Imagine you are the head coach of a soccer team and you only care about not losing (maybe the fans will ask for your head if you lose). With this philosophy, you want to know what’s worse, receiving a goal or a red card.
To answer this question I used data of approximately 4000 matches of European leagues in which there was at least one goal or a red card.
Surprisingly, at least for me, 70% of the matches, with at least one goal, were won by the team that scored first, and 52% of the times where a team received a red card, while the game was tied, ended up loosing the match. With this we could probably assume that is better to receive a red card rather than a goal, but the impact of a red card is not the same at the beginning of the game that on the last minute.
Then is natural to assume that the impact of a goal or a red card depends on which minute of the match the event occurred. This relationship could be analysed using linear models for both cases. For simplicity, I will assume that all the matches finish after 91 minutes.
The following chart shows how two different models assigned the probability of not loosing a match after receiving one goal against,if the match was tied before that goal.
For example, if you are losing 1-0 on the minute 31, the probability of not losing is around 0.32. The main difference of the models is shown after minute 50, mainly because the linear regression model used has an intercept value of 0. (Since is logical to assume that if you receive one goal on the last instant of the match you have probability 0 of not losing)
A similar analysis is run for red cards, but here the assumption of an intercept value equal to 0 doesn’t apply.
The last chart shows the probabiilty of not losing depend when does it happens. The curves intersect on the minute 25, meaning that is the same to receive a red card than receiving a goal.
The gray box represents the area where there isn’t statistical evidence to assume that one event is least worse than the other (this was concluded using bootstrap for both models)
This means that before minute 12 is better to receive a goal, and after minute 42 is better to take a red card than a goal.