The data used in this research is very extensive. The dataset is retrieved from Transfermarkt.com and put together by Cereijo [3]. The data consists of four different datasets that are linked by codes. There is a dataset for players, clubs, games and appearances. Those are put together via player ids, game ids and club ids so that the desired format is reached eventually. A description of how this is done precisely can be found in the Appendix. The dataset that is eventually used needs some explanation. The dataset consists of 44.666 games from July 9th, 2012 (Metalurg Donetsk - Shakhtar Donetsk) until April 11th, 2022 (Antalyaspor - Hatayspor and Bologna - Sampdoria). Four hundred clubs play these games in 35 different domestic competitions in 14 different countries and four different European competitions. In these games, many players participated. The appearances of 17851 players from 171 countries are recorded in the dataset. This means that the number of observations is substantial and accurate results can be achieved. In the following subsections, the data will be further explained.

The cleaned dataset contains numerous variables that will be listed and explained in this paragraph.

In this section, (descriptive) statistics of all variables can be found. First, the calculation of the chemistry variables will be discussed. After that, descriptive statistics of the chemistry and other variables will be shown and discussed.

The most important variables in this research are the chemistry variables. The question remains about how to measure this chemistry variable. Four methods are proposed to measure this abstract variable. As in-group favouritism [14] and, in particular, ethnicity possibly play an important role, this ethnic component will be used as a measure for chemistry. The ethnic component of the measure is divided into strict and loose measures. The strict measure is based on the nationality of the players and the loose measure is based on the part of the world (in this research, the world is divided into eight parts) the players come from. The proposed methods start with all the players that play for a specific team in a specific match (both starters and substitutes). Take, for example, Ajax in the heroic comeback in the Champions League in 2019. The players that played for Ajax in that match are listed in Table 1. From this information a n × n connection matrix A is constructed. The entries in the matrix are either 0 or 1. An entry α_ij will take the value 1 if player i and player j have the same nationality (or region). The diagonal will thus be filled with ones as, there, a player is compared to itself. Two methods to calculate the chemistry of a team in a match will be proposed.

The matrix to calculate the chemistry from the nationalities in Table 1 can be found in Table 2. From this matrix, the number for chemistry will be calculated. The first method to calculate the chemistry from this matrix is the Social Connectedness Index or SCI [4].The Social Connectedness Index counts the number of connections in this matrix the ones, and divides it by the total possible number of connections. This division is done to normalise the measure. The minimal value is zero and the maximum value is one. As a player always will have a connection with itself, the diagonal ones will be omitted in the calculation. In formula form, the Social Connectedness Index looks as follows

In the example, a connection is formed when two players have the same nationality. The number of ones in the example is 48. After subtracting the diagonal ones, 34 connections remain. As n = 14, the SCI for Ajax in this particular match can be calculated.

A connection can also be formed if players come from the same region. Using the SCI to calculate the chemistry thus delivers two different chemistry measures. A possible direction in future research can be to combine the region and nationality measures and weight, for example, the region connections with a factor of 0.5.

The matrix to calculate the chemistry from the regions in Table 1 is shown in Table 3. The second method to calculate the chemistry from these matrices is the largest eigenvalue method. The eigenvalues are calculated with the equation

with x a nonzero vector. The matrix will have n eigenvalues. In this case, the matrices are always symmetric and many rows are the same. The largest eigenvalue will be equal to the largest cluster of players from the same nationality or region. In the Ajax example, there are six Dutchmen, two Moroccans, two Danes, a Serbian, an Argentine, a Brazilian and a Cameroonian. The largest eigenvalue will thus be 6. In order to make the measure comparable over the whole sample with different n, the largest eigenvalue λ₁ is divided by the number of players that played for a team in a certain match. The largest eigenvalue chemistry measure (LE) looks as follows

From the Ajax example in Table 1, the region matrix in Table 3 is constructed. The biggest cluster in the region matrix is from European players. There are nine European players, three Africans and two South Americans. The largest eigenvalue from the region matrix will thus be equal to 9. The region chemistry based on the largest eigenvalue method for Ajax in this match is

To illustrate the differences between the measures, the descriptive statistics in Table 4 and histograms of the different chemistry measures in Fig. (1) are shown below. As can be seen from Table 4 and Fig. (1a), the chemistry measure based on nationality and calculated with the SCI is the most conservative. This can be deducted from the relatively low mean (0.292) and from the histogram that shows almost all mass is below 0.5. The chemistry measure based on nationality and calculated with the largest eigenvalue method is more evenly spread out, as can be seen in Fig. (1c). There is, however, a peak around 0.65. This peak is caused by the occurrence of 8 players from the same country in a match where 13 different players played for a specific team. If this occurs, the chemistry measure calculated with the SCI will lie in the interval (0.442,0.571), explaining the peak around 0.45 in Fig. (1a). The use of the chemistry measure based on region shifts mass to one as there are a lot more connections due to the reduction of possible regions from 171 countries to 8 parts of the world. This shift is both visible in (Table 4 and Fig. 1b and 1d) as the means are above 2/3 and mass is shifted to the right with peaks in the rightmost interval that ends at one. Once again, the SCI is somewhat more conservative than the largest eigenvalue measure as the mean and minimum are somewhat lower (Table 4) in comparison with the largest eigenvalue measure. This is due to the nature of the measure as in the example, the largest cluster of 8 players from the same region out of 13 that participated for that particular team in that particular match will lead to a chemistry measure of

In contrast, the chemistry measure calculated with the SCI in this example cannot exceed 0.571.

From Fig. (2) can be seen that the Goal Difference is centered around zero with 10753 draws which are 24.1% of the data. The number of matches with a negative goal difference is 13730. This means 13730 (30.7%) matches are won by the away team. The remaining 20183 (45.2%) matches are won by the home team. This domination of wins by the home team can be explained by a phenomenon called home advantage [18]; [20]. This home advantage will not be taken into consideration as the data do not provide the necessary detailed information that is needed. It is likely to be captured by the constant term as it is a sort of measure for home advantage already (Table 5). The sign of the constant term will, however, not give any information since most control variables have a positive minimum and maximum value that will be added to the constant term. As can be seen from Table 6, the variables Home Height and Away Height, Home Age and Away Age and Home Position and Away Position all are quite similar in terms of possible home and away the difference. The means of the dummy variables show what part of the sample is classified in that dummy group. This means 13.66% of all matches are played without attendance during COVID and 7.084% of all matches are played in a Dutch competition. In the case of the Netherlands, a match can be played in the Dutch League (Eredivisie), the Dutch Cup (KNVB Beker) or the Dutch Super Cup (Johan Cruijff Schaal).

From Fig. (3) can be seen that most matches are played in a Spanish competition (4141), followed by Italian and English competitions. The least matches are played in Ukrainian (1916) and Danish (1918) Competitions. The matches are evenly spread over the countries, with parts that range from 4.29% to 9.27%. This means the number of matches in the different countries is big enough to detect differences between countries. The variable that will eventually be estimated is the goal difference. The variables of interest are the chemistry variables. To give some more insight into the behaviour of those variables, some interesting correlations will be shown in Table 7. As can be seen from this table, the correlation of height and age with goal difference is very close to zero, although the home and away parts are all opposite. The correlation of position with Goal Difference is quite large, with -0.3726 for Home Position and 0.3616 for Away Position. This is as expected, as the better a team, the bigger the goal difference. As a team is better when the value of position is lower, the sign of the correlation of Goal Difference and Home Position is negative and the sign of the correlation of Goal Difference and Away Position is positive.

The baseline model that will be used is the Two-Way Fixed Effects model. This is a panel regression model that controls for time-fixed effects and team-specific effects. The model is

with y_ijt the goal difference between home team i and away team j played in time period t. Time period t is chosen as a quarter, so t = 0, ..., 39 as the matches are played over ten years or forty quarters time. The match-specific variables x_ijt are the variables of interest, Home Chemistry and Away Chemistry, control variables such as Home Position and Away Age that differ per match as mentioned in the Explanatory Variables section and interaction effects between the variables of interest and the control variables. To check robustness, country-specific dummies will be added. In other words, these dummies check for differences between matches played in different countries. The added dummies are as mentioned in the Explanatory Variables section, with one left out as the baseline country. The England dummy will be used as the baseline country to which the other dummies will be compared. The model to check for these country-specific effects looks as follows

with D_ijt a vector where the entry corresponding to the country the competition is from is equal to one and the rest of the entries equal to zero. These dummies are not redundant as a team can play both in domestic competition and Europe, so there is no one-to-one correspondence between a team and the country dummy. The models, as mentioned above, are, however, implicit. Since for every i and every t, a fixed effect is present, estimating this model will give identification issues due to the large number of parameters that must be estimated. To get rid of those fixed effects ”double-demeaning” is used [21].

Define the team-specific averages over time as

and

the cross-sectional average for each time-period t. The total average is

Define

and similarly

In the equations above t ranges from 0 to 39 but does not necessarily take all those values given a specific t,T therefore, ranges from 1 up to 40. For i and j hold similar reasoning, i ranges from 1 to 400 but does not need to take all values for a specific t. N can therefore range from 1 to 400. The range of j, which differentiates between matches that team i plays in time period t, is from 1 to 13 and can take any value in between. M can therefore range from 1 to 13. The estimates are eventually found by regression of y~_ijt on x~_ijt. The equation to find ^β is

The two-way fixed effect model can be expressed as a Least Squares Dummy Variable (LSDV) estimator that in the formula is

where d_1,i and d₂,_t are the dummy variables for the team-specific and time effects [22].

For the variance-covariance estimator, a clustered variance-covariance estimator is used. This is done to make sure the differences between the time and team clusters are taken into account in calculating the variance-covariance estimator. In formula

where

and

where

and ijt Ɛ G_g indicates that the observation belongs to group g. Groups g are the different clusters. In the case of two-way clustering used in this estimation, there is a cluster for each different i, t and it. For example, there are separate clusters for matches in which AZ Alkmaar is the home team, for matches played in the 29th quarter of the dataset (t = 29) and for matches played in the 29th quarter of the dataset (t = 29) with AZ Alkmaar as the home team.

Predicting matches is the most exciting thing that can be done with all the information gathered from the estimations. These predictions will be made with the model that is estimated and the information available for a new match. A prediction can thus be made shortly before a match since then, the line-ups are known and the corresponding chemistry measures can be calculated. Moreover, the average height of both teams and the average age of both teams also need to be calculated to make the predictions as accurate as possible.

The forecasts that will be made are, however, not likely to be integers. This means the forecasts will be made on the match’s so-called ’toto’ outcome. The ’toto’ outcome means if the home or away team wins the match or if the match ends in a draw. This is less specific than the goal difference, as the goal difference also measures the magnitude of a win. The second part of the predictions is to determine threshold values for the classification in either of the three classes [20]. These threshold values will form a symmetric or asymmetric interval around zero. If the predicted goal difference is in the interval, a draw will be forecast for that match. If the predicted goal difference is outside the interval, either a home win (positive sign) or away win (negative sign) is predicted depending on the sign of the predicted goal difference.

The threshold interval will be determined by k-fold cross-validation [23]. This method splits the dataset into k parts and takes one of those k parts as test set and the remaining k − 1 parts as training set. The training set is used to estimate the model with which the test set will be predicted. Then the predicted outcomes of the test set will be compared to the actual outcomes of the test set. This will be done k-times with all k different parts of the data once used as the test set. Usually, 5 or 10-fold cross-validation is used and in this research, 5-fold cross-validation will be used. A confusion matrix will be created to determine how good the predictions are. This confusion matrix is shown in Table 8.

Table 6. Descriptive statistics for the control variables. The *'s indicate dummy variables where the standard deviation is calculated with

Variable	Mean	Standard Deviation	Min	Max
Goal Difference	0.3313	1.832	-13	10
Home Height (in cm)	181.2	2.318	174.5	189.1
Away Height (in cm)	182.1	1.560	174.4	189.6
Home Age (in years)	26.88	1.547	18.56	37.18
Away Age (in years)	26.22	1.636	18.72	34.59
Home Position	9.010	5.367	1	21
Away Position	9.198	5.390	1	21
COVID/No Attendance*	0.1366	0.3434	0	1
Ukraine*	0.04290	0.2026	0	1
Denmark*	0.04294	0.2027	0	1
Russia*	0.05693	0.2317	0	1
Europe*	0.05024	0.2184	0	1
Turkey*	0.07137	0.2574	0	1
Belgium*	0.05671	0.2313	0	1
Scotland*	0.05049	0.2189	0	1
France*	0.08147	0.2736	0	1
Portugal*	0.07178	0.2581	0	1
Greece*	0.06078	0.2389	0	1
Netherlands*	0.07084	0.2566	0	1
Italy*	0.08980	0.2859	0	1
Germany*	0.07191	0.2583	0	1
England*	0.08913	0.2849	0	1
Spain*	0.09271	0.2900	0	1

The number of correct predicted home wins, draws, and away wins are shown on the diagonal. Off diagonal can be found where the prediction went wrong. The number of predicted draws that turned out to be a home win can be found in entry c₂₁. This confusion matrix thus shows in an exquisite way how the predictions perform. The actual number that eventually determines how good the predictions are is the accuracy. This is measured by the number of matches that are predicted correctly divided by the total number of matches or from the matrix, the sum of the diagonal entries divided by the sum of all the entries. In formula form

Cross-validation will, in this case, provide ten observations for accuracy per researched interval. The mean of these ten observations will be used as the accuracy for a certain interval. The interval with the highest accuracy and the most realistic predictions will be chosen as the optimal threshold interval. The most realistic means that there have to be enough draws as draws are the most difficult to predict and accuracy tends to be higher when draws are not taken into account [24]. Often there is a local maximum and the threshold interval that achieves this local maximum in accuracy will be chosen as ’optimal’. If there is no such local maximum, a cut-off point will be chosen before a decline in accuracy with a wider threshold interval. Deciding which threshold interval will be chosen remains arbitrary as it is not based on a single number, but these methods seem to work quite well.

In the last part, the estimation has been done without taking COVID into account. No difference is made between matches played for a full stadium and matches played without a single visitor because they were not allowed. Tilp and Thaller [5] showed that playing in an empty stadium makes a big difference from playing in front of their own or rival fans. In this part, the estimation results with COVID taken into account will be discussed. COVID is added as a dummy. The dummy’s value is one if there was no attendance and the stadium was empty and zero if there was at least some attendance. Once again, a baseline regression is performed but now with the additional dummy variable COVID/No Attendance (Table 10, second column). The control variables all have the same sign as in the baseline regression without COVID/No Attendance. They are all significant (only Away Height has a p-value above 0.05), as can be seen in Table 10. COVID/No Attendance has a *-significant (p- value between 0.05 and 0.1) negative estimate of −0.0611. This is in accordance with Tilp and Thaller [5] as the Goal Difference is estimated to be more negative when there is no attendance than if there were attendance. In other words, the so-called home advantage becomes smaller. In the estimations with COVID and the different chemistry measures with possible interaction effects, similar things as in the COVIDless estimations occur. The SCI, LE and SCI Region estimations without interaction effects all deliver similar results (Table 10, third, fifth and seventh column). Home Chemistry is estimated to be negative but insignificant and Away Chemistry is estimated to be positive and significant. All effects are the opposite of what is hypothesised, as the estimates suggest that the higher the chemistry, the worse a team performs. The LE Region regression (Table 10, eighth column) also gives similar results as both Home and Away Chemistry have the opposite sign of what is hypothesised. In this estimation, both estimates are insignificant, which is also the case in the LE Region estimation without COVID (Table 9, ninth column). In the case of the chemistry measures based on nationality with interaction effects (SCI and LE with interaction effects), the chemistry effect for the home team is positive (positive on performance) for the ninth and eighth highest-ranked teams respectively and negative for the rest. The estimate for Home Chemistry and the interaction effect is not significant (Table 10, fourth and sixth column). The effect of chemistry for the away team is positive (negative on performance) for both estimations, with the estimate of Away Chemistry negative and the estimate of the interaction effect not significant. The hypothesis comes true in the SCI Region and LE Region estimations where the interaction effects with the position are also estimated (see Table 10, eighth and tenth). In the SCI Region estimation with interaction effects (Table 10, eighth column), the chemistry effect for the home team is 0.0514 − 0.0107X_HP . This means there is a positive (positive on performance) effect of chemistry for the home team if this team is ranked fourth or higher and a negative (negative on performance) effect if this team is ranked fifth or lower. The effect of chemistry for the away team is −0.2004 + 0.334A_AP . Chemistry has a negative (positive on performance) effect on the away team if this team is ranked fifth or higher and has a positive (negative on performance) effect if this team is ranked seventh or lower. Note that the estimates of Home Chemistry and Home Chemistry x Home Position are not significant and the estimates of both Away Chemistry and Away Chemistry x Away Position are significant. If the team is ranked sixth, the effects cancel out as in the no COVID SCI Region estimation with interaction effects. In the LE Region estimation with interaction effects (Table 10, last column), the chemistry effect for the home team is 0.1140 − 0.0159X_HP . This means the effect of chemistry on the home team is positive (positive on performance) for teams ranked seventh or higher and negative (negative on performance) for teams ranked eighth or lower. The effect of chemistry on the away team is −0.2349 + 0.0340A_AP. This means the effect of chemistry on the away team is negative (positive on performance) if a team is ranked in the top six. If a team is not ranked in the top six, the effect of chemistry on the away team is positive (negative on performance). Once again, only the top teams seem to profit from their chemistry, while the lower-ranked teams seem to suffer if they have good chemistry. Note that the estimates of both Home Chemistry and the interaction effect with Home Position are insignificant and the estimates of both Away Chemistry and the interaction effect with Away Position are significant. The estimations in table 9 and 10 show that the chemistry measures used in this research have the hypothesised signs for the top teams. However, the signs for the lower-ranked teams when interaction effects are taken into account and the chemistry is based on the region instead of nationality are different. The estimate of Away Chemistry is significant in all but two estimations. On the contrary, the estimates of Home Chemistry are insignificant. Table 10 displays the mitigating effect of COVID on the home advantage as the estimate for COVID/No Attendance is negative in all nine estimations and *-significant in seven of them. This confirms the hypothesis that COVID has a mitigating effect on the home advantage as Goal Difference is estimated to decrease in times of COVID, holding all other variables equal. A robustness check is done by adding country-specific dummies that tell to what country (or Europe) the competition the match is played belongs. The results from this robustness check are qualitatively the same as discussed here. It should be mentioned that there are some quantitative changes. The results from these robustness checks can be found in the Appendix under Robustness Checks. Regarding the country dummies, the same holds for the estimations with COVID, in which also all country estimates are positive and reference country England thus has the smallest home advantage. Moreover, Scotland consistently has the biggest estimate in the estimations without COVID. An additional robustness check for a selection of the model specifications as in the No COVID section, is done. The results from these Robustness Checks are qualitatively the same as discussed here, with some small quantitative changes.

From this qualitative analysis of team chemistry in football matches onto predictions of football matches. In Fig. (4), graphs for the accuracy achieved by different prediction models are shown. ¹ From these figures, the threshold interval can be determined. There is chosen symmetric threshold intervals as these are more convenient to use. To determine which threshold interval is optimal to use is not straightforward. It usually holds that the smaller the threshold interval, the higher the accuracy. All subfigures in Fig. (4) support this statement. The trend is downwards, which means the closer the threshold values to zero (the smaller the threshold interval), the higher the accuracy. However, predictions should be representative. The smaller the threshold interval, the fewer draws are predicted. Therefore the range of the threshold intervals is chosen to be between 0.1 and 0.3. Here the prediction models still perform well in terms of accuracy and predict a reasonable number of draws. In this section, the performance of two baseline prediction models will be compared to prediction models with the different chemistry measures and possible interaction effects included (chemistry models). The accuracies in Fig. (4) are calculated with 5-fold cross-validation. The confusion matrices are calculated based on one of the five folds from the cross-validation. The prediction model is estimated from 80% (35733 matches) of the data and then applied to the remaining 20% (8933 matches) to get predictions. Then these predictions are compared to the actual outcomes of the remaining 20% and the confusion matrix can be constructed. In Fig. (4a-4c), it is clear that the baseline models outperform the chemistry models with chemistry based on nationality for threshold intervals of (-0.2,0.2) and bigger, although the difference becomes smaller when the interval gets close to (-0.3,0.3). For the intervals between (-0.1,0.1) and (-0.2,0.2), the baseline models and the chemistry models do not differ much in performance on accuracy. Including chemistry based on nationality in the prediction model does not improve the model’s performance, but it could, for smaller threshold intervals, be of interest in predicting football matches. In Fig. (4b and 4d) a different story displays itself. All different models have very similar performance and the chemistry models even sometimes outperform the baseline models. The chemistry models based on the region with interaction effects with position and COVID taken into account perform outstandingly for threshold intervals up to (-0.2,0.2). For chemistry based on region and calculated with both the SCI and the largest eigenvalue method, the accuracy is the largest compared to the other models. The red lines indicating these models are the highest for most of the threshold intervals researched. The other models also perform quite well and to eventually call one of the six the best would be rash. Still, the chemistry models, for a large part, outperform the baseline models with chemistry based on region. Including these chemistry measures in prediction models can help improve the performance of predictions. To show how a threshold interval is chosen, the confusion matrices for the models with ’optimal’ threshold intervals of (Fig. 4b) will be displayed below. In Fig. (4) the purple (baseline) and blue (COVID baseline) lines show the performance of the baseline models over different threshold intervals. After investigating closely for the baseline model, a threshold interval of (-0.21,0.21) is chosen. Since the accuracy increases in comparison with a slightly smaller threshold interval of (-0.2,0.2) and decreases with a slightly bigger threshold interval of (-0.22,0.22), this is a local maximum. For smaller intervals, the accuracy increases, but the number of predicted draws decreases. Since the number of predicted draws is 1570 (17.6%) and the actual number of draws is 2147 (24.0%), this interval is chosen as the optimal one. Draws are

The model that is used for predictions is smaller than in the Estimation section as the accuracy become too low if all control variables are used in the prediction model. In the prediction model only a constant term and Home and Away Position are added as control variables. A Two-Way Fixed Effects model without country dummies is used for the predictions. somewhat underrepresented in the predictions but not too dramatically. The corresponding accuracy for this test set is (1551+475+3067)/8933 = 0.570 (Table 11 or 57.0%. For the COVID baseline model, a threshold interval of (-0.2,0.2) is chosen as there is a local maximum. Moreover, the number of draws that are forecast using this threshold interval is 1521 (17.0%), while the actual number of draws is 2147 (24.0%). Again, draws are somewhat underrepresented. The difference is not too drastic so this interval is chosen as the optimal one in this case and achieves an accuracy of (1568+459+3074)/8933 = 0.571 (Table 12) or 57.1% on the used test set. For the chemistry models, the chemistry measure based on region and calculated with the SCI is used. In Fig. (4b), the yellow line shows the baseline model with chemistry added to it. The yellow line first decreases fast and then almost stops decreasing to decrease faster. In this segment of - almost - no decrease, the optimal value can be found as there the accuracy does not suffer too much from choosing a wider threshold interval. The threshold interval used is (-0.21,0.21). The number of predicted draws is 1561 (17.5%), whereas there are actually 2147 (24.0%) draws. There are not too few draws and still, a reasonable accuracy of (1550+477+3072)/8933 = 0.571 (Table 13) or 57.1% is achieved on the test set. The green line represents the model when adding COVID to the model used to get the yellow line discussed above. The optimal threshold interval of (-0.19,0.19) is a local maximum. Using this threshold interval on the test set leads to an accuracy of (1581+443+3087)/8933 = 0.572 (Table 14) or 57.2%. The number of predicted draws is 1442 (16.1%), which is somewhat lower than the 2147 (24.0%) actual draws. Still, this seems to be the optimal interval to use. The chemistry models can be extended by including the interaction effect of chemistry with the position for both the home and away teams in the model. The confusion matrix of these models with the optimal threshold intervals is in table 15 and 16. The optimal threshold interval for the chemistry model without COVID, but with interaction effects included and chemistry based on region and calculated with the SCI, is (-0.2,0.2). The light blue line represents this model in Fig. (4b) and the chosen interval is a local maximum. The predicted number of draws is 1491 (16.7%), while the actual number of draws is 2147 (24.0%). The predicted number of draws is too low, but still enough to call it a realistic prediction. The model achieves an accuracy of (1572+455+3081)/8933 = 0.572 (Table 15) or 57.2%. The red line in Fig. (4b) represents the model after adding COVID. There is no local maximum, but there is a segment of almost no decrease and there, the threshold interval is optimal. This happens right before a sharp decay of the curve at 0.22. The optimal threshold interval is thus (-0.22,0.22). For this interval, the number of predicted draws on the test set is 1621 (18.1%), while the number of actual draws in the test set is 2147 (24.0%). Draws are underrepresented but not enough to throw this prediction interval away. The accuracy achieved on the test set is 1545+504+3062 8933 = 0.572 (Table 16) or 57.2%. What strikes attention from table 11-16 is that home wins are predicted quite accurately. For example, in Table 16 the number of accurately predicted home wins is 3062 or 75.3%. The number of accurately predicted away wins is 1545 or 56.8% and still reasonable. The number of accurately predicted draws is just 504 or 23.5%. Moreover, as mentioned, the number of predicted draws is just 1621 (18.1%) in this case, whereas there are actually 2147 (24.0%). As can be seen, the number of predicted home wins (4757 or 53.3%) is much bigger than the number of actual home wins (4066 or 45.5%). In contrast, the predicted number of away wins and the actual number of away wins are comparable, 2555 or 28.6% predicted versus 2720 or 30.4% actual. To get rid of this imbalance, the threshold interval for this specific example has been altered. The confusion matrix in Table 16 suggests that the number of predicted home wins should be reduced in favor of draws. With this in mind, the threshold value is changed. By trial and error, an upper bound of the interval of 0.44 has been found to predict the most comparable amount of home wins. The number of away wins was, however, too low, so the lower bound is moved up to produce a balanced prediction. Once again, by trial and error, the lower bound for the threshold interval in this example is established to be -0.17. This interval gives the confusion matrix in Table 17. As can be seen in the table, the number of predicted home wins (4082 or 45.7%), draws (2149 or 24.1%) and away wins (2702 or 30.2%) are approximately equal to the number of actual home wins (4066 or 45.5%), draws (2147 or 24.0%) and away wins (2720 or 30.4%). The percentages of the actual number of home wins, draws and away wins add up to 99.9% due to rounding. The number of accurately predicted home wins is 2745 or 67.5%, the number of accurately predicted away wins is 1607 or 59.1% and the number of accurately predicted draws is 650 or 30.3%. The asymmetric interval has made the predictions more realistic, although the prediction of draws is still challenging. An accuracy of (1607+650+2745)/8933 = 0.560 or 56.0%. This is a tiny bit lower than the accuracy achieved with the symmetric interval (-0.22,0.22) of 57.2% and illustrates the problem that is faced. An asymmetric threshold interval gives more realistic predictions but is outperformed by symmetric intervals that give slightly less realistic predictions. Finding the optimal asymmetric threshold interval is an interesting subject that can be studied in further research. As can be read off table 11-16 there is an obvious difference between predicting home wins, draws and away wins. The models seem to be comfortable predicting home wins with around 75% of the actual home wins that the prediction models correctly predict. Away win prediction is somewhat more difficult for the models, but still, approximately 57% of the actual away wins are correctly predicted by the prediction models. The models struggle clearly with predicting draws, with just roughly 21% of the draws being predicted correctly by the prediction models. Even with fine-tuning of the threshold interval, just 30.3% of the draws are predicted correctly by the prediction models. As accuracy is a weighted average of those percentages, the models can perform very differently in terms of predicting a certain outcome. This is checked by creating accuracy graphs for the three different outcomes (Figs. 5, 6, 7 and 8). From the figures, there are, apart from some small differences, no clear differences in specific outcome forecasting. This means that regarding the outcome, the models perform similarly in terms of predicting specific outcomes. There is no considerable difference between the chemistry measures either. However, the (b) and (d) parts of (Figs. 5-7) suggest that the region chemistry measures perform slightly better than the chemistry measures based on nationality. The lines representing the chemistry models lie a little higher relative to the baseline models with the region chemistry models than the nationality chemistry models. This is confirmed when these lines are summed up according to their weights in Fig. (4) where the region chemistry models clearly outperform the nationality chemistry models.