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Bayesian bivariate Conway–Maxwell–Poisson regression model for correlated count data in sports

  • Mauro Florez ORCID logo EMAIL logo , Michele Guindani and Marina Vannucci
Published/Copyright: August 12, 2024
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Journal of Quantitative Analysis in Sports
From the journal Journal of Quantitative Analysis in Sports Volume 21 Issue 1

Abstract

Count data play a crucial role in sports analytics, providing valuable insights into various aspects of the game. Models that accurately capture the characteristics of count data are essential for making reliable inferences. In this paper, we propose the use of the Conway–Maxwell–Poisson (CMP) model for analyzing count data in sports. The CMP model offers flexibility in modeling data with different levels of dispersion. Here we consider a bivariate CMP model that models the potential correlation between home and away scores by incorporating a random effect specification. We illustrate the advantages of the CMP model through simulations. We then analyze data from baseball and soccer games before, during, and after the COVID-19 pandemic. The performance of our proposed CMP model matches or outperforms standard Poisson and Negative Binomial models, providing a good fit and an accurate estimation of the observed effects in count data with any level of dispersion. The results highlight the robustness and flexibility of the CMP model in analyzing count data in sports, making it a suitable default choice for modeling a diverse range of count data types in sports, where the data dispersion may vary.

Keywords: Conway–Maxwell–Poisson distribution; intractable likelihood; random effects; Markov chain Monte Carlo; soccer and baseball; COVID-19
Corresponding author: Mauro Florez, Department of Statistics, Rice University, Houston, TX, USA, E-mail:

Acknowledgments

We would like to express our gratitude to Scott Powers for his invaluable assistance and suggestions. We also extend our thanks to the two anonymous referees and the editors, for their insightful comments and recommendations, which have greatly improved the quality of this paper.

  1. Research ethics: Not applicable.

  2. Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Competing interests: The authors state no conflict of interest.

  4. Research funding: None declared.

  5. Data availability: Data available at: https://www.football-data.co.uk/ and https://www.retrosheet.org/.

Appendix A: Proof of Equation (3)

Proof.

Let δ i = (δi1, δi2), where δ ij = exp(b ij ), then δ i = exp(b i ) ∼ LN2(μ, Σ) with μ = exp(0.5 ⋅ diag(D)), and Σ = diag(μ)(exp(D) − 11 T ) diag(μ). Also, assume that λ i j = exp x i j T β j , then y ij |λ ij , δ ij , ν ij ∼ CMP(λ ij δ ij , ν ij ). Finally, let λ ̃ i j = λ i j μ , λ ̃ i = ( λ ̃ i 1 , λ ̃ i 2 ) , and Λ i ̃ = diag ( λ ̃ i ) . By law of iterated expectations and using the asymptotic approximations of Shmueli et al. (2005), we have that E ( y i | β , γ , D ) ≈ λ ̃ i + 1 2 ν i − 1 2 , and var ( y i | β , D ) ≈ V i − 1 Λ ̃ i ( exp ( D ) − 11 T ) Λ ̃ i , V i = diag(νi1, νi2). Thus

cov ( y i 1 , y i 2 ) ≈ λ ̃ i 1 ( exp ( d 12 ) − 1 ) δ ̃ i 2 ≈ λ i 1 ⁡ exp ( 0.5 d 11 ) ( exp ( d 12 ) − 1 ) λ i 2 ⁡ exp ( 0.5 d 22 ) .

Clearly, (3) can be positive or negative depending on the sign of d12, i.e., the non-diagonal element of D.

Appendix B: Prior sensitivity analysis

We assess the performance of the bivariate Conway–Maxwell–Poisson (CMP) model under various prior specifications. Specifically, we focus on the Over-dispersed scenario of 1 Season discussed in Section 3 and examine different combinations of values for B0, G0, ν0, and R0. These combinations are summarized in Table 5. To monitor convergence, we calculate the multivariate potential scale reduction factor ( R ̂ ) Brooks and Gelman (1998) using 3 chains and also check the sampler’s efficiency by calculating the effective sample size (ESS).

Table 5:

Prior sensitivity: values considered in each scenario to assess the sensitivity of the model’s performance to the specification of prior hyperparameters.

Scenarios A B C D
B 0 0.1I I 3I 10I
G 0 0.1I I 3I 10I
ν 0 30 10 10 5
R 0 I I 0.1I 0.1I

Moving on to Table 6, we present the mean squared error (MSE) values for μ j across all scenarios. Consistent with expectations, scenario A demonstrates the best recovery of the observed values. Again, this outcome can be attributed to the small variance of the priors employed in this scenario. Conversely, scenarios C and D exhibit the poorest performance in terms of recovering the observed values.

Table 6:

Mean squared error (MSE) of μ – prior sensitivity analysis.

Scenarios A B C D
μ 1 0.71 1.01 1.19 1.2
μ 2 0.25 0.25 0.22 0.545
Table 7:

Prior sensitivity analysis: multivariate potential scale factor ( R ̂ ) .

Scenarios A B C D
R ̂ y 1 y 2 y 1 y 2 y 1 y 2 y 1 y 2
β 1.07 1.06 1.15 1.16 1.2 1.21 4.22 1.82
γ 1.05 1.05 1.19 1.17 1.22 1.24 4.74 5.04
b 1.02 1.02 1.03 1.03 1.22 1.24 4.22 3.91
Table 8:

Prior sensitivity analysis: effective sample size (ESS).

Scenarios A B C D
R ̂ y 1 y 2 y 1 y 2 y 1 y 2 y 1 y 2
β 350.44 408.54 123.25 131.13 98.48 97.13 7.26 10.71
γ 506.3 511.11 195.37 192.4 106.45 111.09 47.51 43.48
b 9,366 9,555.62 9,175.4 9,286.9 2,324.947 2,351.463 2,306.58 2,329.84

Scenarios A and B exhibit more stable estimations, with all R ̂ values below 1.2 (Table 7). Indeed, some of the trace plots for chains in scenarios C and D demonstrated some mild convergence issues (plots not shown). Additionally, the effective sample size (ESS) analysis (Table 8) indicates higher algorithm efficiency in scenario A, while scenarios C and D show lower efficiency. These findings are likely connected to the specified values of ν0 and R0, as well as the larger covariance matrices B0 and G0. Notably, the chains for random effects display larger R ̂ values and smaller ESS in scenarios C and D compared to scenarios A and B. This discrepancy may be attributed to the expected value of the inverse of the random effects’ covariance matrix D, which is determined by ν0R0. Consequently, larger values are permitted for the random effects in scenarios C and D, despite the expectation of low heterogeneity in this simulation scenario.

Appendix C: Shape parameters γ

In Figure 18, we observe the shape parameters estimation for the Premier League data analysis. In the left sub-figure we have the estimation for parameters associated with the Home Goals. Strong teams, such as Chelsea, Liverpool, and Manchester City, are located below 0, indicating that when other teams play against them, the dispersion parameter will be lower, implying a larger variance in their scored goals (over-dispersion). We observe a similar effect analyzing the Away Goals. Conversely, weaker teams such as Norwich or Brighton have their shape parameter above 0, indicating that when teams play against them, the goals will be more under-dispersed. On the other hand, in the x-axis, Manchester United and Manchester City are below 0, which means that the number of goals they score tends to be more over-dispersed compared to the other teams. We see the opposite trend with Liverpool, where the goals scored tend to be more under-dispersed. We can compare this with respect to over or under-dispersion because the intercept effect was 0. It is noticeable that the effects are different in the Home Goals and Away scores, supporting the modeling approach we assumed.

Figure 18: Analysis of premier league data: shape parameters.
Figure 18:

Analysis of premier league data: shape parameters.

Similarly, we plot the estimations for the MLB analysis in Figure 19. One of the noticeable cases is Miami Marlins (MIA), with a negative effect on the x-axis in the Home Points and a positive effect in the Away Points. This means that at Home, MIA tends to have scores more dispersed, while Away, their scores will be less dispersed in comparison to the others. This highlights the ability of our model to adapt to data of different types and to accommodate different phenomena and mechanics observed in the different teams and sports.

Figure 19: Analysis of MLB data: shape parameters.
Figure 19:

Analysis of MLB data: shape parameters.

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Received: 2023-06-14
Accepted: 2024-07-06
Published Online: 2024-08-12
Published in Print: 2025-03-26

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. Thoughts from the Editor
  4. Research Articles
  5. European football player valuation: integrating financial models and network theory
  6. On the efficiency of trading intangible fixed assets in Major League Baseball
  7. Expected goals under a Bayesian viewpoint: uncertainty quantification and online learning
  8. Bayesian bivariate Conway–Maxwell–Poisson regression model for correlated count data in sports
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https://doi.org/10.1515/jqas-2024-0072
Keywords for this article
Conway–Maxwell–Poisson distribution; intractable likelihood; random effects; Markov chain Monte Carlo; soccer and baseball; COVID-19

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. Thoughts from the Editor
  4. Research Articles
  5. European football player valuation: integrating financial models and network theory
  6. On the efficiency of trading intangible fixed assets in Major League Baseball
  7. Expected goals under a Bayesian viewpoint: uncertainty quantification and online learning
  8. Bayesian bivariate Conway–Maxwell–Poisson regression model for correlated count data in sports
  9. Success factors in national team football: an analysis of the UEFA EURO 2020
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