In soccer knockout ties which are played in a two-legged format the team having the return match at home is usually seen as advantaged. For checking this common belief, we analyzed matches of the UEFA Champions League knockout phase since 1995. It is shown that the observed differences in frequencies of winning between teams first playing away and those which are first playing at home can be completely explained by their performances on the group stage and - more importantly - by the teams' general strength

Eugster, Manuel J. A.

Gertheiss, Jan

Kaiser, Sebastian

English

Open Access LMU

Manuel J.A. Eugster, Jan Gertheiss & Sebastian KaiserHaving the Second Leg At Home – Advantage inthe UEFA Champions League Knockout Phase?Technical Report Number 080, 2010Department of StatisticsUniversity of Munichhttp://www.stat.uni-muenchen.deHaving the Second Leg At Home – Advantage inthe UEFA Champions League Knockout Phase?Manuel J. A. Eugster∗, Jan Gertheiss∗† & Sebastian Kaiser∗Department of Statistics, Ludwig-Maximilians-Universität Munich, Germany.Firstname.Lastname@stat.uni-muenchen.deAbstractIn soccer knockout ties which are played in a two-legged format the team havingthe return match at home is usually seen as advantaged. For checking this com-mon belief, we analyzed matches of the UEFA Champions League knockout phasesince 1995. It is shown that the observed differences in frequencies of winningbetween teams first playing away and those which are first playing at home canbe completely explained by their performances on the group stage and – moreimportantly – by the teams’ general strength.Keywords: Home Field Advantage, Knockout Matches, Logit Model, UEFAChampions League1 IntroductionIn soccer knockout ties with first and second leg the team having the return matchat home is usually seen as advantaged. When matches are determined, players,coaches and managers usually hope that they will have the return match at home.And for example in the UEFA Champions League round of 16, group winners areallowed to play the first match away, because it is believed that – on average –the chance of reaching the next round is higher for the team which has the deci-sive second leg at home. However, in the present 2009/2010 Champions Leagueseason, three of four teams that reached the semi-finals had the first quarter-finalmatch at home. So in the present paper we analyze UEFA Champions Leagueknockout matches and search for statistical evidence which supports the common∗All authors contributed equally to this work.†Corresponding author.1belief that the team which is first playing away is advantaged. In the litera-ture studies about home field advantage in all sports are collected for examplein Strauß (2002, in German) or Nevill and Holder (1999). In Pollard and Gomez(2009) all European soccer leagues are examined about home field advantageand in current soccer studies (Dilger and Geyer, 2009; Franck and Nüesch, 2010)there is always a correction for home field advantage found in the model. Up toour knowledge there is no study dealing with the advantage of home field advan-tage in the second leg of a two leg contest. We analyze data from the 1994/1995season (when the first time Champions League quarter-finals were played) until2009/2010, and consider matches from the knockout phase following the groupstage. All in all, 152 ties (each consisting of first and second leg) are taken intoaccount. And indeed, the naive analysis, where it is just counted how many timesa team which is allowed to play the second leg at home reaches the next round(85 out of 152), results in an estimated 56% probability of winning in favor ofthe team having the return match at home. Moreover, the result is – or seems tobe – statistically significant on a 10% α-level (using a binomial test).However, the naive analysis is severely biased, because it ignores that in theround of 16 (or quarter-finals before 2003/2004) due to UEFA regulations groupwinners automatically play runners-up and have the return match at home. Sincea group winner tends to be a stronger team than a second-place finisher it canbe assumed that it is more likely that a group winner reaches the next round. Soeven if there was no effect of having the second leg at home/away, just countingthe number of ties where the team succeeds which is playing the return match athome, would produce results in favor of those teams. With other words, in theround of 16 the effect of having the return match at home cannot be separatedfrom the teams’ performances on the group stage. Therefore we will provide arefined data analysis in the following section.2 Data AnalysisTo provide some more insight into the considered data we will first present someslightly modified frequency tables. Then the data are analyzed using a so-calledlogistic regression model which allows to quantify the effect of having the secondleg at home and to separate it from the teams’ group stage performances.The data are exemplified by two ties of the quarter-finals in 2009/2010 (Ta-ble 1): In the match FC Bayern München (T1) versus Manchester United FC(T2) Bayern first played at home (therefore labeled as T1). On the group stage,Bayern had been a runner-up (G1 = 2), whereas Manchester United had beena group winner (G2 = 1). First and second leg taken together, Bayern was thewinner of this tie. When Arsenal FC (T1) was playing against FC Barcelona (T2),both teams had been group winners (G1 = G2 = 1), and Barcelona reached thesemi-finals.2T1 T2 G1 G2 WinnerFC Bayern München Manchester United FC 2 1 T1Arsenal FC FC Barcelona 1 1 T2Table 1: Example data of two ties (each consisting of first and second leg) of the ChampionsLeague quarter-finals in 2009/2010.2.1 Refined Frequency TablesJust considering quarter-finals (after 2002/2003) and semi-finals is the simplestway to eliminate the bias that is induced by the rule that in the round of 16 (orquarter-finals before 2003/2004) group winners automatically play the first legaway. If this is done the estimated probability that a team succeeds which hasthe return match at home falls at 50% (and of course no significance is givenanymore). The observed chance of winning is 30/30.However, a more sensible way of summarizing the data (which does not disre-gard most of the data) is a frequency table where the outcomes of knockout tiesare conditioned on the performance of the competing teams on the group stage.The results are shown in Table 2. In the right panel the percentage of ties is givenwhere the winning team was the home team in the second leg. It is seen thatjust in the cases where the home team in the second leg was a group winner andthe away team was a runner-up, this value is clearly above 50%. This finding,however, is not surprising, since a group winner can be expected to be a strongteam. Moreover, if both teams are group winners or runners-up, by contrast,there is apparently no advantage for the home team of the second leg. The onlyrow in Table 2 that may support the hypothesis that the second leg home team isadvantaged is the second row. The chance for a runner-up to beat a group winneris 50/50 if the runner-up is allowed to play at home in the return match. However,a runner-up may only play at home in a return match, if he had already beatena group winner before, because (due to UEFA regulations) in the knockout tiedirectly following the group stage a runner-up has the first leg at home and has toplay a group winner. So a runner-up who is playing at home in the return matchcan also be expected to be a strong team. In Figure 1 the number of ties in theround of 16, the quarter- and semi-finals are shown where the winner was playingat home in the first leg or second leg, respectively. Quarter-finals are separatedinto seasons until/after 2002/2003, because until 2002/2003 there were quarter-finals following a second group stage with group winners automatically playingrunners-up and having the second leg at home. It is seen that in the round of16 there are much more ties where the winning team was the home team of thesecond leg. However, all these teams had also been group winners before. Inquarter- and semi-finals there are (almost) no differences in bar heights anymore.The finding that also in quarter-finals of seasons before 2003/2004 group winners3Away Team Home Team Away Team Home Team Percentage(2nd Leg) (2nd Leg) Winner WinnerGroup Winner Group Winner 11 11 50.00 %Runner-Up 6 6 50.00 %Runner-Up Group Winner 44 65 59.63 %Runner-Up 6 3 33.33 %Table 2: Frequency of observed combinations, and percentage of ties where the winning teamwas the home team in the second leg.Round of last 16 Quarter−finals until 2003 Quarter−finals after 2003 Semi−finalsHome Team 2nd LegAway Team 2nd LegRoundNumber of Wins010203040Figure 1: Frequency of ties where the winning team was the home/away team in the secondleg, separated into round of 16, quarter- and semi-finals.and runners-up performed (almost) equally, may be explained by the fact thatall the teams found in these quarter-finals had to survive a challenging secondgroup stage before. Hence all these teams can be expected to be quite strong. Toget more insight into the data we provide a model based analysis in the followingsubsections.2.2 Logistic ModelingIn order to quantify the effect of having the first/second leg at home and toseparate it from the effect of group stage performances, we fit a logistic regressionmodel with effect coding and interactions. For an introduction to this type ofmodel, see for example Fahrmeir and Tutz (2001). In short, the fitted modelcompletely explains the observed data; that means, fitted probabilities using this4model equal the observed relative frequencies (i.e. percentages in Table 2). Themodel, however, allows to fit the probability that the team which first plays away(T2) succeeds (first and second leg taken together), while conditioning on theteams’ performances on the group stage. Using an adequate model specificationthe effect of group stage performance can be separated from the effect of havingthe return match at home. The model is defined as follows:P (T2 wins | G1 = i, G2 = j) = exp{η(i, j)}1 + exp{η(i, j)} , (1)withη(i, j) = γ + λ1(i) + λ2(j) + λ1,2(i, j), (2)where γ specifies the mean effect of having the first/second leg at home; γ > 0means that the probability of winning is higher for the team which is first playingaway, and γ < 0means that the probability of success is higher if a team first playsat home. Since the probability of success is conditioned on the performance of thetwo considered teams on the group stage, additional parameters are introduced:λ1(i) specifies the main effect of team T1 being a group winner (i = 1) or second-place finisher (i = 2). In an analogue way λ2(j) is the main effect of the successof team T2 on the group stage (j = 1, 2). Finally, interactions of the group stageindicators G1 (team T1) and G2 (team T2) are modeled via λ1,2(i, j). To makethe model identifiable, symmetric restrictions (well known from ANOVA models)need to be fulfilled:∑iλ1(i) =∑jλ2(j) =∑iλ1,2(i, j) =∑jλ1,2(i, j) = 0 (3)If model (1) is fitted, the estimate of γ from (2) isγˆ = −0.08,which is negative and almost zero. That means, (in case of the UEFA ChampionsLeague) there is no statistical evidence for the common belief that having thesecond leg at home increases the chance of winning. If just a main effects modelis fitted, i.e. λ1,2(i, j) from (2) are omitted, the estimated γ is γˆ = −0.003,which is quite close to zero as well. Moreover, none of the fitted coefficients issignificantly nonzero, which means that it cannot be stated that teams which arefirst playing at home/away are disadvantaged/advantaged.2.3 Including Team CoefficientsIn the previous subsections team strengths have only been taken into account viathe performances of the competing teams on the group stage. To measure teamstrengths on a continuous scale, however, there is also a so-called UEFA team5coefficient available, which is commonly used to rank the teams. Each season thisquantity is drawn up on the basis of a combination of 20% of the value of therespective national association’s coefficient for the period of the previous 5 seasonsand the teams’ individual performances in the UEFA club competitions duringthe same period. Each team retains the cumulative number of points obtainedduring this period (UEFA, 2009). These coefficients are now used to correct fora possibly unbalanced draw where the better teams are more often chosen asthe team with home field advantage in the second leg. More precisely, we addedanother variable to explicitly include strength differences of the competing teamsinto the model. The new variable C is defined as the difference of the UEFA teamcoefficients between teams T2 and T1. In addition, the variable is standardizedby dividing by the coefficient of the best team, i.e.C =UEFA Coefficient (T2)− UEFA Coefficient (T1)max(UEFA Coefficient). (4)So the difference is always between −1 and 1. Table 3 shows the UEFA teamcoefficients and the corresponding C values for our exemplary ties. Barcelona wasthe leader of the 2009 ranking, so 121.853 is the maximum coefficient for thatyear. It can be clearly seen (positive C value) that Barcelona and Manchesterperformed better than Arsenal and Bayern over the previous 5 seasons.UEFA Coefficient (T1) UEFA Coefficient (T2) C98.339 (FC Bayern München) 111.899 (Manchester United FC) 0.111106.899 (Arsenal FC) 121.853 (FC Barcelona) 0.123Table 3: Extension of Table 1 with 2009 UEFA Coefficients and (scaled) differences thereof (C,right).The probability that team T2 succeeds is now also conditioned on the newcovariate C (in addition to G1 and G2). The formula in (2) is changed toη(i, j, C) = γ + λ1(i) + λ2(j) + λ1,2(i, j) + ξC, (5)where the influence of covariate C is measured by regression parameter ξ. Esti-mates of parameters are shown in Table 4.For the interpretation of the estimated model parameters, it should be keptin mind that, due to restrictions from (3),λ1(1) = −λ1(2), λ2(1) = −λ2(2) (6)andλ1,2(1, 1) = λ1,2(2, 2) = −λ1,2(1, 2) = −λ1,2(2, 1). (7)The results show that no variable except the difference in strength between teamT2 and T1 has a statistically significant influence on the probability of success of6Parameter Estimate Std. Error z value Pr(>|z|)γ −0.0838 0.2610 −0.32 0.7483λ1(1) 0.0074 0.2611 0.37 0.7090λ2(1) 0.2708 0.2610 1.04 0.2995λ1,2(1, 1) −0.1796 0.2631 −0.68 0.4947ξ 1.5981 0.6254 2.56 0.0106Table 4: Results of the Logit Model with correction on team strengthteam T2. As expected, the probability of success is higher for teams with higherUEFA coefficients, since ξˆ = 1.6 is positive. With γˆ = −0.08 the mean effect ofhaving the return match at home is close to zero again (and negative). In Figure2 (left) the estimated probability that the home team of the second leg (T2) suc-ceeds is plotted as a function of the (standardized) difference in the UEFA teamcoefficients of the competing teams for all possible combinations of group win-ners and runners-up (in the form of T1 :T2). It is seen that only in the case thatrunners-up play against group winners and runners-up have the second leg away(blue line) the probability that the second leg home team succeeds is clearly above50% at the point of zero-difference in the UEFA coefficients. In the right panelof Figure 2 a heatmap is shown for the UEFA coefficients of two former groupwinners playing against each other. Colors correspond to the estimated proba-bility that the home team of the second leg wins. It is seen that this probabilityis about 50% if both teams have similar UEFA coefficients. With the presentedresults it is clearly shown that there is no relevant advantage/disadvantage ofplaying at home/away in the second leg, and that the outcome rather dependson the strength differences between the competing teams.3 Summary and DiscussionIn soccer knockout ties with first and second leg the team which is playing athome in the return match is generally seen as advantaged. For checking thiswe analyzed matches of the UEFA Champions League. The UEFA ChampionsLeague is usually seen as the most important competition for football club teamsin the world (or at least in Europe), and hence a natural reference. Our anal-ysis showed that there is no statistical evidence for the common belief that thechance of winning is higher if a team is playing away in the first leg and havingthe second leg at home. For players, however, there may also be other reasonsfor hoping that the return match will be at home. In the case of winning, forexample, they can celebrate together with all the fans at their home stadium.7−1.0 −0.5 0.0 0.5 1.00.00.20.40.60.81.0Standardized Difference in UEFA Coefficient (C)Probability of Winning (Home Team 2nd Leg) Group Winner : Group WinnerRunner−Up : Runner−UpGroup Winner : Runner−UpRunner−Up : Group Winner0.20.30.40.50.60.70.80.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Standardized UEFA Coefficient (Home Team 2nd Leg)Standardized UEFA Coefficient (Away Team 2nd Leg)Figure 2: Probability curves (T2 wins) for the standardized difference in the UEFA coefficientsfor all combinations T1 :T2 of group winners and runners-up (left), and heatmap for the compet-ing teams’ UEFA coefficients against each other if both teams are former group winners (right;with colors corresponding to the estimated probability that the home team of the second legwins).As a consequence of our analysis, we propose to change UEFA regulationsin the sense that in the round of 16 group winners do not automatically play athome in the second leg, but may choose depending on their individual preferences.Though on average there is apparently no effect of having the return match athome, for some teams effects may be differing. If for a certain team playing athome in the second leg is advantageous or not, may for example be examined byfurther statistical analysis.Computational DetailsAll computations and graphics have been done using the statistical software R2.10.1 (R Development Core Team, 2009). R itself and all functions used arefreely available under the terms of the General Public License from the Compre-hensive R Archive Network at http://CRAN.R-project.org.ReferencesDilger, A. and H. Geyer (2009). Are three points for a win really better thantwo? Journal of Sport Economics 10 (3), 305–318.Fahrmeir, L. and G. Tutz (2001). Multivariate Statistical Modelling Based onGeneralized Linear Models (2nd ed.). New York: Springer.8Franck, E. and S. Nüesch (2010). The effect of talent disparity on team produc-tivity in soccer. Journal of Economic Psychology 31 (2), 218 – 229.Nevill, A. M. and R. L. Holder (1999). Home advantage in sport: An overviewof studies on the advantage of playing at home. Sports Medicine 28, 221–236.Pollard, R. and M. A. Gomez (2009). Home advantage in football in south-westEurope: Long-term trends, regional variation, and team differences. EuropeanJournal of Sport Science 9 (6), 341 – 352.R Development Core Team (2009). R: a language and environment for statisticalcomputing. Vienna, Austria: R Foundation for Statistical Computing. ISBN3-900051-07-0.Strauß, B. (2002). Über den Heimvorteil. Spectrum derSportwissenschaften 14 (2), 70–90.UEFA (2009). Regulations of the UEFA Champions League 2009/2010.9

Are three points for a win really better than two?

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Home advantage in football in south-west Europe: Long-term trends, regional variation, and team diﬀerences.

Home advantage in sport: An overview of studies on the advantage of playing at home.

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Having the Second Leg At Home - Advantage in the UEFA Champions League Knockout Phase?

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Having the Second Leg At Home - Advantage in the UEFA Champions League Knockout Phase?

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