# Usefulness of Dismissing and Changing the Coach in Professional Soccer

**ABSTRACT** Whether a coach dismissal during the mid-season has an impact on the subsequent team performance has long been a subject of controversial scientific discussion. Here we find a clear-cut answer to this question by using a recently developed statistical framework for the team fitness and by analyzing the first two moments of the effect of a coach dismissal. We can show with an unprecedented small statistical error for the German soccer league that dismissing the coach within the season has basically no effect on the subsequent performance of a team. Changing the coach between two seasons has no effect either. Furthermore, an upper bound for the actual influence of the coach on the team fitness can be estimated. Beyond the immediate relevance of this result, this study may lead the way to analogous studies for exploring the effect of managerial changes, e.g., in economic terms.

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**ABSTRACT:**Statistics of soccer tournament scores based on the double round robin system of several countries are studied. Exploring the dynamics of team scoring during tournament seasons from recent years we find evidences of superdiffusion. A mean-field analysis results in a drift velocity equal to that of real data but in a different diffusion coefficient. Along with the analysis of real data we present the results of simulations of soccer tournaments obtained by an agent-based model which successfully describes the final scoring distribution [da Silva et al., Comput. Phys. Commun. 184, 661 (2013)]. Such model yields random walks of scores over time with the same anomalous diffusion as observed in real data.Physical Review E 08/2013; 88(2):022136. · 2.31 Impact Factor - SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**Scoring goals in a soccer match can be interpreted as a stochastic process. In the most simple description of a soccer match one assumes that scoring goals can be described by a constant goal rate for each team, implying simple Poissonian and Markovian behavior. Here a general framework for the identification of deviations from this behavior is presented. For this endeavor it is essential to formulate an a priori estimate of the expected number of goals per team in a specific match. The analysis scheme is applied to approximately 40 seasons of the German Bundesliga. It is possible to characterize the impact of the previous course of the match on the present match behavior. This allows one to identify interesting generic features about soccer matches and thus to learn about the hidden complexities behind scoring goals.07/2012; - SourceAvailable from: ncbi.nlm.nih.gov
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**ABSTRACT:**Scoring goals in a soccer match can be interpreted as a stochastic process. In the most simple description of a soccer match one assumes that scoring goals follows from independent rate processes of both teams. This would imply simple Poissonian and Markovian behavior. Deviations from this behavior would imply that the previous course of the match has an impact on the present match behavior. Here a general framework for the identification of deviations from this behavior is presented. For this endeavor it is essential to formulate an a priori estimate of the expected number of goals per team in a specific match. This can be done based on our previous work on the estimation of team strengths. Furthermore, the well-known general increase of the number of the goals in the course of a soccer match has to be removed by appropriate normalization. In general, three different types of deviations from a simple rate process can exist. First, the goal rate may depend on the exact time of the previous goals. Second, it may be influenced by the time passed since the previous goal and, third, it may reflect the present score. We show that the Poissonian scenario is fulfilled quite well for the German Bundesliga. However, a detailed analysis reveals significant deviations for the second and third aspect. Dramatic effects are observed if the away team leads by one or two goals in the final part of the match. This analysis allows one to identify generic features about soccer matches and to learn about the hidden complexities behind scoring goals. Among others the reason for the fact that the number of draws is larger than statistically expected can be identified.PLoS ONE 11/2012; 7(11):e47678. · 3.53 Impact Factor

Page 1

Usefulness of Dismissing and Changing the Coach in

Professional Soccer

Andreas Heuer1*, Christian Mu ¨ller2, Oliver Rubner1, Norbert Hagemann3, Bernd Strauss4

1Institute of Physical Chemistry, University of Muenster, Muenster, Germany, 2Institute of Organic Chemistry, University of Muenster, Muenster, Germany, 3Institute of

Sports Sciences, University of Kassel, Kassel, Germany, 4Institute of Sports Sciences, University of Muenster, Muenster, Germany

Abstract

Whether a coach dismissal during the mid-season has an impact on the subsequent team performance has long been a

subject of controversial scientific discussion. Here we find a clear-cut answer to this question by using a recently developed

statistical framework for the team fitness and by analyzing the first two moments of the effect of a coach dismissal. We can

show with an unprecedented small statistical error for the German soccer league that dismissing the coach within the

season has basically no effect on the subsequent performance of a team. Changing the coach between two seasons has no

effect either. Furthermore, an upper bound for the actual influence of the coach on the team fitness can be estimated.

Beyond the immediate relevance of this result, this study may lead the way to analogous studies for exploring the effect of

managerial changes, e.g., in economic terms.

Citation: Heuer A, Mu ¨ller C, Rubner O, Hagemann N, Strauss B (2011) Usefulness of Dismissing and Changing the Coach in Professional Soccer. PLoS ONE 6(3):

e17664. doi:10.1371/journal.pone.0017664

Editor: Bard Ermentrout, University of Pittsburgh, United States of America

Received August 6, 2010; Accepted February 10, 2011; Published March 22, 2011

Copyright: ? 2011 Heuer et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: These authors have no support or funding to report.

Competing Interests: The authors have declared that no competing interests exist.

* E-mail: andheuer@uni-muenster.de

Introduction

Fred Everiss, responsible for the soccer team of West Bromwich

Albion (UK) coached his team over 46 years (1902–1948) without

any interruption. This is probably the all time world record for

coaches in professional soccer. In Germany, for instance, Volker

Finke is the record holder. He coached the professional soccer team

of SC Freiburgfor almost16 years (1991–2007) without interruptions

(German record), although due to the relegation into the Second

German soccer league his team had to leave the Premier German

Soccer league (the so called ‘‘Erste Bundesliga’’, established 1963)

four times.However,such loyalty isveryunusual in professional team

sports. Frequently, the usual response to a continuing series of recent

lost matches is to dismiss and replace the coach mid-season. For

example in the German Bundesliga the club ‘‘Eintracht Frankfurt’’ is

leading in dismissing a coach during mid season (20 times in 47 years

of the German Premier soccer league). Fired coaches are often hired

by competitors who also dismissed the coach. For example, Gyula

Lorant as well as Joerg Berger are the most often dismissed head

coaches in the German Bundesliga (six times each).

The reason to fire a coach mid-season [1] is often due to

disappointed expectations in comparison to the team wage bill [2]

and to the widespread assumption of clubs, fans, and the media

that changing the coach has a major positive effect on a

subsequent team’s performance (one-way causality hypothesis)

[3]. This is opposed to the Ritual Scapegoating Hypothesis, i.e.

dismissing the coach will have no effect on a team’s performance

(the nil hypothesis) [3]. The latter follows the assumption that a

coach has only a small impact on the performance of the team

which the coach is responsible for.

Already in 1964 [3] preferred the hypothesis of ritual

scapegoating. However, a closer inspection of their empirical

findings in professional Baseball could not clearly support any of

their presented hypotheses. Not surprisingly, whether mid-season

coach dismissals have effects on the subsequent team performance

has long been a subject of controversial discussions, mainly in the

Sport Sciences [4] and Economic Sciences as well [1,5].

Many of these studies focused on coach dismissals in

professional soccer in different national leagues. These studies

disagree with respect to the final result as well as the used research

design. Partly these results have to be questioned due to design

problems like a sub-optimal choice of the performance criterion

[1,6–15], the use of a very small data basis (e.g., Dutch soccer

[8,9], Spanish soccer [10]), missed control teams [1,8,10], or a

biased choice of control teams (English soccer [11,12], German

soccer [13–15], Dutch soccer [16]).

Methods

Team Fitness in Soccer from a Statistical Perspective

N Heuer and Rubner [7] have recently shown theoretically that

the mathematically optimal measure of a soccer team’s fitness

is the goal difference (DG). Therefore, to optimize the

predictability it is essential to use DG rather than the number

of points or the rank as a characteristic of the team fitness (as

almost always used by the studies mentioned above, a rare

exception is [8]). Stated differently, the number of points

contains a larger random contribution than the goal difference.

Qualitatively, the superiority of goal differences as compared

to points expresses the fact that a 5:0 and a 1:0 win is counted

identically in terms of points although in general this difference

indicates the presence of different fitness values for both teams.

Quantitatively, the identification of random contributions can

be achieved via a straightforward correlation analysis of

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subsequent sets of matches (e.g., by comparing the first and the

second half of the season).

Most importantly, a team’s fitness remains just about constant

throughout a season. Any variations during the season are due to

temporal fluctuations (like weather conditions, red cards) whereas

systematic variations mainly occur between different seasons [6,7].

This observation already gives a hint to formulate our main

hypothesis in line with [3] that changing the coach during the

season is useless and would have no effects in the subsequent team

performance. Using optimized statistical approaches to avoid the

design problems mentioned above these questions will be

answered in this work. Additionally, to classify these mid-season

dismissal effects on subsequent performances we will also analyze

the effects of changing the coach between seasons.

Analysis of Coach Dismissals (CDs)

We analyze the Premier German soccer league (as we already

mentioned, the so-called German ‘‘Erste Bundesliga’’) which

started in the season 1963/64. We consider all mid-seasonal coach

dismissals (CDs) for all 46 seasons until 2008/09. Almost in each

season every team has to play 34 games (except the three seasons

1963/64, 64/65 as well as 1991/1992). The entire data set covers

14,018 games. Since during the first decades of the Bundesliga

several matches have been adjourned due to weather conditions

etc. it is essential to take into account the correct order of matches

for each team. The key procedure of our approach can be

summarized as follows

1. To be able to quantify possible fitness variations due to the CD

we require that before and after the CD the team plays at least

m=10 matches in that season, i.e. 10#tCD#24 where tCDis

the match day just before the CD. During the m=10 matches

before the CD no other CD is allowed. Our final data basis

contains 154 CDs out of 361 mid-seasonal CDs in total. To first

approximation the CDs are equally distributed in the time

interval 10#tCD#24 with an average value of around 17.

2. To quantify the effect of a CD we choose an appropriate

control group. For a specific CD event, occurring after match

day tCD(by construction tCD$10), we identify all events where

some other or the same team during any season displays a

similar goal difference (more specifically with a difference of the

goal difference DG between control team and CD team in the

interval [0.185,20.215]) during tCDsubsequent matches and

has still at least 10 matches to play after this time interval. The

minor asymmetry of the selection interval for control teams

guarantees an identical average value of DG of control and CD

teams and just reflects the Gaussian-type distribution of DG

-values around zero [7]. We use always normalized goal

differences (per match). In this way we obtain approximately

100 control teams per CD, except for a single extreme case in

the year 1965/66 where no control teams could be found.

Additionally, we have chosen a control group by two separate

conditions. First, during the matches 3 to 10 before the CD

event the deviation of the average goal difference DG between

control team and CD team had to be in the interval

[0.196,20.204] and second, during the two matches before

the CD event (matches 1 and 2) a per-match-deviation of the

goal difference by 60.5 was allowed. The reason for these

different choices is discussed in the main part.

3. Going beyond most previous studies we have also corrected the

home/away-asymmetry [7,8], i.e. the match results are

projected on the fictive results in a neutral stadium, in order to

extract the respective team fitness without the home/away-bias.

More specifically, we have substituted DG by DG6Dh (2: home

match;+:awaymatch)whereDh (.0)denotes the average home

advantage. It turns out that the home advantage depends on the

season, but is independent of the specific team [7].

Our procedure implies some important methodological aspects

that have to be kept in mind:

1. The value of m=10 has been selected by the condition that the

final result displays a minimum error. In case of a larger

interval the number of CDs would be smaller, in case of a

smaller interval the characterization of the team fitness would

be worse.

2. A few times it occurs that within the m=10 matches a new

coach is already replaced by another coach. Sometimes this is

planned (in case of a caretaker coach) or is the consequence of

successive bad performance. As implied by our approach we

have in that case incorporated the first CD but not the second

one. This is motivated by the fact that otherwise we cannot

judge the team quality during the short time (less than m

matches) between the first and the second CD. In any event,

our setup implies that the results exactly hold for all CDs where

the coach was active for at least m matches.

3. Previous studies (see above) have restricted the control group to

teams which did not dismiss the coach during the relevant

period. This, however, introduces a bias towards a more

positive expectation because teams with a bad future

performance tend to be excluded. To overcome this statistical

problem it is essential to use unbiased control groups.

4. The identification of control teams via all tCDmatches before

the CD is motivated by our previous observation that the

change of the team fitness during the season is neglible so that

as many matches as possible should be taken into account for

the estimation of the team fitness. However, based on the

subsequent results we will conclude that a minor modification

of the selection process might be appropriate. In any event, this

will be discussed further below.

Analysis of Changes of Coaches (CCs)

We have also studied all cases where a coach was changed (as a

regular change or a dismissal) during the summer break. This

event is denoted as CC (change of coach). We have considered

those 141 cases (starting 1966/67) where the corresponding team

played in the German Premier League in both seasons before and

after the CC. Here we start somewhat later in order to have

enough seasons to estimate the team fitness before the CC (see

below).

An important aspect for the CC analysis deals with the

prediction of the expected outcome of a season. If during one

season the goal difference is given by DG (old) the expected

average fitness F(est) in the next season can be consistently

estimated via F(est)=cF+dFDG (old) [7]. Here F(est) can represent

the expected goal difference or the number of points in the new

season. The parameters cFand dFare calculated from a regression

analysis for all teams which are not relegated. An even better

estimator is obtained by averaging (for all teams where this is

possible) the outcome over the previous three years with weighting

factor 1.0, 0.7 and 0.5 for the determination of DG (old). These

parameters have been estimated by optimizing the prediction

process. If a team was not playing in the Bundesliga in the second

and/or third last season, these seasons were just omitted from the

calculation of DG (old). Note that our results are insensitive to the

specific choice of these weighting factors.

The Usefulness of Coach Dismissal and Change

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Results

CD: Analysis of Possible Effects

The temporal evolution of CD and CC events is explicitly

shown in Fig. 1. Interestingly, the total number does not show any

significant time dependence. It seems, however, that the number

of CC events was larger during the initial period of the Bundesliga

whereas at the same time the number of CD events during the

initial or final period of the season was smaller. This might be a

consequence of the increased presence of media and the

corresponding pressure to act in case of a bad performance.

In Fig. 2 we show the goal difference of an average CD team vs.

time (measured in units of matches). There is a naive interpre-

tation of this plot. First the teams, which later on will dismiss the

coach, display an average value of DG=20.5. Then the fitness

further deteriorates down to DG=21.3 which prompts the CD.

Afterwards the average value of DG is 20.25, suggesting a

significant improvement.

As already noted in literature [1,8,10–15] a group of teams with

an average negative goal difference will on average also have

experienced bad luck. After the selection procedure, i.e. in the

prediction period (final 10 matches), any positive or negative

random effects will average out. To quantify this effect we analyse

the performance of the control teams as introduced in the method

section. By our construction we obtain an identical average value

of DG of control and CD teams ((DG=20.53960.002 and

DG=20.53960.035, respectively. Time resolved average DG

-values are also displayed in Fig. 2. For the prediction period we

obtain an average DG -value of 20.25760.044 for the CD teams

and of 20.28760.002 for the control teams, yielding D

(DG)=0.03060.044, supporting the nil hypothesis. A more

detailed error analysis which takes into account the statistical

uncertainty of DG in the selection period, yields a slightly larger

statistical error of 0.046 as compared to 0.044. With an optimistic

estimationof a residualimprovement

0.046=0.076 our result amounts to a total improvement during

half a season, i.e. 17 matches, of DG<1.3.

Repeating this analysis for different values of m, i.e. different

time intervals to define the selection and prediction period, the nil

hypothesis is supported for all choices, albeit with larger statistical

errors. An objective approach to judge the size of this effect is to

compare the square of this maximum possible improvement with

the variance of the fitness distribution which is 0.27 (see also [7] for

a similar value determined for the last 23 seasons). Thus we obtain

0.0762/0.27=0.02. This again clearly shows that any possible

improvement is absolutely negligible. Using a different measure of

of

D(DG)=0.030+

the effect size, as standard in statistical literature, yields a similarly

small value [17].

This apparent improvement in Fig. 2 is known as regression

towards the mean [13–15]. Qualitatively, this effect reflects the

fact that a subgroup which is selected based on a negative

accomplishment during a finite time interval will seemingly

improve in the future. This is just a direct consequence of the

presence of statistical fluctuations and is fully reflected by the

behavior of the control teams. In the present case it can be

expressed as the ratio rDG of the average DG value in the

prediction period and the DG value in the selection period. For the

control teams one empirically obtains rDG=0.53. Previous work

has developed a general formula stating the rDGis approximately

equal to 1/(1+f/tCD),1 with f<13; see [7]. With this expression at

hand we can perform a consistency check of our approach.

Additionally taking into account the distribution of tCDvalues as

well its average value of 17 the relevant factor here is c(13.5,

17)<0.56 which is indeed close to 0.53. The slight variation of f

reflects the difference between ,1/tCD. and 1/,tCD..

Please note that there is no gradual improvement during the m

matches after the CD event. First, this result is consistent with the

Figure 1. The number of CD and CC events. In particular we show the intra-seasonal CD events after the 10thmatch and before the 25thmatch.

doi:10.1371/journal.pone.0017664.g001

Figure 2. Comparison of the CD (coach dismissal) with the

control teams based on the average goal difference. The time

axis is shifted with respect to the time of the CD (occurring directly after

match day tCD) to enable comparison of different events. The average

values for the prediction period are included as solid lines. No effect of

the CD is present within statistical errors.

doi:10.1371/journal.pone.0017664.g002

The Usefulness of Coach Dismissal and Change

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Page 4

general observation that the team fitness does not change during

the season. Second, this also implies that the cases where a

carekeeper coach is replaced after less than m=10 matches does

not yield a further significant positive (or negative) shift.

We have repeated the analysis by restricting ourselves to the last

23 years of the Bundesliga. Here we find D (DG)=0.0860.06.

Within the error bars this result is identical to that of the whole

period and is thus again compatible with the nil hypothesis. Thus,

there is no significant time dependence in the efficiency of CD

events.

Interestingly, the CD teams play worse during the last two

matches before the CD event. Thus one might speculate that the

CD event at least helps to stop this emerging negative streak. This

hypothesis can be checked by selecting control teams which also

have two worse results at the end of the selection period (see above

for details). The results are shown in Fig. 3. Except for 14 CD

teams it was always possible to find appropriate control teams,

albeit with a smaller number (due to the more detailed

constraints). This shows up in larger fluctuations. Again the

average results in the prediction period are basically identical. This

result is compatible with our previous finding [7] that two

consecutively lost matches are not sufficient to identify the

beginning of a negative streak (in contrast to four consecutively

lost matches).

Furthermore we checked that the CD events are not related to

any effects of the home/away-asymmetry. Since we have corrected

out this asymmetry no effects should be present. However, we

explicitly checked that within statistical noise the number of

home/away and away/home matches before the CD event is

nearly equal and the fraction of two subsequent home or two

subsequent away matches before the CD event is both less than

7%.

The results, reported so far, deal with the average effect of a CD

event. In particular they are still compatible with the hypothesis

that the CD has a positive effect for some teams and a negative

effect for other teams. This can be tested by analyzing the variance

of DG -values. Results are shown in Fig. 4. The variance increases

by 0.0560.1. This result is compatible with a zero effect. Of

course, the value of 0.05 would still allow for the (extreme)

scenario that half of the CD events result in an improvement of

DG<0.2 (<!0.05) and the other half in a deterioration of DG<

–0.2. This explicitly shows that the resulting effect, if present at all,

is very small effect.

In practice one is particularly interested in points P rather than

in the goal difference DG. Because of the important implications of

our results we have repeated the same analysis as in Fig. 2 by using

points to characterize the fitness of teams. To standardize all

games beginning in 1963 we have always used 3 points for a win

and 1 for a draw following the worldwide established FIFA rules. It

should be noted, that in the German Premier League 2 points

were used for a win until 1994/95.

As seen in Fig. 5 the qualitative behavior is fully identical as

discussed in the context of Fig. 2. The average values in the

prediction period are P=1.34760.036 and P=1.32960.004 for

the CD teams and the control teams, respectively. Their difference

reads DP=0.01860.036. Note that DP=0.018 per match

corresponds too much less than one point per half season. In

any event, the nil hypothesis is fully supported.

However, comparisons of the results for DG and P explicitly

show that the information content of the goal difference is by far

superior. As shown in [7] an approximate scaling of DG to P

values can be performed by using a factor of approx.1.6 (see [7]).

Indeed, this factor is approximately recovered when comparing D

(DG)=0.030 with DP=0.019. However, the relative statistical

error is significantly larger for DP, i.e. 0.2, as compared to D(DG),

i.e. 0.13. Furthermore, also the strong apparent fitness decrease

during the two matches before the coach dismissal is significantly

more pronounced for DG as compared to P. This is partly due to

the fact that points are (trivially) bounded from below whereas no

such bound exists for DG.

Motivation for a CD

A further important question deals with the motivation to

dismiss a coach. Naturally, an unsatisfactory performance is

expected to be the main reason. As already discussed above the

data in Fig. 2 suggest that beyond this general performance

argument (see below for a closer discussion) the occurrence of two

bad results trigger the dismissal of the coach. This observation has

consequences for the consistency of our approach. Based on our

previous results [6] we expect that fitness fluctuations are very

small during a season. Due to the relative shifting of the data (via

Figure 3. Analogous representation as in Figure 2, but with the

additional constraint that the control teams also display

correspondingly bad results during the two matches before

the CD event. Again no effect of the CD is present.

doi:10.1371/journal.pone.0017664.g003

Figure 4. Comparison of the CD teams with the control teams

based on the variance of the goal differences. In analogy to

Figure 1 the average values over the prediction period are given as solid

lines. Again no effect is present.

doi:10.1371/journal.pone.0017664.g004

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t–tCD) we systematically identify two matches where the teams just

had particular bad luck. It is consistent to exclude these two

matches from the fitness estimation of a team because these two

data points are biased. As a consequence the control teams on

average should have the same DG for t–tCD,21.

This argument can be rationalized with a simple example. In

the ‘‘dice throwing premier league’’ a coach is dismissed after 2

times throwing a 1. Of course, in principle all teams have equal

properties (average fitness 3.5). However, if the 10 matches before

a CD event were analyzed exactly in analogy to our procedure one

finds an average fitness of 3.3. The reduction is due to the

systematic inclusion of the final two results with a 1. Thus, the

fitness estimate is lower than the true fitness of 3.5. Excluding the

last two results for the CD from the analysis yields a fitness value of

3.6. Now the value is larger as the true fitness because in our

approach no second (1,1)-pair is allowed to occur during the 10

matches before the CD event. Thus we conclude that a better

fitness estimate is obtained if we omit the two matches before the

CD event. However, since this estimation would be slightly too

optimistic, the optimum estimation lies in between both approach-

es (with and without the final two matches) as exemplified above.

Adapting the choice of control teams to this condition (omission

of the last two matches) the average value of DG in the selection

period reads 20.431 instead of 20.539. Correspondingly the

optimized set of control teams also plays better in the prediction

period (20.235 instead of 20.287). Thus the effect of the CD gives

rise to a negative value of D(DG)=20.02260.048 rather than

D(DG)=0.03060.046 (as mentioned above). As a consequence

our finding of a nil effect is further corroborated by this self-

consistently modified procedure. As discussed in the previous

paragraph for general reasons the ‘‘true’’ value is expected to lie

between the original (0.030) and the new estimate (20.022) which

even better agrees with the nil hypothesis.

It is to be expected that beyond this triggering effect also the

performance in the whole season is unsatisfactory. To quantify this

effect we determine the expected number of points in a season

P(est) as well as the expected goal difference DG(est) for all CD

teams with the procedure, introduced in the method section. Then

one can assess the degree of frustration of a team from comparison

with the actual outcome. For this comparison we choose the

number of points, i.e. P(true) – P(est), since this observable is

relevant for managerial decision processes. Since the CD does not

change the fitness of the team we can use the outcome of the total

season to get an optimum statistical accuracy. To obtain an even

more specific correlation we additionally correlate the difference

P(true) – P(est) with DG(est), the latter representing the fitness of a

team. In this way we can distinguish between the motivation of a

CD for good and bad teams.

The results are displayed in Fig. 6. Obviously, most (82%) of all

teams have indeed performed worse than the pre-season

expectation. Thus, the motivation to dismiss a coach is not only

pure imagination but is indeed backed by a bad performance

(which, unfortunately, does not change after the CD). Interesting-

ly, the deviations from expectation are stronger for good teams (on

average up to 9 points for the whole season) as compared to bad

teams with approximately half of the number of points. This may

have a simple psychological explanation. Even with a somewhat

poorer performance good teams are still significantly distant from

the relegation positions. Thus, for these teams the need for action

results from the mere comparison with the expected outcome. For

bad teams, however, already a minor negative deviation will push

these teams to positions very close to relegation. This may

immediately increase the pressure to act and thus to dismiss the

coach as the most simple action.

We have repeated the analysis with evaluating the number of

points after midseason, i.e.at the average time of the coach

dismissal. The graph looks similar albeit with slightly smaller

values for the number of points (because only half of the season is

over). In any event, the interpretation remains exactly the same as

before.

CC: Analysis of Possible Effects

Having found no signature of the in-season CDs one may

wonder whether changing the coach during the summer break, i.e.

a CC, has an influence on the team performance. This question

has two facets. First, independent of the quality of the coach the

mere act of changing a coach may bring in a systematic shift in

fitness. Of course, this shift may be positive (e.g. due to bringing in

new stimulus in saturated structures) or negative (e.g. due to

corrosion of well-established team structures). Second, beyond this

systematic effect the different qualities of coaches might lead to the

effect that some teams profit whereas other teams may suffer from

this change (relative to the average). Whereas the systematic effect

Figure 5. Analogous to Figure 2, using points rather than the

goal difference as the observable of interest. Again no effect of

the CD is present within statistical errors.

doi:10.1371/journal.pone.0017664.g005

Figure 6. Correlation of the deviation from the expectation of

points with the expected fitness in a season where a CD takes

please. The solid line is the regression line. From this graph the

motivation to dismiss a coach can be extracted.

doi:10.1371/journal.pone.0017664.g006

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can be studied from the first moment of the appropriate

performance distribution, the variance of this distribution contains

additional information about the quality variation of different

coaches, as already discussed in the context of CD.

In analogy to above we start by correlating P(true) – P(est) with

DG(est); see Fig. 7. It turns out that the average value of P(true) –

P(est) is 20.360.6. Thus, no significant overall improvement of

deterioration is seen. Furthermore, no significant correlation with

DG(est) is observed since the relative error of the slope of the

regression line is approx. 70% of the slope itself. Thus we may

conclude that a possible systematic effect of a CC is less than one

point per season, i.e. totally negligible. Repeating the same analysis

for DG(true) - DG(est) (as before defined as the average goal

difference per match) we obtain 20.0260.04 which again indicates

that any effect, if present at all, is very small. We may conclude that

changing the coach has no systematic positive or negative effect.

In the next step we study the variance of DG(true) - DG(est) of

the CC teams. In what follows we restrict ourselves to the

distribution of goal differences due to its superior properties as

compared to the number of points. For the variance we obtain the

value of 0.19760.026. Here the statistical error is smaller than in

the CD analysis because we include information from a complete

season rather than just from 10 matches. To identify the statistical

contribution (due to the random effects in a soccer match beyond

the actual team fitness) we also determine the variance for all

teams. We take the same seasons as for the CC teams and, of

course, also require that the team was playing in the Bundesliga in

the previous season (for the determination of DG(est)). Here the

variance is given by 0.21260.013. The difference of the variances

thus reads 20.01560.029. Within the statistical error no

difference to the variance of the CC teams is present. Note that

a significant quality variation among the coaches would have

resulted in a positive value of that difference. In any event, the

hypothesis that all coaches basically have the same or similar

quality (or their quality is irrelevant for the team performance) and

that a CC has no direct effect cannot be ruled out by studying the

data of more than 40 years Bundesliga.

Taking into account the size of the statistical error one may

estimate the possible relevance of the specific coach on the team

performance. With an optimistic view the maximum increase of

the variance is given by 20.015+260.029<0.04. The value has to

be compared with the fitness variance of all teams in the

Bundesliga which is 0.27 (see above). This implies that with this

optimistic estimation the relative contribution of the coach to the

team fitness is 0.04/0.27, i.e. 15%. Most likely, however, this

contribution is even smaller. This small value also reflects the fact

that the group of coaches, which is considered to be hired in the

Bundesliga, fulfills already high quality criteria so that the quality

variation within this group is quite small.

Discussion

This work can support the results of some previous studies [11–

15], but now ruling out several methodological weaknesses and

covering a very large data set with respect to effects of coach

dismissals. The underlying team fitness does not improve due to

coach dismissal. The increase immediately after the coach

dismissal can be completely traced back to a simple statistical

selection effect (regression towards the mean). The idea to dismiss

a coach emerges from a bad performance as compared to

expectation (see Fig. 6) and the actual dismissal is triggered by two

particularly unfortunate matches. Furthermore, for teams below

the average a smaller deviation from the pre-season expectation

may be sufficient to dismiss the coach as compared to better teams

where typically a larger deviation is required.

Changing the coach during the summer break results in the

same nil effect. Most interestingly, even the variance of the

appropriate distribution of teams changing the coach during two

seasons does not show any effect. This has the immediate

consequence that the impact of coaches as ‘‘fitness producers’’

for the teams is limited and is most likely (on average) much

smaller than 15% as compared to other factors (like the team wage

bill [18]), determining the quality of a soccer team. Stated

differently, the quality of coaches, working in the Premier German

Soccer league and hired successively by a team is either quite

similar or does not have much impact on the quality of the team as

already assumed before [3]. Our results do not exclude the

possibility that it is favorable to work with a coach several years in

a row. This aspect will be studied in future work along similar

lines.

Author Contributions

Conceived and designed the experiments: AH CM OR NH BS. Performed

the experiments: AH CM OR NH BS. Analyzed the data: AH CM OR

NH BS. Contributed reagents/materials/analysis tools: AH CM OR NH

BS. Wrote the paper: AH CM OR NH BS.

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doi:10.1371/journal.pone.0017664.g007

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The Usefulness of Coach Dismissal and Change

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