Partes del AFM

We fitted several versions of both the basic and the extended model varying the set of fixed effects used, i.e with or without: home advantage parameter, attacking team ability

and defending team ability. Based on the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) the best version of the shots count model included the home advantage and the defending ability of the opposing team as fixed effects for the basic model and additionally the position of the player and the time played for the extended model. The attacking ability of a player’s team was dropped from each version of the model. We comment on this in the discussion section of this chapter.

Table 5.1 presents parameter estimates for the two specifications of the shots count model. Note that in both cases the estimates of team ability to prevent shots are with respect to the 2006/07 champions Manchester United. The lower the parameter value, the fewer shots the given team allows.

On the natural scale the global mean and home advantage parameter estimates for the basic model imply that an average player makes exp(−1.06+0.25)≈0.45 shots per game against Manchester United on the home field and exp(−1.06)≈0.35 away from home with a statistically significant difference between the two numbers. These numbers are in agreement with empirical observations with the average players (players with estimatedb_k within the 25th and 75th percentiles) producing roughly 0.4 shots per game against Manchester United during that season.

For the extended model, the estimates of the effect a player’s position has on the number of shots he makes are made with respect to the central attackers. For exam- ple, the parameter value of−0.27 for the central midfielders means that they shoot on average exp(−0.27)≈76% as often as the centre forwards. The fact that we estimate τ=0.86<1 is interesting as it implies that players tend to shoot less with each addi- tional minute they spend on the pitch, which may be due to fatigue.

Note also how much lower the estimate of the players’ random effects variance (σ_p2) is for the extended model. This stems from the fact that, in this model, a consider- able proportion of between player variability in the number of shots is explained by the players’ position and the number of minutes they spend on the pitch.

Table 5.2 presents results of the likelihood ratio test with H₀ : τ = πG =πCD = πLRD =πCM=πLRM=0. The null hypothesis is easily rejected even at very low signif- icance levels indicating that the extended model provides a better fit for the data.

Recall figure 5.2 in which we have observed over-dispersion of the empirical dis- tribution of the number of shots relative to the Poisson distribution with single mean parameter. The left panel of figure 5.3 contrasts it with the model implied distribution which assumes different values of the mean parameter for different player-game obser- vations due to various player, opponent and home advantage configurations. The model distributions were obtained from 1000 simulations of the shot counts for each player-

CHAPTER 5. SIGNAL AND NOISE IN GOALSCORING STATISTICS 52

Basic model Extended model

γ(n) −1.06(0.08)∗∗∗ −0.09(0.08) ν(n) 0.25(0.02)∗∗∗ 0.25(0.02)∗∗∗ Liverpool −0.31(0.09)∗∗∗ −0.25(0.09)∗∗ Chelsea −0.08(0.08) −0.08(0.08) Arsenal −0.03(0.08) 0.01(0.08) Wigan Athletic 0.01(0.08) 0.00(0.08) Manchester City 0.03(0.08) 0.04(0.08) Tottenham Hotspur 0.06(0.08) 0.09(0.08) Aston Villa 0.15(0.08) 0.16(0.08)∗ Bolton Wanderers 0.15(0.08)∗ 0.18(0.08)∗ Sheffield United 0.15(0.08) 0.17(0.08)∗ Portsmouth 0.17(0.08)∗ 0.18(0.08)∗ Middlesbrough 0.19(0.08)∗ 0.20(0.08)∗∗ Blackburn Rovers 0.20(0.08)∗ 0.22(0.08)∗∗ Newcastle United 0.23(0.08)∗∗ 0.25(0.08)∗∗ West Ham United 0.26(0.08)∗∗∗ 0.27(0.08)∗∗∗ Fulham 0.27(0.08)∗∗∗ 0.27(0.08)∗∗∗ Reading 0.29(0.08)∗∗∗ 0.32(0.08)∗∗∗ Everton 0.32(0.08)∗∗∗ 0.32(0.08)∗∗∗ Charlton Athletic 0.34(0.08)∗∗∗ 0.36(0.08)∗∗∗ Watford 0.34(0.08)∗∗∗ 0.32(0.08)∗∗∗ τ - 0.86(0.03)∗∗∗ πG - −4.23(0.28)∗∗∗ πCD - −1.06(0.08)∗∗∗ πLRD - −0.88(0.07)∗∗∗ πCM - −0.27(0.05)∗∗∗ πLRM - −0.21(0.05)∗∗∗ σ_p2 0.95 0.32 AIC 20841.03 19545.20 BIC 20999.09 19746.36 Num. obs. 9744 9744 Num. players 506 506 ***_p<0.001,**_p<0.01,*_p<0.05

Df logLik deviance Chisq Chi Df Pr(>Chisq) Basic model 22 -10398.51 20797.03

Extended model 28 -9744.60 19489.20 1307.83 6 <0.0001

Table 5.2: Likelihood ratio test for the basic and extended shot count models.

game observation given the fitted values of the mean parameter. The fitting sample mean and variance of the number of shots per player per game are 0.75 and 1.29 respec- tively indicating a considerable over-dispersion relative to the Poisson distribution with a single mean parameter.

Basic model Extended model

Figure 5.3: Histogram of shots per player per game (black bars) and model based frequencies.

In our basic GLMM the extra variance from the random effects and the covariates increases the dispersion of the model distribution relative to the one with a single param- eter with the simulated sample mean and variance of 0.75 and 1.22 respectively meaning that approximately 95% of the over-dispersion is accounted for by the model.

There does still appear to be some excess of zero counts left unexplained by the basic model. This may be due to the fact that some players, e.g. goalkeepers or those playing very few minutes, have very little chance to take any shot. These two factors are taken into account by the extended model (presented in the right panel of figure 5.3) with the marginal distribution providing an excellent fit to the data (sample mean 0.75 and variance 1.27). The Chi-square test for the goodness of fit does not find evidence to reject the hypothesis that the data is generated by the model (p-value =0.4497). In the absence of any residual over-dispersion (which would have suggested a more complicated model based on the negative binomial distribution would be appropriate), we conclude that it was reasonable to assume a mixed effects Poisson model.

CHAPTER 5. SIGNAL AND NOISE IN GOALSCORING STATISTICS 54