Down to Ten: Estimating the Effect of a Red Card in Soccer
Author(s): G. Ridder, J. S. Cramer and P. Hopstaken
Source: Journal of the American Statistical Association, Vol. 89, No. 427 (Sep., 1994), pp. 11241127
Published by: American Statistical Association
Stable URL: http://www.jstor.org/stable/2290942 .
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Down to Ten: Estimatingthe Effectof a
Red Card in Soccer
G. RIDDER,J. S. CRAMER,and P. HOPSTAKEN*
We investigate
the effectof the expulsionof a playeron the outcomeof a soccermatchby means of a probabilitymodel forthe
score.We proposeestimators
oftheexpulsioneffect
thatare independentoftherelativestrength
oftheteams.We use theestimates
theexpulsioneffect
on theoutcomeofa match.
to illustrate
KEY WORDS:
Conditionallikelihood;Poissonprocess;Soccer; Unobservedheterogeneity.
1. INTRODUCTION
and 90 minutes.Recordedtime is measuredfromthe beginningof the matchand fromits resumptionafterthe inProfessionalsoccer(knownoutsidethe United Statesas terval,however.As a result,theremay be some minutes
football)is popularall overtheworld;in Europe and South whenthereis no playat all, whereasthe45thand 90thminAmericait is the dominantspectatorsport.Because soccer utes may lastlongerthana fullminute;but thisis a minor
is a low scoringgame,the ruleshave been oftenrevisedso distortion.
as to raisethenumberofgoalsscoredbyeithersideand thus
Let
increasetheplay'sappeal.Since 1990,playerscan be expelled
ri = minutein whicha playeris expelledfromteam 2,
forthe restof a matchforillegaldefensiveactions,such as
goal
by
Nij = totalnumberof goals scoredin matchi by teamj,
repeatedflagrantfoulsand preventingan adverse
showing
him
Kij = numberofgoals scoredbefore-ri,
illegalmeans.The refereeexpelstheplayerby
Mij = numberofgoals scoredafterTi,
a redcard.
of teamj in matchi at the
theeffect
ofsuchan expulsion Xij(t) = scoringrateor intensity
In thisarticlewe investigate
tth
minute
of
play,
holds
widely
on the outcome of a match.Popular opinion
= multiplicative
effect
on Xij(t)ofexpulsionofplayer
oftheredcard,butas far
different
viewson theeffectiveness
from
team
2,
and
to
not
been
submitted
empirical
as we knowthequestionhas
of teamj in matchi as compared
,yij= relativestrength
research.We proposea model forthe effectof the redcard
with
the
overall
in
of
the
the
teams
strengths
thatallowsforinitialdifferences
averagescoringrate,X(t).
in
the
match.
for
the
during
and
variation
scoringintensity
We make thefollowingthreeassumptions:
Poisson
we proposea time-inhomogeneous
Morespecifically,
1. The two teams score accordingto two independent
forthescoreofeitherside.
effect
modelwitha match-specific
Poisson
processes.As a consequence,the numberof goals
a
card
by
conof
the
red
effect
We estimatethedifferential
scored
team 1 is stochastically
ofthenumber
by
independent
inis
estimator
that
ditionalmaximumlikelihood(CML)
of
2.
scored
team
the
time
intervalsbegoals
by
Moreover,
effects.
This
estimator
was
dependentof the match-specific
tween
are
The
subsequent
goals
stochastically
independent.
and
GrilHall,
in
by
Hausman,
introduced econometrics
are
not
the
thus
intensities
constant
scoring
during
match;
of
Andersen
(1973).
liches(1984), buildingon ideas
In Section2 we specifythemodel,in Section3 we discuss thePoisson processesare nonhomogeneous.
2. The ratioofthescoringintensities
ofthetwofullteams
and in Section4 we givetheresults.We consider
estimation,
=
is
a
each
constant
for
game;
that
is,
in
Xij(t)
yijX(t)formatches
some implicationsoftheestimates Section5.
of 11against11players,
withX(t) theaveragescoringintensity
2. A MODEL FOR THESCORE
at thetthminuteofplayoffullsidesof 11 against11.
IN A SOCCER MATCH
3. Afterthe red card,fort > Ti, team 2 has 10 players,
and thescoringintensities
are OjyijX(t),j = 1, 2.
i denotes
First,we introducesome notation.The subscript
In Assumption1 we describethe score in a matchas a
a match,and j = 1, 2 denotesthetwo sides in thatmatch;
a teamin a matchis thusidentified
bytwosubscriptsij. We randomphenomenonthatis onlypartlypredictable.It deof the
restrict
attentionto matcheswitha redcard,and we always pends on the playingtime,on the relativestrength
take it thatthe red card is givenagainstthe second side,j teams,and on the effectof the red card. As we show,the
= 2. Time is measuredin minutesfrom0 to 90, whichis scoringintensity
increaseswiththetimeplayed.Ifwe do not
oftheredcardwillbe overstated,
the officialdurationof a match.In soccerthe clock is not allowforthis,thentheeffect
but the refereecan allow because we confoundit withthe timeeffect.Of coursethe
stoppedwhenplay is interrupted,
affected
oftheteams.
and secondhalfs,after45 scoreis strongly
bytherelativestrength
forlosttimeat theend ofthefirst
The incidenceof red cards may be relatedto the relative
so thata comparisonof red card games to unin* G. Ridderis Professor,
Free University, strength,
Departmentof Econometrics,
In adgamesgivesa biased estimateoftheeffect.
Amsterdam,The Netherlands.J. S. Crameris Professor,Departmentof terrupted
Economics,and P. Hopstakenis SeniorResearchFellow,Foundationfor
EconomicResearch,University
of Amsterdam,
The Netherlands.The authorsthankGusta Renes forhelpfulcomments,
Tony Lancasterforspotting
an embarrassing
errorin a previousversion,and theeditorand tworeferees
forcommentsthathave improvedthearticleconsiderably.
? 1994 AmericanStatisticalAssociation
Journalof the AmericanStatisticalAssociation
September 1994,Vol. 89, No. 427, Statisticsin Sports
1124
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All use subject to JSTOR Terms and Conditions
Ridder, Cramer, and Hopstaken: The Effectof a Red Card in Soccer
1125
Table 1. Goals Scored in the 1991dition,the timingof red cards may also be relatedto the
1992 Season by 15-MinuteIntervals
relativestrength
oftheteams,and againthisbiasestheeffect.
The thirdfactoris the effectof the red card,whichby AsTimeinterval
Numberofgoals
(min)
sumption3 is measuredby 01 and 62.
It is notouraim to predicttheoutcomeofsoccermatches,
0-15
128
whichrequiresan estimateof yij.Our estimateoftheeffect
16-30
140
31-45
147
oftheredcard is independentof yij,whichis ofgreathelp,
46-60
169
a good estimateof yj is difficult
as experience
becausefinding
61-75
170
shows.
76-90
198
The Poisson assumptionand its implicationsformAssumption1. It is notdifficult
to relaxthePoissonassumption
at thecostofa morecomplexstatistical
model,butour limscoredbythesame teambeforeand aftertheredcard.More
itednumberof observationswillnot supportthis.
precisely,we considerthe fractionof the goals scoredafter
the red card,whichwe denoteby yij.It is intuitively
3. STATISTICALANALYSIS
clear
thatthisfraction
is independentofthetime-constant
match3.1 Estimation of the Average Scoring Intensity
specificeffect.
Under Assumptions1 to 3 (withP denotingthe Poisson
In Table 1 goals scored in 340 fullmatchesin the two
professional
soccerdivisionsin theNetherlandsin the 1991- distribution),
1992 season are classifiedby 15-minuteintervalsof play.
This showsthatthe rateof scoringincreasesmonotonically Kij-P(y j fX(t)dt) and MijP(QP fjX(j
(t)dt).
I
over the match,as has also been observedin Englandby
Morris(1981).
(3)
If we assume thatthe averagescoringintensity
increases
linearlyduringthematch,thentheexpectednumberofgoals In the sequel we denote
scoredby a teamj in matchi in intervals is
= A(t) dt and Bi =
A(t) dt.
(4)
s = 1, . ,6, (1)
E(Nij,) = yij(15a + 112.50(2s -1))
Ai
so thatthe averagenumberof goals scoredby one team in
intervals is
E(Ns) = 15a + 112.53(2s- 1)
s = 1,..., 6,
(2)
definesa scale for yij;
wherewe take y = 1. This implicitly
forexample,if yiy= 2, thenteamj has a scoringintensity
in matchi thatis two timestheaverage.
The averagenumberofgoals per minutein timeinterval
s equalstheentryin Table 1dividedby680,twicethenumber
ofcontests.Estimatesofa and : are theneasilyobtainedby
ordinaryleast squares (OLS) regression.With X(t) as the
fora 90-minutegame,we find(R2 = .95;
scoringintensity
standarderrorsin parentheses)
f
f
The conditionaldistribution
ofMi, givenNij, is
MijINij - B(Nij, gij(0)),
(5)
whereB denotesthebinomialdistribution
and
g,1(O)= A +BB
(6)
The conditionaldistribution
is degenerateifNij = 0, andy1j
is definedonly ifNij 2 1. In the CML procedurewe omit
observationswithNij = 0. The estimatorof the red card
effect
is notbiasedbythisrestriction,
as we shallsee presently.
In theconditionaldistribution
(5) and in theconditional
effectsyiYcancel. Up to an
likelihood,the match-specific
a = 1.050(.024) and ,=.00776(.00072).
additive constantthat does not depend on Oj, the logNote thatthe reportedstandarderrorsare consistentin the likelihoodis
Inclusionofa quadraticterm
presenceofheteroscedasticity.
nj
did notimprovethefit.In thesequel we ignorethesampling
+ (Ni - Mij)log(l -giJ(Oj)),
logLJ= z M0jlog(g,1(61))
thecomputation
varianceoftheseestimates.This simplifies
i=l
as theyare
ofvariancesand is an acceptableapproximation,
(7)
small.The estimatesimplythatthescoringintensity
increases
duringa 90-minutegame from1.05 in the firstminuteto withnl, n2denotingthe numberof observationson teams
1.75 in thefinalminute.
thatdo notand do receivea redcard.Because we condition
on thetotalscores,Nij,we can treatthemas nonstochastic
3.2 A Conditional Maximum Likelihood Estimator
constants.Hence omittingobservationswitha giventotal
Because the incidenceof red cardsis probablyrelatedto score-in particular,observationswithNij = 0-does not
the relativestrengthyij,a comparisonof red card matches affecttheCML estimator.
The likelihoodequationis
withothermatchesmay give a biased estimateof the red
card effect.For thatreason,we proposean estimatorthat
does not depend on the -ybJ's
or on theirdistribution.
This
= E N,1yij.
(8)
E N,jg,j(OaCMLJ)
estimatoris based on a comparisonofthe numberof goals
i=l
i=l
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Journal of the American Statistical Association, September 1994
1126
This is a momentequation,equatinga weightedaverageof
the yij's to a weightedaverage of theirexpectations.The
weightsare thetotalscoresNij, whichby conditioningcan
be treatedas knownconstants.
In derivingthepropertiesoftheCML estimator,
we note
in thelogit
thatthebinomialparametergij(0j)can be written
is globallyconcavein log(0),
form.Hence thelog-likelihood
so thatthe CML estimatorforOjis uniquelydefined.The
asymptoticvarianceof theCML estimatorcan be obtained
in theusual way.
3.3
Table 3. Probabilitiesofthe Outcomeofa Match
witha 15-MinuteExclusionStartingat r
Minuteofstart
ofpenaltyr
Pr(teamof 11 wins)
.42
.42
.43
.43
.43
.44
0
15
30
45
60
75
Pr(draw)
Pr(teamof 10 wins)
.24
.24
.24
.24
.24
.24
.34
.34
.33
.33
.33
.32
OLS Estimation
Withan additionalassumption,we can estimatetheeffect
and 0CML2 = .95 (.20).
OCML1 = 1.88 (.29)
From (3),
ofthe redcard by linearregression.
Accordingto the CML estimates,the scoringintensityincreasesby 88% forthe team with 11 players;thiseffectis
Kij = _jAi+ (7ij - -j)Ai + (Kij - E(KiiLIij))
The scoringintensityforthe team
statistically
significant.
= 7jAi + vij
with 10 players(team 2) hardlychanges;the effectis not
from1.
significantly
different
and
The OLS estimatorgivesratherdifferent
results(withthe
in
=
+
+
standard
errors
consistent
the
of
heteroscedaspresence
Mij _j6jBi (7ij
1j)0jBi (Mij E(MijLI0i))
ticity):
= VjOjBi + V2ij
(9)
In (9) we allowtheaveragerelativestrength
in redcardgames
to differ
fromthe overallaverage1. 1 and 2 indicatethe
of the teamsbeforea playerof team 2 is
averagestrengths
and
are independent,
expelled.The disturbances
vIijand V21J
an additionalassumptionis requiredforconsistent
estimates,
viz. cov(yij, A ) = cov(yij, B ) = 0. A sufficient
condition
forthisis thatTi and -iyarestochastically
Under
independent.
thisassumption,we can estimateQbytheratiooftheregresin (9).
sion coefficients
4.
ESTIMATIONRESULTS
We applyCML estimationto data on 140 redcardgames
in the seasons 1989-1990, 1990-1991, and 1991-1992 in
bothdivisionsof the Dutch professionalfootballleague. In
13 of thesematches,two or moreredcardsweregiven.Beofbeingone playerup or down,
cause we estimatetheeffect
the partafterthe second expulsionis omitted.In onlytwo
matcheswere a red card and a penaltykick givenjointly.
Because forthe CML estimatorwe mustomitobservations
numberof
wherea team has not scoredat all, the effective
observationsis 112 forteams with 11 playersand 93 for
results(stanteamswith10players.We obtainthefollowing
dard errorsin parentheses):
Table2. Probabilities
oftheOutcomeoftheMatch
oftheRed Card
byMinute
of
Minute
redcard r
Pr(teamof11 wins)
0
15
30
45
60
75
90
.65
.62
.58
.54
.49
.44
.375
6OLS1
=
1.43 (.03)
6OLS2
=
1.14 (.03).
The estimatedincreasein thescoringintensity
forteam 1 is
much smallerthan forthe CML estimator(but highlysignificant).More surprisingly,
theOLS estimatorshowsa staincreasein thescoringintensity
forteam
tistically
significant
2. Hence usingbetween-game
information
givesratherdifferentestimatesthat moreoverare hard to interpret.
The
oftheOLS estimatorshowthatteams
first-stage
regressions
thatreceivethered card have the same scoringintensity
as
theaverage( 2 = 1.03(.09)), buttheopposingteamis much
stronger(^Y1= 1.33 (.09)). Hence the red card usuallyis
givento thealreadyweakerteam.
By stratifying
our sample,we can investigate
whetherthe
estimates
arerobustagainstchangesin thespecification.
First,
we testwhethertheredcard effect
dependson the venue of
play.Thiscaptures,amongotherthings,
thehomeadvantage,
and theestimateshouldbe invariantto thisdistinction.
The
LR statisticis .53 forthe team with 11 playersand .66 for
theteamwith10 players;hencewe can notrejectinvariance.
The estimatesare 61,home = 2.00 (.36), 61,away = 1.56 (.46),
and 02,home = .73 (.28), 62,away = 1.07 (.27). The estimates
are also invariantto stratification
on the total score in a
match.In thesequel we use theCML estimatesto illustrate
of theredcard on the outcomeof a match.
theeffect
OF THEESTIMATES
5. IMPLICATIONS
We can use the resultsto illustratethe effectof the red
card on a soccermatch.In Table 2 we givetheprobabilities
Pr(draw) Pr(teamof10 wins) ofthethree
possibleoutcomesofthematchbetweenequally
teams
as a functionof 1-. The last row of the table
.17
strong
.18
.18
.20
of a drawbetweentwo teamsof
showsthatthe probability
.20
.22
average
strength
is
.25.
This
is an indicationof the role of
.25
.21
chance
in
the
outcome
of
a
soccer
match.The roleofchance
.28
.23
.24
.32
was also stressedby Osmond(1993) . A redcardearlyin the
.375
.25
matchincreasesteam 1l'sprobability
ofvictorysubstantially.
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Ridder, Cramer, and Hopstaken: The Effectof a Red Card in Soccer
1127
Table 4. Expected Numberof Goals in Match
by Minuteof Red Card
Minuteofred card T
Expected numberofgoals
0
15
30
45
60
75
90
3.95
3.80
3.63
3.45
3.25
3.03
2.80
APPENDIX:DATAUSED IN ANALYSIS
The symbols are introduced in Section 2.
Team 2's probability
ofvictory
decreasesevenmore,whereas
thechangein theprobability
of a drawis relatively
small.
Witha redcard,a playeris expelledfortheremainderof
thematch.In indoorsoccerand ice hockey,a playercan be
excludedfora certainperiod.In Table 3 we showtheeffect
of a 15-minutetime penaltyforequally strongteams. Althoughthe effect
dependson thetimeat whichthepenalty
is imposed,thisdependenceis ratherweak.
As notedin Section1,a motivationforthemorefrequent
use oftheredcardis to increasethenumberofgoals scored
in a match.Table 4 showsthatit has thedesiredeffect.
We also considerthe dilemmaof a defenderwho facesa
playerwho threatensto break throughthe defense.If the
opposingplayerhas a clearwayto thegoal,trippingup the
playerresultsin a red card forthe defender.If the player
goes past thelastdefender,he willscorewitha highprobability.In our calculationwe assumethattheobjectiveofthe
oflosingthematch.
defenderis to minimizetheprobability
Thereis a unique momentin a contestat whichtheoptimal
actionofthedefenderchanges.Afterthatmoment,it is optimal to tripup the opposingplayer.These times,which
depend on the probabilitythatthe attackerwill score and
on therelativestrength
of thedefender'steam (withtheattacker'steam of averagestrength,
-y= 1), are reportedin
incentiveto resort
Table 5. The weakerside has a stronger
to illegaldefense.This is consistentwithour observation
thatthered card is usuallygivento theweakerside. It may
also induce a correlationbetweenX and y,and such a correlationbiases theOLS estimates.
K] K2 Ml
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82
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43*
45*
60*
63*
64*
69*
71*
78*
* Matches withtwo or more red cards. Second red card in 77, 85, 78, 73, 60, 89, 68, 67, 65,
88, 76, 77, 82.
Table 5. Time(Minuteof Game) AfterWhicha DefenderShould Stop
ofScore and Relative
a Breaking-Away
Playerby Probability
Strengthofthe Defender'sTeam
ofscore
Probability
Relativestrength
of teams, y
.3
.6
1
.5
1
2
70
71
72
42
48
52
0
16
30
[ReceivedJanuary1993. RevisedMarch 1994.]
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Andersen,E. B. (1973), ConditionalInferenceand ModelsforMeasuring,
Copenhagen:MentalHygiejniskForlag.
Hausman,J.A., Hall, B. H., and Grilliches,
Z. (1984), "Econometric
Models
forCount Data Withan Applicationto thePatents-R&D Relationship,"
Econometrica,52, 909-938.
Morris,D. (1981), The Soccer Tribe,London: JonathanCabe.
Osmond,C. (1993), "Random Premiership?,"
RSS News,November,5.
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