By Luca Caviezel '24
Introduction
It’s June 13, 2004; the Group Stage of the UEFA European Championships. Bitter rivals France and England are facing off and are tied 1-1 deep into stoppage time. It looks as if the game will end in a draw when France star Thierry Henry is taken down in the England box after pouncing on a loose pass from Steven Gerrard. The whistle blows–the penalty is awarded to France! Manager Jacques Santini is faced with a dilemma: should he allow Henri to take the penalty, being one of the best players in the world and the player who earned the penalty; or does he instead choose team captain and global superstar Zinedine Zidane to take it instead. It was a crucial moment in the game, as a goal would almost certainly seal the win for the French. In the end, Zidane stepped up to take the kick, and coolly powered the ball into the bottom-left corner of the goal after sending the keeper in the wrong direction. The stadium erupted in cheers as the defending European champions won the game at the last moment.
Jacques Santini made the right choice in having Zidane take the penalty kick, and in this particular instance both the player who was fouled (Henri) and the one who ended up taking the kick (Zidane) would have been great options to take the potentially game-winning penalty. But the question of whether the player who is fouled for the penalty should be the one to take it is one that has plagued soccer coaches and players at every level for a long time. One argument for the player who was fouled taking the kick is that they earned the penalty and therefore it is rightfully theirs to take. On the other hand, emotions might be heightened if they just had an easy goal-scoring opportunity stolen from them. Additionally, players skill levels and experience must be considered–older players might not be as unsettled by the intensity of the situation as younger ones. Many teams have a designated penalty taker, for example Jorginho for Chelsea FC, but what if the player fouled is the designated taker? All of this must be considered by coaches when deciding who should take the penalty.
There have been few previous studies that attempt to answer the question of whether the fouled player should take the penalty kick. Eichler (2002) analyzed 5 seasons of German top flight soccer and found that fouled players scored 12% more often than non-fouled players. Drösser (2003), on the other hand, analyzed 10 seasons of the German top division and found that fouled players scored 4% fewer penalties, giving a slight advantage to non-fouled players. Kuss, Kluttig, and Stoll (2006) found that there was no statistically significant difference between the penalty conversion rates for players who were fouled and players who weren’t, controlling for several variables including age and number of goals scored. In short, there is no consensus on what the correct approach is. Additionally, there have been no recent studies on the subject. This paper seeks to answer the question: should the player who wins a penalty be the one to take it? It will look at penalties taken in the English Premier League from the beginning of the 2018-19 season through the end of November of the 2021-22 season. I hypothesize that non-fouled players will have a higher penalty average than fouled players.
Data and Methods
The data was collected from the results section of the Premier League website. Game logs were filtered to find games that included a penalty kick, and the player fouled, player taking the kick, and whether the penalty was scored was recorded. Among other variables of interest was the time of the penalty, player’s team, opposition team, whether the game was Home or Away, and each team’s score at the time of the penalty. This dataset includes 349 penalty kicks across the 2018-19 to 2021-22 seasons (2021-22 season through November 30, 2021). Additionally, player-level statistics were collected from Football Reference to be used as controls, including age, career goals scored, career starts, career penalties taken and scored, as well as the accompanying season-level statistics. This data was then compiled to calculate season and career penalty scoring averages for each penalty taker. Finally, penalties occurring because of a handball were filtered out of the data, as there was no fouled player in these instances.
To determine whether the predicted relationship exists, I will be running a series of logistic regressions, some bivariate and some multivariate. I use logistic regression rather than linear regression because of the binary nature of the outcome variable. The goal is either scored or not scored, and logistic regression gives the probability of these outcomes happening. With linear regression, this probability could be greater than 1 or less than 0, a problem which is avoided by using logistic regression. These regressions take the form shown in Equation 1, where P(x) is the probability of the outcome variable occurring (from 0 to 1, where 1 means the penalty is scored and 0 means the penalty is not scored) and β0 and β1 are unknown coefficients that are estimated to best fit the equation. In this instance, β0 is the value of the log odds of the outcome variable when the dummy variable x1 is equal to 0 and β1 is the difference between the log odds of the outcome when the dummy variable x1 is 1 rather than 0. In regressions with multiple explanatory variables, more βnxn terms are added to the exponents of both e terms.
Using a little bit of algebra, this equation corresponds to the following (Eq. 2), where PS is the log odds that the penalty is scored, β0 is the intercept coefficient and is equivalent to the log odds of the outcome variable when the penalty taker is different from the fouled player, β1 is the slope coefficient and is equivalent to the change in log odds of the outcome when the fouled player takes the kick, and x1 is a dummy variable with a value of 0 when the fouled player does not take the kick and a value of 1 when he does:
The log odds of the penalty being scored can be interpreted using Equation 3, where p is the probability that the penalty is scored and log(odds) is equivalent to PS from Equation 2:
The key dependent variable of this project is whether the penalty is scored. This is a straightforward choice to determine the success of the player taking the penalty. The key independent variable is whether or not the player taking the kick was the one who was fouled, represented in the regression equation as a dummy variable that takes the value of 0 when the player taking the kick is not the player who was fouled and 1 when the player taking the kick is the one who was fouled. To ensure that this relationship is not spurious, I will then perform several multivariate regression tests controlling for other variables that might influence the outcome of the penalty kick. The first variable I will be controlling for is the penalty taker’s career penalty average, through the end of the season in which the penalty was taken. This variable is controlled for because players who are better at taking penalties should score more often, regardless of whether they won the penalty. Another performance statistic I will control for is career penalty attempts. A further regression test controls for whether the player was at their home stadium or at an opposing stadium. I will also control for the penalty taker’s age, used as a proxy for experience.
For all statistical calculations and visualizations, I used RStudio Version 1.3.1093, including the “tidyverse” and “pscl” packages.
Results
Logistic Regression 1: Regressing outcome of the penalty on whether the fouled player takes the kick
The first regression is bivariate, regressing the outcome of the penalty on the dummy variable indicating whether the player taking the penalty was the player fouled. This results of this regression are promising: statistically significant at a 90% confidence level (p = 0.0857), the log odds of the penalty being scored are 1.34 when the fouled player does not take the kick and 2.05 when the fouled player takes the kick. Plugging this into Equation 3, we find that the probability of the goal being scored when the fouled player takes the kick is 0.886 and 0.793 when someone else takes the kick, appearing to give a slight advantage to the fouled player.
Logistic Regression 2: Regressing outcome of the penalty on whether the fouled player takes the kick, controlling for age
When controlling for age, the results are similar: both age and the dummy variable indicating whether the penalty taker was the player fouled are statistically significant at a 90% confidence level (p = 0.00875 and p = 0.0502, respectively), and age is significant even at a 99% confidence level. This indicates that both variables have an impact on the outcome of the penalty kick: as the kick-taker gets older, the penalty is more likely to be scored, and there is an even greater positive effect of the kick-taker being the player fouled on the likelihood of the penalty being scored than in Regression 1.
To test the how well the overall model predicts the probability of the penalty being scored, I use a McFadden pseudo-R2 value. This tests the proportion of variance in the dependent variable that is caused by the independent variable(s) in a manner similar to a standard R2 value, except it can be used for regressions that are not linear, including logistic regression. The McFadden pseudo R2 is interpreted in the same way as a standard R2 value, having a range from 0 to 1 where 0 means that no variation in the dependent variable is explained by the independent variable(s) and 1 means all of the variation is explained by the independent variable(s). The McFadden R2 value for Logistic Regression 2 is 0.0381, meaning that the model explains very little of the variation in the independent variables.
Logistic Regression 3: Regression outcome of the penalty on whether the fouled player takes the kick, controlling for age, career penalty average, whether the penalty taker is in his home stadium or visiting an opponents’, and career penalty attempts
When controlling for these 4 variables, the effects of age and whether the fouled player takes the penalty kick become insignificant. Career penalty attempts and whether the player is Home or Away is also insignificant. The only significant variable in the regression is career penalty average, significant at a 99% confidence level (p < 0.01). The effect of career penalty average is positive, which, as is logical, indicates that as the penalty taker’s career penalty average increases, the likelihood of the penalty being scored increases. The McFadden R2 for this regression, 0.249, is still very low, meaning the model does not explain most of the variance in the outcome variable.
Discussion
While Logistic Regressions 1 and 2 seemed to indicate that the player who won the penalty has a higher likelihood of scoring it than players who weren’t fouled. However, when controlling for other variables, particularly career penalty average, this effect disappeared. Instead, career penalty average has the biggest impact on the outcome of the penalty kick. This could potentially be due to the fact that the fouled player might only take the penalty if they are one of the team’s designated penalty takers. A further experiment could investigate whether this is the case. Another potential explanation for this is that the coaches only allow the fouled player to take the kick in specific instances, including if they are one of the designated takers, as previously indicated, but also potentially including moments where the penalty might give the player a hat trick, and therefore the player could be entitled to taking the kick. Again, a further investigation of penalty situations in the English Premier League is necessary to resolve this question. Two important potential sources of error in this research project are the small sample size and high penalty averages. The overall penalty average for all penalties in the dataset was 81.5% which is extremely high, regardless of whether the penalty taker is the player who was fouled (Figure 1). This means that any effect that might exist will be very limited. Additionally, the sample size of 287 non-handball penalties is smaller than is desirable for a completely accurate statistical analysis.
In Logistic Regression 3, I initially planned to include season penalty average as an additional control variable. However, after running a correlation test between season penalty average and career penalty average, I decided against this because of the high correlation between the two variables (correlation coefficient = 0.669) at a 99% confidence level (p < 0.01). This could lead to multicollinearity which could skew the results of the regression, and so I had to choose only one of the two to include in the regression. One of the major flaws when controlling for these averages is many of the players have taken very few penalties and thus their averages can be highly skewed by one or two shots. When controlling for season-level statistics, the problem is even more apparent than for career statistics, as in each season, the median number of penalties taken per player is 4 or below, meaning 50% of penalty takers took 4 or fewer penalties every season (Figure 2). Because of this, I chose to control for career penalty average instead.
Finally, there might be some correlation between a penalty taker’s age and the number of penalty kicks he has taken across his career because logically, older players should have taken more penalties than younger players. I ran a correlation test between the two variables and found that there is high correlation (coefficient = 0.649) at a 99% confidence level (p < 0.01). Thus, there is a high likelihood of multicollinearity between these two variables in Logistic Regression 3. Therefore, I reran the regression without controlling for career penalty attempts to avoid this multicollinearity, which had no significant impact on the relationship between the independent and dependent variables. The McFadden pseudo-R2 changed slightly, from 0.249 to 0.245, but none of the explanatory variables changed in statistical or substantive significance levels.
Conclusion
Without controlling for other variables, whether the fouled player takes the penalty kick appears to have an impact. However, when controlling for other variables, this effect disappears, and career penalty average instead is the most significant determinant of the outcome of the penalty, with a higher career penalty average indicating a higher likelihood that the penalty is scored. This tells us that there is no significant difference on the probability of the penalty being scored when the fouled player takes the kick instead of another player on the team. This is an important finding for players and coaches in important penalty kick scenarios, indicating that penalty takers should be chosen based on how successful they have been at penalty kicks in the past rather than allowing the player who won the penalty to take the kick.
References
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Drösser, C. (2003). “Stimmt’s? Verschossen.” Die Zeit, 32, 25.
Eichler, C. (2002). “Lexikon der Fußballmythen.” Frankfurt am Main: Eichborn.
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Kuss O, Kluttig A, Stoll O. (2007). "The fouled player should not take the penalty himself": an
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