# Impact of One Additional Substitution on Player Load and Coaching Tactics in Elite Football
## 2 Materials and Methods
### Methodology  
Evaluation of TOS  
In this section, we present the relevant parameters and calculations for the evaluation of this change. The cumulative load (_CL_) of a team per match corresponds to the product of the match duration and the number of players on the pitch. In line with previous research, goalkeeper substitutions were excluded, and therefore, the number of players observed for the calculation was reduced to 10 [1; 9]. Additionally, the average stoppage time of 3.4 min in the Bundesliga was added to the match time for a more realistic evaluation. In summary, a consolidated _CL_ in football of 934 min was calculated with an average of 93.4 min of match time and 10 players. The maximum number of substitutions (_max_NS_) in one season accounted for a total of 1836 possible substitutions in the German Bundesliga (34 match days \(\times\) 9 matches per match day \(\times\) 6 substitutions per match) and 1456 possible substitutions in the German U-19-Bundesliga West (26 match days \(\times\) 7 matches per match day \(\times\) 8 substitutions per match). Moreover, the average number of substitutions per match and the utilisation rate (average percentage of substitutions used per match) were used in the calculations. Load reduction per substitution in minutes Equation (1) and the relative load reduction per substitution in percent Equation (2) were calculated as follows:  
\[(MD-OTOS)=LRpS. \tag{1}\]  
**Remark 1**. : _MD: match duration; OTOS: average time of substitution; LRpS: load reduction per substitution._\[\frac{LRpS}{CL}=rLRpS. \tag{2}\]  
LRpS: load reduction per substitution; CL: cumulated load; rLRpS: relative load reduction per substitution. The relative load reductions of the Bundesliga and U-19 Bundesliga West were calculated with the following formula:  
\[\frac{(MD-OTOS)*ONS}{CL}=rLR. \tag{3}\]  
MD: match duration; OTOS: average time of substitution; ONS: average number of substitutions; CL: cumulated load; rLR: relative load reduction. The 90 th minute was added as the time of the fourth substitution to the average TOS with three substitutions in regular time, because this related to the earliest possible TOS with a fourth substitution in extra time, therefore leading to the greatest possible load reduction. To calculate the maximum benefit of an additional substitution in extra time on load reduction, the following equation was used:  
\[\left(\frac{3}{4}*OTOS3r\right)+\left(\frac{1}{4}*90\right)=min\_OTOS4x. \tag{4}\]  
In extra time, the CL fundamentally added up to 1234 min (123.4 min \(\times\) 10 players). The relative maximum load reduction with an additional substitution in extra time (\(max\_rLRx\)) was calculated as follows:  
\[\frac{(MD-min\_OTOS4x)*NS}{CL}=max\_rLRx. \tag{5}\]  
min_OTOS4x: earliest possible average time of substitutions with added substitution in extra time; MD: match duration; NS: number of substitutions; CL: cumulated load; max_rLRx: maximum relative load reduction with added substitution in extra time. The relative maximum load reduction per season with an additional substitution in extra time was calculated for a team playing in the German Bundesliga (34 matches), in the national DFB-Cup (6 matches), and in an International European Cup (13 matches) as follows:  
\[\frac{max\_LRx*NGx}{NG*CLr+NGx*CLx}=max\_LRx\text{ per season}. \tag{6}\]  
max_LRx: maximum load reduction with additional substitution in extra time; NGx: number of matches ending in extra time; CLr: cumulated load of matches ending in regular time; CLx: cumulated load of matches ending in extra time; max_LRx per season: maximum relative load reduction per season with added substitution in extra time. Additionally, the load reduction with four substitutions was compared to the load reduction with three substitutions to assess the benefit that could be gained with an additional substitution. This increase of load reduction (_ILR_) as calculated as follows:  
\[\frac{LR4S-LR3S}{LR3S}=ILR. \tag{7}\]  
**Remark 6**.