract also omits the actual parameter estimates, standard errors, and any sensitivity checks, so a reader cannot yet judge how stable the trajectory is under different imputation or specification choices. The scope is narrow—one match, one team’s supporters—which is fine for an initial demonstration but limits claims about broader emotional dynamics. Readers working on intensive longitudinal modeling in psychology or sports science would get the most from this, especially if they want a concrete example of applying latent growth plus autoregression to wearable streams. The work shows clear thinking about the temporal structure and honest use of imputation and adjusted fit indices. It deserves a serious referee because the data source is fresh and the modeling framework matches the research question, even though the authors will need to supply the missing specification details, estimates, and at least preliminary validation of the indicators before publication.

Referee Report

2 major / 1 minor

Summary. The manuscript applies a time-dependent structural equation model with latent growth components and autoregressive effects to intensive longitudinal smartwatch data (heart rate and stress readings) collected from Arminia Bielefeld fans during the 2025 DFB Cup final. It constructs a latent 'football fever' variable, aggregates results over multiple imputations of missing data, applies dimensionality-adjusted fit measures, and reports a V-shaped trajectory (high at kick-off, decline through the first half, renewed arousal in the second half) together with substantial between-fan heterogeneity in baseline levels and temporal dynamics.

Significance. If the measurement model proves valid and the reported dynamics are not artifacts of unvalidated indicators or imputation choices, the work offers a concrete demonstration of how consumer wearables can be integrated into latent-variable modeling of real-time emotional processes in naturalistic settings. It underscores the value of combining growth-curve and autoregressive structures for capturing both long-term trends and short-term carry-over in physiological data, potentially informing future studies of spectator affect in sports and other high-arousal contexts.

major comments (2)

[Abstract] Abstract: the central V-shaped claim is presented without any reported parameter estimates, standard errors, significance tests for the growth factors (intercept, slope, quadratic), or autoregressive coefficients. Because the trajectory shape is defined by these parameters, their absence prevents verification that the decline-then-rise pattern is statistically supported rather than an artifact of the chosen functional form or imputation procedure.
[Methods] Methods/Results (implied by abstract description): the measurement model treats heart-rate and stress readings as sufficient indicators of the latent football-fever construct, yet no external validation (concurrent self-reports, behavioral observations, or physiological benchmarks) or sensitivity checks for indicator validity are described. Any systematic mismatch between the wearable signals and the intended psychological process would propagate directly into the estimated growth factors and AR coefficients, undermining the substantive interpretation.

minor comments (1)

[Abstract] Abstract: 'e. g.' should be written as 'e.g.' for standard abbreviation style.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments on our manuscript. We address each major point below and indicate the revisions we will make to improve the presentation and transparency of our results.

read point-by-point responses

Referee: [Abstract] Abstract: the central V-shaped claim is presented without any reported parameter estimates, standard errors, significance tests for the growth factors (intercept, slope, quadratic), or autoregressive coefficients. Because the trajectory shape is defined by these parameters, their absence prevents verification that the decline-then-rise pattern is statistically supported rather than an artifact of the chosen functional form or imputation procedure.

Authors: We appreciate this observation. The full Results section contains the parameter estimates, standard errors, and significance tests for the intercept, linear and quadratic growth factors, and autoregressive coefficients, along with the multiple-imputation aggregation. However, we agree that the abstract should explicitly summarize these key statistics to allow readers to directly assess the statistical support for the V-shaped trajectory. We will revise the abstract to include the relevant growth-factor estimates, their standard errors, and p-values. revision: yes
Referee: [Methods] Methods/Results (implied by abstract description): the measurement model treats heart-rate and stress readings as sufficient indicators of the latent football-fever construct, yet no external validation (concurrent self-reports, behavioral observations, or physiological benchmarks) or sensitivity checks for indicator validity are described. Any systematic mismatch between the wearable signals and the intended psychological process would propagate directly into the estimated growth factors and AR coefficients, undermining the substantive interpretation.

Authors: We acknowledge that the manuscript does not report external validation of the latent construct. The choice of heart-rate and stress indicators draws on prior physiological literature linking these wearable signals to emotional arousal in sports contexts. Because concurrent self-reports or behavioral observations were not collected in this naturalistic field study, we cannot add new validation data. In the revision we will (i) expand the Methods section with additional citations justifying the indicators, (ii) report sensitivity analyses (e.g., alternative single-indicator and two-indicator specifications), and (iii) explicitly discuss the lack of external validation as a limitation. These textual additions will clarify the scope of the claims. revision: partial

Circularity Check

0 steps flagged

No significant circularity; results emerge from data-driven SEM fitting on observed indicators

full rationale

The paper specifies a time-dependent SEM with latent growth factors and autoregressive terms, fits it to smartwatch heart-rate and stress readings, and reports the resulting V-shaped trajectory plus heterogeneity as model outputs. No step reduces by construction to its inputs: the latent construct is identified from the observed physiological indicators via the measurement model, the growth parameters are estimated rather than presupposed, and no self-citation chain, uniqueness theorem, or renaming of known results is invoked to force the V-shape. The derivation remains self-contained against the external data.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Abstract-only review prevents exhaustive identification; the model rests on standard SEM assumptions plus data-specific choices for imputation and dimensionality adjustment.

free parameters (2)

latent growth parameters (intercept, slope, quadratic)
Estimated to produce the reported V-shaped average trajectory and between-fan heterogeneity.
autoregressive coefficients
Fitted to capture short-term carry-over effects in the time-dependent model.

axioms (2)

domain assumption Wearable heart rate and stress indicators are valid reflections of the latent football fever construct.
Invoked when constructing the latent variable from smartwatch data.
domain assumption Missing measurements can be imputed without biasing the estimated trajectory or heterogeneity.
Stated via aggregation across multiple imputations.

pith-pipeline@v0.9.0 · 5534 in / 1379 out tokens · 67862 ms · 2026-05-09T23:13:22.530274+00:00 · methodology

discussion (0)

Reference graph

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