Home advantage and away goals rule: An analysis from Brazil Cup

3.1 Database

The main objective of this study is to analyse the advantage of deciding football play-offs (consisting in two matches) at home, especially to measure the influence, if any, of the away goals rule on this advantage. Our database consisted of 1662 knock-out rounds of Brazil Cup (Copa do Brasil) from 1994 to 2017 1 .

For the analysis we considered 1093 play-offs (2186 individual matches). From the original 1662, we excluded: 2

288 observations in which the confrontation was decided in only one game, since it was not the scope of this study.

3.2 Data collection and model

The database was collected and organized by the authors. Data are widely disclosed, and can be consulted on the CBF website and also in specialized websites such as http://www.bolanaarea.com/. Throughout the paper, we will call the team playing at home the second game as second leg home team, SLHT, and the team playing at home the first game as the first leg home team, FLHT. Differently of most of the consulted papers, the home advantage is not measured as a percentage of points earned at home on the total points earned, as knock-out matches are being analysed in these study: instead, we will be interested in analysing a binary response variable y which represents the outcome, after both games, of the second leg home team (1 if it qualifies and 0 otherwise); this is similar to the approach adopted in (Page and Page 2007). Besides, we believe that this methodology could also be used to measure the home advantage in one leg championships, not only in knock-out competitions, considering as home advantage only the cases that the home team wins the match.

One of our hypotheses is that the main component explaining the outcome of a play-off is the ability spread 3 between the teams participating in the match. Therefore, based on the 2016’s criterion of the Brazilian Football Confederation (CBF 2014) we created a proxy variable that measures the overall ability of each team in the sample, on a yearly basis.

In some years, at the first stages, the seeding system allocates the “better” teams as the SLHT. The fact the draw is not totally random means we should expect the SLHT qualifies with a higher percentage in the play-offs. However, as did (Page and Page 2007), we can accurately assess the real effect of the second leg home advantage controlling for the differences in teams’ ability.

The proxy variable ability was constructed as follows. Each team receives a score according to its final classification in the Brazilian Championship and the Brazil Cup. In addition, they receive a bonus score as compensation if they have been prevented from competing in the Brazilian Cup due to conflict of dates with their participation in the South American and Libertadores Cups. The score is calculated at the end of each year as a weighted average of the last five years’ points. It should be emphasised that the score of a team in any given year does not depend on his result on that year, only on the previous ones – this prevents endogeneity of the variable ability.

A simple points in the standings or a prior year finish in a championship has not been used because the teams that play the Brazilian Cup change from year to year. Also, since Brazil is a continental country, it is needed to travel long distances sometimes to play, for this reason it is common that a team prioritizes one championship over the other. Therefore a ranking that looks to just one championship may not reflect the reality of a team that decided to not prioritize that competition. Overall looking to a bigger period of time permit us to look not only to the “moment” of a team but also how well he is being administered, how much competitive he is over the previous years. Off course, the fact that the proxy gives more weight on the achievements of the year just before it is computed means that last results obtained are the most important element in the ability, however it is not the only one.

In the adjusted models, we used a standardisation of the difference between the teams’ scores, as a proxy for their ability spread, similar to the idea of (Page and Page 2007). It is noticed that this variable, besides representing the ability spread, also captures the variation between the phases of the competition, since the spread in abilities decreases as one advances from 32th to 16th to 8th and so on. Likewise, the variation between the years is mainly due to changes in the ability of each team. Another reason to use this variable is to analyze the hypotheses cited above.

The unstandardised ability spread, AB i U , for each two-leg match i = 1, ...  , n, where n = 1093, is given by AB i U = Ability of SLH T i - Ability of FHL T i

The standardised ability spread (hereafter simply ability spread) is then defined by A B i = AB i U s d where s _d is the sample standard deviation of the unstandardised ability spread. We chose not to subtract the mean in the standardised ability spread, AB _i , so that the values were centred at zero 4 . In this way, negative values represent play-offs in which the ability of the SLHT is inferior to that of the FLHT, while positive values correspond to play-offs where the SLHT has superior ability. An ability spread equal to one standard deviation represents a difference of 5694.89 points in the original index 5 .

In order to estimate the probability of classification of the SHLT in the Brazil Cup, relating to the ability spread and criteria used to decide the winner, we adjusted a logistic regression model. For this, the response variable is the classification of SLHT and the explanatory components considered in the model are the ability spread (AB) and the type of decision, which is a polytomic categorical variable. Therefore, three dummies were introduced, for goal difference, away goals rule and penalties shoot-out (respectively GD, AGR and PN), maintaining the classification by points as a reference category. In addition, we included the interactions between the ability spread and each dummy in the adjustment.

In preliminary studies, we also adjusted two additional models. The first, a logistic regression with the reduced database, containing only the confrontations of the final stages – round of 16, quarter-finals, semi-finals and final. However, the results were similar to those obtained by the regression that will be described next, and thus we chose to use all data to avoid discarding information. In the second model, the dependence between the same response and covariates as above was estimated through a non-parametric fit to the data, yielding once again results that were ‘close’ to those obtained through the logistic regression approach,indicating that the parametric adjustment is suitable; results are presented in the Appendix A.1, more details about this methodology in (Hayfield and Racine 2008) and (Barbosa and Brandão 2017). Since both preliminary fits yielded results similar to the logistic regression adjustment described below, we refrain from providing the detailed analysis.

Therefore, the final adjusted model is g ( x i ) = ln ( π ( x i ) 1 - π ( x i ) ) = β 0 + β 1 A B i + β 2 G D i + β 3 AG R i + β 4 P N i + β 5 A B i × G D i + β 6 A B i × AG R i + β 7 A B i × PN i ,

i = 1,  ...   n, where π(x) is the probability of the SLHT winning the two-leg knock-out match given the value of the covariate vector x, that is π (x) = Pr(y = 1|x). This model is useful to understand the phenomenon of the advantage of playing at home the second match of a play-off, as well as some features that may have an influence on this advantage.

3.3 The independence assumption: A discussion

Independence between observations is an important feature because it ensures good statistical properties for the maximum likelihood estimators of the model parameters. In the case of the Brazil Cup, the participating teams might be repeated over the years and, more specifically, some are repeated along different phases during the same year, as they advance in the competition. For example, the information that a certain team has already won a play-off as FLHT certainly alters the likelihood that the same team will win a new showdown at a later stage in which they play as FLHT: there are indications that this team has competence to qualify when he does not play the second game at home. Therefore, it is reasonable to assume that there is some correlation between certain characteristics of the sample units. However, in each phase, new play-offs are formed, defined by a previous draw, so that the home teams at the second leg are largely randomised. In addition, the ability of each team is maintained, but the ability spreads between the participants are different, so the characteristics of the new observation, at worst, have weak dependence on the characteristics of the previous observation, not enough to violate the hypotheses that legitimise the estimation procedures adopted herein. Moreover, in the present study the observational units are the two-leg matches, not the particular participants of each confrontation and the response variable considered here is the classification, or not, of the SLHT.

We argue that the characteristics that are correlated between the confrontations, as the participating teams, do not cause problems since they are not used in the adjusted models. Even when a team repeats itself in two different play-offs of the same year, the ability spread of that confrontation (which is likely the main factor influencing the classification probabilities) is bound to change, as said team will necessarily face a different opponent. Thus, even supposing there is prior information on whom the participants in the confrontation are, it is not known who the FLHT will be (because of randomised assignment), so there is no indication as to the likelihood of the response 6 .

Finally, we chose to use logistic regression, despite the knowledge that there are insurmountable limitations to this approach because of the possible existence of a correlation between the observational units, even though the responses are independent. A possible alternative is the use of mixed models, which would add a random-effects term for the year of the clashes, so that the correlation between clashes of the same team would be controlled by the year of the competition. However, in a preliminary study we found it has led to results similar to those obtained by the fixed-effects-only logistic regression that we adopt herein; additional results are presented in the Appendix A.2. Comparing the AIC of the two regressions the difference is 1.65, being 1255.32 for the mixed model and 1253.67 for the fixed-effects-only logistic regression. In the same way, the difference between the estimated coefficients is not relevant and both give the same conclusions about significances. Therefore, for convenience of the reader, we chose to adopt the more parsimonious model.

4.1 Descriptive

In the tournament, 44.19% of the play-offs have the direct qualification of the SLHT, that is without tie in the aggregated result. However, when looking only the observation of the final stages, this percentage decreases to 37.25%. On the other hand, the percentage of tied play-offs (which will be defined by some of the criteria) remains around 36% when considering the whole tournament or only the final stages, divided into about 19% of SLHT qualified and 17% of the FLHT.

The SLHT has qualified in 63,31% of the play-offs, significantly more than half (p-value <0.0001), with CI = 0.6331±0.0286. On the other hand, considering only the play-offs from the sixteenth rounds, this percentage decreases to 56.96%, and it is also significantly more than half the clashes (p-value = 0.0004), with CI = 0.5696±0.0385. Although both are larger than 0.5, it can be noted that the estimates of the intervals are quite different. This is because, in the final stages, the disparity in the ability of the teams is smaller, so that the visitor can win more often, even though the SLHT persists with the advantage.

Table 1 presents the percentages of aggregate results and classification. The Fig. 1 presents the percentages of the aggregated results (Chart (a)) and the percentages of classification (Chart (b)) over the years, in order to analyse if there are also differences between the years.

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Fig. 1

Aggregate results and qualification of SLHT by year.

In the two graphs, it is identified that there is a variation between the years, but mainly in the proportions of the results. This is due to the phases that were disputed. In the years in which the championship had more stages, there were more observations in the initial phases, in which there was no draw to decide the SLHT, often it was the one with better ability, consequently, happened more victories of the SLHT. In the classification chart, the behaviour seems to be more constant, especially after 2002, when it started the phase thirty-second round. This indicates that, although the points in the play-offs is considerably variable, the final outcome, the qualification, is relatively constant over the years.

Figure 2 presents the percentages of qualification in each round of Brazil Cup. It is possible to identify that in the two initial rounds, the SLHT wins approximately 72% of the play-offs, with CI = 0.7225±0.0414, significantly more than half (p-value<0.0001). In addition, this percentage in the early stages is also significantly higher than in the final stages (p-value <0.0001), whose percentage is 56.96%, which is also significantly higher than 0.5 (p-value = 0.0004), with CI = 0.5696±0.0385. This again evidences that the ability spreads between the teams of final stages play-offs is smaller than in the beginning. So, even if the SLHT team has an advantage, the FLHT manages to win more than in the early stages. However, if only post eighth rounds are considered, then the percentage is 51.99%, matching the SLHT and FLHT rating percentages (p-value = 0.4563), CI = 0.5199±0.0525.

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Fig. 2

Percentage of qualification by round.

In the cup finals, the percentage of victory of the SLHT is 31.58%, much lower than in the other phases. It is the only one that shows an advantage for the FLHT, but it is important to note that this figure is calculated with only 19 observations.

The estimated coefficients and their respective significance, obtained by the Wald’s test, after the estimation of the logistic model, are described in the Table 6. It is found that only the spread in ability is individually significant. However, the interactions of the ability spread with each type of classification are significant. Thus, there is evidence that the probability of victory of the SLHT, given each type of qualification, changes according to the ability spread between the teams of the play-off. Therefore, these results demonstrate that all the variables of the model are important to explain the probability of qualification of the SLHT in the Brazil Cup.

Although in the significance tests the classification types are not significant according to Table 6, we see in Table 7 that the model with these variables, even without the interactions, presents a gain with respect to the ROC curve considering the model that uses only the ability spread. In order to check this, two tests were done comparing the deviance’s of the models. For that, three models were adjusted: the first one is the complete one, previously described; in the second, all variables were maintained, but the interactions were withdrawn; the third model considers as co-variate only the ability spread.

The result of this analysis is that the model 1 is significantly more informative than model 2, which is significantly more informative than model 3. Therefore, removing the interactions reduces the likelihood of the model, and removing the indicators causes it to decreases further. Therefore, the variable qualification type, when considered globally, is in fact significant to explain the probability of classification of the SLHT. The comparison between models 1 and 3 informs that, together, the type of qualification and the interaction with the ability spread significantly improves the explanation of the model.

In addition, in order to ascertain that ability spread is the most important variable for classification, we have done tests comparing the values of the areas under the ROC curve, the DeLong test. The values are described in Table 7. It is observed that for the first three models, which include the ability spread, the predictive capacity is above 70%. However, by removing this variable from the model, the predictive capacity has a large reduction, decreasing to 58.41%.

The results of the tests, which allow us to identify the significance of the decrease in predictive capacity of the model, are presented in Table 8. It is verified that the interactions, the type of qualification and the ability spread are important variables, since the withdrawal causes a significant decrease in the area below the curve. However, ability spread is the variable whose inclusion that results in greater predictive capacity increase, corroborating the hypothesis that this is the most important variable for the model.

The estimated probabilities are represented in Fig. 3. Each point in the graph represents an observation: the cases where the SLHT qualified are arranged in row 1, whereas in the 0 line are those whose result was defeat of the SLHT. The horizontal position of each point represents the ability spread of the teams of this play-off, so the closer to 0 a value is, the greater the equivalence in the qualities of the teams.

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Fig. 3

Estimated probability of classification of the SLHT.

Analysing the probabilities as AB becomes more positive, that is, the more ability the SLHT has over the FLHT, it turns out that, for the SLHT, the criterion that leads to the highest probabilities of classification is the penalty shoot-out until the difference is – 0.43 standard deviation. Then the highest probability is for the goal difference, until the SLHT is 0.09 standard deviation the better. Above this value the most likely criterion is points.

For the FLHT, the highest probability happens when the decision is by points while this team is much better than the principal. When the difference is – 0.42 standard deviations, the best criterion for the FLHT is the away goals rule, this is maintained until the ability spread is 1.50 standard deviations, after that the penalty shoot-out becomes the criterion in which the qualification of the FLHT is more likely.

The away goals rule is never the best criterion for the SLHT because the probabilities are always lower. It should be noted that this does not mean that these teams can not use these criterion, only that there is always another one that provides a higher qualification probability. Although the teams do not choose the criterion used, at most it can set up a strategy that aims at a certain type of decision, but it does not depend only on the team.

To complement the analysis, the Fig. 4 presents a chart with four graphs, one for each curve exposed in the previous figure, estimated by the complete model. The curves were separated into four graphs only for easy visualisation. In each graph, the centreline represents the estimation of the pointed of probability, while the dotted lines represent the corresponding confidence intervals. According to Table 2, the goal difference was used in 196 play-offs, the away goals rule in 113 and the penalty shoot-out in 79, which influences the precision of the confidence intervals. The other 705 observations were decided by the points and therefore did not use any tie-breaker criterion.

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Fig. 4

Estimated probability of classification of the SLHT and confidence intervals.

In the matches defined by away goals rule or penalty shootings, the estimates are less accurate because of the small number of observations. It can be noticed that the confidence intervals are wide for all ability spreads. However, for values of the ability spread close to 0, the confidence intervals have the smallest amplitudes. This is because most of the observations that used these criterion had small ability spreads, that is, when teams have similar qualities. Despite the large amplitude, these estimates bring relevant information. Since the range contains the value equal to 0.5, for all ability spreads, then for the two criteria, teams have the same qualification probabilities, regardless of the ability spread.