Bivariate Dixon and Coles Model in Football

• The paper introduces the bivariate Dixon and Coles model that refines joint probabilities for low-scoring football outcomes to improve prediction accuracy.
• The model extends the independent Poisson framework by integrating tactical covariates and advanced dependence structures for more realistic score modeling.
• Empirical findings demonstrate that incorporating these extensions aids in simulating match results and informs strategic decisions in competition structuring.

The bivariate Dixon and Coles model is a statistical framework for modeling association football (soccer) match outcomes that explicitly accounts for the dependence between the number of goals scored by each team. Originating as an extension of the independent bivariate Poisson model, it is distinguished by its targeted modification of joint probabilities for rare, low-scoring results—most notably, draws. Recent research has demonstrated versatile extensions and applications: the integration of tactical covariates drawn from network cluster analysis of team playing styles (Diquigiovanni et al., 2018), generalizations to flexible dependence structures (Petretta et al., 2021), embedding in the Sarmanov family for broader marginal and correlation modeling (Michels et al., 2023), and scenario-based quantitative guidance for strategic club management in new tournament formats (Winkelmann et al., 27 Aug 2025). The following sections elucidate foundational principles, model evolution, methodological innovations, and contemporary practical uses.

1. Foundational Principles and Original Formulation

The Dixon and Coles model posits that the goals scored by home $(X_k)$ and away $(Y_k)$ teams in match $k$ are conditionally modeled as Poisson random variables:

$$X_k \sim \text{Poisson}(\lambda_k), \quad Y_k \sim \text{Poisson}(\mu_k)$$

where scoring intensities are parameterized as:

$$\log(\lambda_k) = \gamma + \alpha_{i(k)} + \beta_{j(k)}, \qquad \log(\mu_k) = \alpha_{j(k)} + \beta_{i(k)}$$

Here, $\gamma$ represents home advantage, $\alpha_{i(k)}$ and $\alpha_{j(k)}$ are team-specific attack strengths, while $\beta_{i(k)}$ and $\beta_{j(k)}$ are defensive strengths. The classical Poisson independence assumption often fails for low-scoring outcomes. To address this, Dixon and Coles introduce a dependence parameter $\rho$ and a correction factor $\tau_\rho(x, y)$, modifying the joint probability mass function for specific $(x, y)$ pairs:

$$\tau_\rho(x,y) = \begin{cases} 1 - \lambda\mu\rho & (x,y)=(0,0) \ 1 + \lambda\rho & (x,y)=(0,1) \ 1 + \mu\rho & (x,y)=(1,0) \ 1 - \rho & (x,y)=(1,1) \ 1 & \text{otherwise} \end{cases}$$

Thus, only the outcomes $(0,0)$, $(0,1)$, $(1,0)$, and $(1,1)$ depart from independence, capturing empirical features such as higher-than-expected frequencies of draws and close wins.

2. Model Extensions: Tactical Covariate Integration

Recent advancements have enabled the incorporation of team tactical information, especially via network-based cluster analyses. In (Diquigiovanni et al., 2018), playing styles are represented as undirected weighted networks, clustered into tactical groups. A specific style, termed “on-the-wings,” is hypothesized to influence goal scoring rates. The bivariate Dixon and Coles model is thus extended with a covariate $c_i(k)$ signaling the use of this tactic:

$$c_i(k) = \begin{cases} 1 & \text{if team } i \text{ in match } k \text{ uses the on-the-wings style} \ 0 & \text{otherwise} \end{cases}$$

Resulting intensity functions:

$$\log(\lambda_k) = \gamma + \alpha_{i(k)} + \beta_{j(k)} + \delta c_{i(k)}$$

$$\log(\mu_k) = \alpha_{j(k)} + \beta_{i(k)} + \delta c_{j(k)}$$

where $\delta$ quantifies the effect of the playing style. Estimation proceeds via composite (profile) likelihood, with uncertainty in $\delta$ assessed by construction of the Godambe information matrix, yielding time-specific confidence intervals. Empirical results indicate uniformly positive and statistically significant $\delta$ values, confirming that the on-the-wings tactic increases expected goals. Temporal analysis reveals a decreasing trend, suggesting tactical adaptation or diminishing returns over time.

3. Dependence Structure Generalizations: Mar-Co Model and Sarmanov Family

While the original Dixon and Coles framework adjusts only four low-score outcomes, broader dependence structures have been introduced. The Mar-Co model (Petretta et al., 2021) specifies conditional distributions for one team's score given the other's, dispensing with the Poisson marginal requirement. The away team's conditional mean is:

$\psi_Y(\mu_k, h) = \exp\left[ \theta_1 + \theta_2 \log(\mu_k) + \theta_3 \logit(F_{\mu_k}(h)) \right]$

where $F_{\mu_k}(h)$ is the Poisson CDF and $\theta_3$ controls dependence. A symmetric model applies to the home team, and the joint pmf is constructed as an equal mixture of both conditionals. This approach reshapes outcome probabilities more broadly than the four-point shifts produced by $\rho$, enabling improved modeling for bet types sensitive to distributional tails (e.g., Under/Over bets).

Separately, (Michels et al., 2023) embeds the Dixon and Coles model within the Sarmanov family of bivariate distributions:

$$P(X_1 = x_1, X_2 = x_2) = P_1(x_1) P_2(x_2) [1 + \omega q_1(x_1) q_2(x_2)]$$

with constraint $\sum_{x_i} q_i(x_i) P_i(x_i) = 0$, for $i=1,2$. The original Dixon and Coles model emerges as a special case with suitable $q$-functions for Poisson marginals. Extensions allow for alternative $q$-functions, general score-dependent probability shifts, and use of negative binomial or mixed marginals. These constructions facilitate modeling of stronger negative correlations and overdispersion, as observed in women's football.

4. Estimation Procedures and Inference

Estimation in the bivariate Dixon and Coles model commonly proceeds via composite or profile likelihood, owing to the computational efficiency and tractable inference even for modest sample sizes. The log-likelihood function integrates nuisance parameters (team abilities, home effect, dependence), with covariates included when present. For models augmented with tactical effects (Diquigiovanni et al., 2018), uncertainty quantification for parameters of interest (e.g., $\delta$) leverages the sensitivity and variability matrices to form the Godambe information, with confidence intervals derived using Satterthwaite’s approximation. Such approaches enable sequential updating as new match data arrives, supporting rolling statistical inference over time.

When the model is embedded in larger frameworks (e.g., Sarmanov family or Mar-Co), maximum likelihood estimation adapts to the broader parameter space—including coefficients governing dependence structures. Model comparison metrics (such as Ranked Probability Scores for betting contexts) are used to evaluate predictive performance, with enhancements in tail regions or correlation modeling highlighted in empirical studies (Petretta et al., 2021, Michels et al., 2023).

5. Applications in Competition Structuring and Strategic Decision-Making

The bivariate Dixon and Coles methodology has been instrumental in analyzing outcomes in novel tournament formats, such as UEFA’s incomplete round-robin league phase (Winkelmann et al., 27 Aug 2025). Here, the model—calibrated for decreased draw frequencies with $\rho>0$—supports simulation-based estimation of qualification thresholds. Team strengths are proxied via standardized Elo ratings, with expected goals modeled as:

$$\lambda = \exp(\beta_0 + \beta_1 \, \text{EloDiff} + \beta_2), \qquad \mu = \exp(\beta_0 - \beta_1 \, \text{EloDiff})$$

Simulations (e.g., 10,000 runs of the league phase) yield probabilistic forecasts for direct qualification and play-off probabilities, which influence strategic decisions regarding resource allocation, transfer policy, and in-game tactics.

6. Empirical Findings and Theoretical Implications

A consistent theme in empirical findings is the necessity of flexible dependence modeling. In men’s football, the Dixon and Coles specification effectively accounts for the surplus draws and tight scorelines; however, in women’s football, features such as underrepresentation of 0–0 draws, abundant 2–0/3–0 wins, strong negative goal correlations, and overdispersion necessitate Sarmanov-based extensions (Michels et al., 2023). Tactical covariates extracted from network clustering have been shown to exert statistically significant effects on goals scored (Diquigiovanni et al., 2018), while dynamic adjustment of dependence and marginal structures (Mar-Co, Sarmanov) facilitates improved predictive performance, especially in betting contexts (Petretta et al., 2021).

The broader implication is that the bivariate Dixon and Coles model and its extensions represent a unifying statistical formalism for football score modeling that is adaptable to diverse league characteristics, tactical variables, and usage scenarios. Its integration of tactical information, marginal flexibility, and dependence sophistication supports both theoretical exploration (e.g., competition balance, playing style impact) and practical forecasting (e.g., qualification probability estimation, betting accuracy).

Table: Key Extensions and Their Contexts

Extension	Context/Goal	Notable Features
Tactical Covariates	Playing style effect on scoring	Cluster-informed $\delta$
Mar-Co Dependence	Enhanced outcome dependence	Mixture conditional modeling
Sarmanov Embedding	Flexible marginals/correlation	General $q$-function, NB/Poi
Elo-integrated Forecasting	UEFA league phase qualification	Simulation, low draw modeling

Each extension adapts the foundational model either through covariate inclusion, marginal generalization, or expanded dependence, thereby enriching the modeling toolkit for researchers and practitioners.

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The bivariate Dixon and Coles model and its advanced derivatives constitute an essential methodology for football score analysis, offering precise outcome modeling, tactical effect quantification, and actionable competition guidance. Continued research in this area is expected to further illuminate the interplay between strategy, model specification, and predictive accuracy across varied footballing contexts.