As a football game progresses, players get more and more tired. While players are usually fresh at the beginning of a game, they are usually fatigued by the end after having played for hours. Even though we know that NFL players get tired as a game goes on, we do not know how their fatigue impacts their performance, mainly for quarterbacks and running backs.
The Impact of Fatigue on QBs
Quarterbacks are the most used players in a football game. They touch the ball on every play and are almost never subbed out barring injury. Especially in games where a team is trailing, quarterbacks have to throw the ball up to 30 or 40 times a game, avoiding the pass rush, moving in the pocket, and finding open targets. One would think that an increased usage for a quarterback would cause him to be less efficient, and at first glance, this seems to be true. After plotting the pass number of a quarterback (a QB’s first pass would be a pass number of 1, his 20th pass would be a pass number of 20) in a game against the average yards per play, there seems to be a negative correlation between usage and efficiency.
While there initially seems to be evidence that quarterbacks get less efficient as a game progresses, a deeper look actually shows that this is not the case. For each pass, there is a certain probability of a completion based on the depth of the target, whether the receiver is in the middle of the field, how much time is left, and more. Using Ben Baldwin’s model for expected completion percentage, we can find the average expected completion percentage and completion percentage over expected based on the pass number for a quarterback in a game.
From the graphs above, we can determine the actual reason for why quarterbacks average fewer yards per play as they throw more passes. The reason is not accuracy, since the pass number in a game and the average completion percentage over expected have no correlation, but it is rather that quarterbacks attempt more difficult passes further down the field as they pass more and more.
There is a negative correlation between expected completion percentage and pass number and a positive association between pass number and average depth of target (in yards), meaning that quarterbacks often attempt shorter, easier to complete passes early in a game and more difficult passes down the field later in the game. The reason is likely because quarterbacks are trying to score more urgently towards the end of a game in order to tie or take the lead. Therefore, we can conclude that quarterbacks do not lose efficiency in passing during a game, but rather just get more aggressive.
The Impact of Fatigue on Running Backs
Just like quarterbacks, running backs are also used a lot in NFL offenses and are likely to be less efficient as a game goes on due to fatigue. In order to investigate these affects, I used a similar approach of plotting the rush attempt number against the average yards.
From the graph above, we can see that there is a negative correlation between rush number and average yards per rush. Running backs often start games averaging close to 4.75 yards per carry, but on a running back’s 20th carry, the average falls to around 4.1 yards per carry.
Using a linear regression to predict rushing yards based on a running back’s rush number, the slope was -0.034 with a p-value of 0.0000000002. This means that rushes average 0.034 fewer yards every carry for a running back, and the extremely low p-value indicates there is a very high chance that an increased number of rush attempts leads to lower efficiency.
Because an increased number of rush attempts leads to a lower average yards per carry, NFL teams should try to use multiple running backs and split carries between them to maximize rushing efficiency. By having multiple running backs split carries, a team would never get into a situation where one running back has a high number of rush attempts, reducing the chance that their rushing efficiency suffers due to fatigue. Of course, this does not apply to certain teams with exceptional running backs, like the Browns and Titans (explained further in the next seciton).
Unfortunately, there was no way for me to break down the rushing yards using expected rushing yards or yards after contact because they are not included in play-by-play data, so it is impossible to know for sure if the decrease in rushing efficiency is due to worse offensive line play or worse running back play.
Ranking RBs in Efficiency and Stamina
Using the average yards per rush and the correlation between average rushing yards and rush number, we can find which running backs are the most efficiency while getting even better as a game goes on. The graph below displays these values for all players with at least 300 rush attempts between the 2018 and 2021 NFL seasons. The players located in the top right of the graph are the best running backs since they are both efficiency and get better as they are used more. Those in the top left are inefficient on average but get better with more usage. Those in the bottom right are efficient but get worse as they are used more, and players in the bottom left of the graph are the worst runners since they are inefficient and get even worse as they rush more.
The best two running backs in the NFL in terms of rushing efficiency and stamina are Derrick Henry and Nick Chubb. Both players average at least 5 yards per carry and tend to average more yards as they run more. Meanwhile, players like Josh Allen, Lamar Jackson, and Raheem Mostert are very efficient, averaging close to or more than 6 yards per carry, but get worse and worse with more carries.
The worst running backs are Le’Veon Bell, Mike Davis, David Montgomery, and Josh Jacobs. Each of these players average only about 4 yards per carry and get worse with more usage. As seen from the graph below, each of these players are above average NFL rushing efficiency for their first 5 carries at most (where their line crosses the dotted red line for the average).
Comparing Running Backs
Another useful way to utilize the relationship between rush number and efficiency for each running back is to compare running backs on the same team to see how teams should use their players. For example, we can explore how Nick Chubb and Kareem Hunt of the Cleveland Browns perform as their usage increases. From the graph below, we can see that Chubb gets better further in the game, while Hunt is only efficient for his first few carries. Therefore, the Browns should elect to use Hunt more often for rushing plays earlier in the game, then lean on Nick Chubb more heavily later in the game.
We can look at similar scenarios for other teams such as the Buccaneers and Jaguars. For the Buccaneers, Ronald Jones actually gets far better with more attempts while Leonard Fournette’s efficiency decreases slightly, so they should use Fournette earlier in the game and use Jones more often in the latter parts of a game. For the Jaguars, James Robinson is actually more efficient than Carlos Hyde no matter how many rush attempts he has, so the Jaguars really should try to use Robinson as much as possible and they should only use Hyde briefly.
The last team’s running backs that I chose to analyze was the Titans. The Titans are in a unique situation after the Derrick Henry injury. Henry was the source of most of the Titans’ offense, as the Titans relied heavily on the run game and play action to gain yardage and score. With Henry out for possibly the rest of the season, it will be interesting to see how Tennessee’s offense will adjust.
Even though the Titans signed Adrian Peterson, their run game will likely be far less efficient without Henry. Peterson averages less than the NFL average yards per carry, and has almost no association between rush number and yards per carry. Meanwhile, Derrick Henry was always more efficient than the NFL average yards per rush, and he got even better as he run more. In fact, Henry only averaged about 4.5 yards per carry on his first rush, but by his 30th rush attempt, he was averaging almost 6 yards per carry. One thing is for sure: the Titans run game will be a lot worse without Derrick Henry.
In general, football players get more tired as a game progresses. This is seen by both quarterbacks and running backs at first glance. The average yards per pass attempt for quarterbacks seems to fall as the quarterback’s pass number in a game rises. However, after further investigation, this falls is caused by increased aggressiveness as the completion percentage over expected remains the same throughout the game for quarterbacks while their average depth of target increases, on average.
Meanwhile, running backs do become less efficient as they rush more often. Using a linear regression, it is estimated that the average yards per carry falls by 0.034 yards for each rush attempt. This means that after 30 carries, the average runner’s yards per carry decreases by over a full yard! Because running backs lose efficiency with high usage, NFL teams should utilize a running back by committee approach, which means that they rotate several running backs instead of leaning on one, unless they have a premier running back like Derrick Henry.
The two best running backs at being efficient while also getting better with more carries are Derrick Henry and Nick Chubb. The worst are Le’Veon Bell, Mike Davis, Josh Jacobs, and David Montgomery because they their average yards per carry is lower than the NFL average and their efficiency decreases with more usage. In addition, the trend between usage and efficiency can help to determine how teams should use their running backs. The Browns should run with Nick Chubb more often than with Kareem Hunt because Chubb gets better with more carries while Hunt doesn’t. Meanwhile, the Jaguars should try to always use James Robinson because his average yards per carry is better than Carlos Hyde no matter how many carries he has. Overall, though, NFL teams should try to use running backs by committee instead of a feature running back (unless it is Derrick Henry or Nick Chubb).