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.

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

Our procedure implies some important methodological aspects that have to be kept in mind:

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 ΔG (old) the expected average fitness F(est) in the next season can be consistently estimated via F(est) = c_F+d_F ΔG (old) [7]. Here F(est) can represent the expected goal difference or the number of points in the new season. The parameters c_F and d_F are 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 ΔG (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 ΔG (old). Note that our results are insensitive to the specific choice of these weighting factors.

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 interpretation of this plot. First the teams, which later on will dismiss the coach, display an average value of ΔG = −0.5. Then the fitness further deteriorates down to ΔG = −1.3 which prompts the CD. Afterwards the average value of ΔG is −0.25, 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 ΔG of control and CD teams ((ΔG = −0.539±0.002 and ΔG = −0.539±0.035, respectively. Time resolved average ΔG -values are also displayed in Fig. 2. For the prediction period we obtain an average ΔG -value of −0.257±0.044 for the CD teams and of −0.287±0.002 for the control teams, yielding Δ (ΔG) = 0.030±0.044, supporting the nil hypothesis. A more detailed error analysis which takes into account the statistical uncertainty of ΔG in the selection period, yields a slightly larger statistical error of 0.046 as compared to 0.044. With an optimistic estimation of a residual improvement of Δ(ΔG) = 0.030+0.046 = 0.076 our result amounts to a total improvement during half a season, i.e. 17 matches, of ΔG≈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 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 r_ΔG of the average ΔG value in the prediction period and the ΔG value in the selection period. For the control teams one empirically obtains r_ΔG = 0.53. Previous work has developed a general formula stating the r_ΔG is approximately equal to 1/(1+f/t_CD)<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 t_CD values 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/t_CD> and 1/<t_CD>.

Please note that there is no gradual improvement during the m matches after the CD event. First, this result is consistent with the 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 Δ (ΔG) = 0.08±0.06. 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 ΔG -values. Results are shown in Fig. 4. The variance increases by 0.05±0.1. 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 ΔG≈0.2 (≈√0.05) and the other half in a deterioration of ΔG≈–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 ΔG. 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 = 1.347±0.036 and P = 1.329±0.004 for the CD teams and the control teams, respectively. Their difference reads ΔP = 0.018±0.036. Note that ΔP = 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 ΔG and P explicitly show that the information content of the goal difference is by far superior. As shown in [7] an approximate scaling of ΔG to P values can be performed by using a factor of approx.1.6 (see [7]). Indeed, this factor is approximately recovered when comparing Δ (ΔG) = 0.030 with ΔP = 0.019. However, the relative statistical error is significantly larger for ΔP, i.e. 0.2, as compared to Δ(ΔG), i.e. 0.13. Furthermore, also the strong apparent fitness decrease during the two matches before the coach dismissal is significantly more pronounced for ΔG as compared to P. This is partly due to the fact that points are (trivially) bounded from below whereas no such bound exists for ΔG.

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 t–t_CD) 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 ΔG for t–t_CD<−1.

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 approaches (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 ΔG in the selection period reads −0.431 instead of −0.539. Correspondingly the optimized set of control teams also plays better in the prediction period (−0.235 instead of −0.287). Thus the effect of the CD gives rise to a negative value of Δ(ΔG) = −0.022±0.048 rather than Δ(ΔG) = 0.030±0.046 (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 (−0.022) 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 ΔG(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 ΔG(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). Interestingly, 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.

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 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 ΔG(est); see Fig. 7. It turns out that the average value of P(true) – P(est) is −0.3±0.6. Thus, no significant overall improvement of deterioration is seen. Furthermore, no significant correlation with ΔG(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 ΔG(true) - ΔG(est) (as before defined as the average goal difference per match) we obtain −0.02±0.04 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 ΔG(true) - ΔG(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 0.197±0.026. 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 ΔG(est)). Here the variance is given by 0.212±0.013. The difference of the variances thus reads −0.015±0.029. 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 −0.015+2×0.029≈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.