Statistical method

The approach we propose is designed to take the above factors into consideration. We measure the performance of the teams in the season by means of a paired-comparison model as in Maher (1982). The performance of the teams is obtained only using the outcomes of matches that have already been played in the same season. On the basis of this measured performance, we determine the final standings using the model-implied expected number of points in the remaining games. We should emphasize that our analysis is based on a paired-comparison model that has been widely used to model and predict football matches by including regression variables as well as time-variation in the strength of the teams. In particular, the paired-comparison model for outcomes of football matches of Maher (1982) is adopted in many studies, including Dixon and Coles (1997), Koopman and Lit (2015), and Koopman and Lit (2019) We denote the outcome of a football match between the home team $i$ and the away team $j$ as a pair of counts $(X_i,Y_j)$ for $i,j\in \{1,\dots,n\}$, $i\neq j$, where $X_i$ is the number of goals scored by the home team $i$, $Y_j$ is the number of goals scored by the away team $j$, and $n$ indicates the number of teams in the competition. We describe the match outcome by means of a bivariate Poisson paired-comparison model $$(X_i,Y_j)\sim \mathcal{BP}(\lambda_{ij}^x,\lambda_{ij}^y, \gamma),$$ where $\mathcal{BP}(\lambda_{ij}^x,\lambda_{ij}^y, \gamma)$ denotes a bivariate Poisson distribution with intensity $\lambda_{ij}^x$ for the home team count, intensity $\lambda_{ij}^y$ for the away team count, and coefficient $\gamma$ for the dependence between the two counts. The probability mass function (pmf) of the bivariate Poisson $\mathcal{BP}(\lambda_{ij}^x,\lambda_{ij}^y, \gamma)$ is given by \begin{equation}\label{pmf} \mathbb{P}(X_i=x, Y_j=y)=\frac{\lambda_{ij}^x \, \lambda_{ij}^y}{x! \, y! \, e^{\lambda_{ij}^x+\lambda_{ij}^y+\gamma}}\sum_{k=0}^{\min(x,y)}\binom{x}{k}\binom{y}{k}k!\left(\frac{\gamma}{\lambda_{ij}^x\lambda_{ij}^y}\right)^k, \end{equation} for $x,y \in \{0,1,\dots,\infty\}$. The intensities $\lambda_{ij}^x$ and $\lambda_{ij}^y$ determine the difference in expected goals between the home and away teams. The intensities are specified as $$\lambda_{ij}^x=\exp(\delta+\alpha_i+\beta_j), \quad \text{and} \quad \lambda_{ij}^y=\exp(\alpha_j+\beta_i),$$ where $\alpha_k$ represents the attacking ability of team $k$, $k=i,j$, $\beta_k$ represents the defending ability of team $k$, $k=i,j$, and $\delta$ is the home ground advantage. The specification for the intensities originates from Maher (1982). It accounts for the different strength level of the teams $(\alpha_k,\beta_k)$, $k=1,\dots,n$, as well as the home ground advantage $\delta$ to determine the probability distribution of the match outcome.
In most applications of the bivariate Poisson model for sports data, the objective is to specify $\alpha_k$ and $\beta_k$ to best predict the outcomes of future matches. For instance, the model can extended with other covariates that may explain the strength levels of the team. However, in our current study the objective is to determine a final ranking that reflects the performance of the teams in the matches that have been played earlier in the season. We achieve this by estimating the parameters of the model $(\alpha_1,\dots,\alpha_n,\beta_1,\dots,\beta_n,\delta,\gamma)$ using the method of maximum likelihood (ML) and only based on the data of the current season. In this way, the estimated intensities only reflect the performance of the teams in the current season. Once the parameters have been estimated, we can obtain the expected number of points of each team in the remaining games and construct the final table by using the model-implied expected number of points at the end of the season. In particular, first we calculate the winning probability of the home team $p_h$, the winning probability of the away team $p_a$, and the probability that the match ends with a draw $p_d$ as follows $$p_h=\sum_{y=0}^\infty\sum_{x=y+1}^\infty \mathbb{P}(X_i=x, Y_j=y),$$ $$p_a=\sum_{x=0}^\infty\sum_{y=x+1}^\infty \mathbb{P}(X_i=x, Y_j=y),$$ and $$p_d=\sum_{z=0}^\infty \mathbb{P}(X_i=z, Y_j=z),$$ where the expression of the pmf is given above. Based on these probabilities, we calculate the expected number of points of the home and away teams. We consider the system of assigning 3 points to the winning team, 0 to the losing team, and 1 point to each of the teams if the game ends with a draw. This system is the standard in most football competitions. The expected number of points of the home team $ep_{h}$ and the one of the away team $ep_{a}$ are $$ep_{h} = 3 p_h + p_d, \quad \text{and} \quad ep_{a} = 3 p_a + p_d.$$ Finally, the final table is obtained by summing up the expected number of points of each team in the remaining games and adding these expected points to the points of the incomplete table.