## 5 Results
### Bootstrap Results  
While the overall ratings produced from the new rating model is strongly correlated with the ratings produced by earlier versions of plus-minus ratings, it remains to be seen whether additional insights can be gained by splitting the overall rating into a defensive and an offensive component. To examine the effects of the coefficients of the model not directly related to the player ratings, a bootstrap procedure was applied, and its results are discussed in the following. First, Figure 5 shows the effect of a player's age on their offensive rating, whereas Figure 6 shows the effect of age on the defensive rating. There are few observations of players at the extreme ends of the age spectrum, so the regularization makes the age effects closer to zero for those. In general, the confidence intervals are narrower in the middle of the age range, which makes sense since there are more players in the data set with such ages, so the estimation is more reliable. The effect of age is much more pronounced for the defensive ratings, compared to the offensive ratings. In particular, younger players are relatively worse when it comes to defensive ratings. This could indicate that it is more important to use peak age players in defense, and less important to avoid young players up front. Table 3 provides bootstrap estimates for the home field advantage parameters. The first observation is that overall, the home field advantage is in line with previous studies, for example [PERSON] and [PERSON] [7] who found an overall advantage of 0.388 goals per 90 min for the home team. However, the offensive-defensive rating model uses the home field advantage to shift the baseline for goals scored by both teams--that is, the positive numbers for offensive advantages indicate that home teams score more goals than the baseline, whereas the negative numbers for defensive advantages indicate that away teams score more goals than the baseline. The last column in Table 3 indicates the difference of the median in favor of the home team. This implies that certain leagues behave differently in terms of the total number of goals: While the overall home field advantage is similar for Italy and Portugal, with 0.33 goals in favor of the home team per 90 min, in Portugal, both teams score 0.27 goals more per 90 min than in Italy, all else being equal. Although the confidence intervals are wide, this is in line with the reputation that Italian football is defensively oriented. However, factors such as the distribution of playing strength between teams in a league may also contribute towards how the home field advantage is estimated by the model. Figure 4: Ablation study for the new rating system, with evaluation metrics as in Figure 3.  
The overall home field advantage also seems to differ between competitions. It is highest for the UEFA Champions League (UCL) and Europa League (UEL), with 0.44 goals per 90 min. These competitions are special, since many of the rounds are played as two matches, one at home and one away, leading to the elimination of the overall losing team. The matches are also associated to longer travel distances than the domestic leagues, which may be another factor in explaining the larger home field advantage. There could potentially be an additional change in the home field advantage when either team has players being sent off. However, the bootstrap estimation suggests that this change is very minor: when the home team has one player sent off, it scores 0.38 (with a 95% confidence interval [0.32, 0.44]) goals less and concedes 1.07 ([0.97, 1.18]) goals more per 90 min; on the other hand, when the away  
Figure 5: Effect of age on offensive ratings, showing median effect (black line) and 95% confidence interval (gray area) from bootstrap tests. Figure 6: Effect of age on defensive ratings, showing median effect (black line) and 95% confidence interval (gray area) from bootstrap tests. team has one player sent off, it scores 0.30 ([0.25, 0.36]) goals less and concedes 1.14 ([1.03, 1.22]) goals more per 90 min.