# Skill Curves in the NFL: Unlocking the Interactions between Passing & Rushing

The NFL has clearly turned to a pass-first league during the past decade or so, and for a good reason.  Passing is overall more efficient!  Indeed using play-by-play data from the 2014-2016 NFL seasons and the Winston-Sagarin-Cabot model for expected points, a passing play adds on average 0.25 expected points, while a rushing play adds on average only 0.06 points. This has led many people to condemn teams that still exercise a run-heavy game. In this work we revisit the importance (or not) of the rushing game, by examing the passing skill curve in NFL.  In particular, borrowing from Dean Oliver’s seminal work on basketball analytics we explore the efficiency of a passing play as a function of its utilization.  Dean Oliver identified that the efficiency of a basketball player (e.g., true shooting) declines with an increase of his utilization (e.g., fraction of total team shots taken by him).  Hence, the central hypothesis in this work is that passing efficiency exhibits a similar skill curve, which consequently means that we cannot blindly increase the number of passing plays and expect the same efficiency.

In order to explore our hypothesis we used play-by-play data from the 2014-2016 NFL seasons.  In particular, for every game in our dataset we calculated for each team the utilization of passing as the fraction of passing plays over the total number of its offensive snaps. We further calculated the average expected points added for the passing plays for each team and each game.  We have adjusted the expected points for strength of defense.  The following figure presents the results (binned for better visualization), where to reiterate passing efficiency is the expected points added per passing play.  As we can see there is a declining trend for the passing efficiency as we increase its utilization.

The correlation coefficient is $\rho = -0.37 ~ (p-value < 0.001)$.  These results, while they account for quality of passing defense, they do not account for the quality of the rushing game as well as the overall passing ability of the team that can impact the results.  Therefore, we build a regression model where the independent variable is the average expected points added per passing play $p$ (adjusted for defense) within a game, while the dependent variables include:

• Passing utilization $u$.
• Passing rating of the offense $\pi_{rtg}$. This captures the performance expectations of the passing offense.
• Average expected points added per rushing play $r$ during the game under consideration. This captures how well the ground game has performed during the game under consideration.
• An interaction term between $u$ and $r$.

The table above presents our results where we can see that the utilization is still negatively correlated with the expected points added per passing play.  The interaction term also shows that this correlation depends on the rushing ability of the offense.  In particular, the effect of passing utilization on its efficiency is $-0.85+0.67\cdot r$, namely, if the offense runs the ball better the negative relationship between $u$ and $p$ is less strong.  In particular, with $r=0.65$ (the maximum observed value in our dataset), the corresponding coefficient is -0.42 — compared with a coefficient of -1.33 for the minimum value of $r$ in our dataset, i.e., -0.73.

So how much should a team run?  Obviously the question depends on many factors but it should be evident that calling passing plays all the time is going to have diminishing returns.  While the passing efficiency might still be greater than that of rushing even when $u \rightarrow 1$, this does not mean that it is the best the team can do. What we a team is interested is maximizing the efficiency on a per-play basis regardless of the type of play, i.e.,

$\max_{u \in [0,1]} ~ u\cdot p + (1-u)\cdot r$

The following figure presents the passing utilization that maximizes the above equation for different values of $p$ and $r$.  As one might have expected for teams with better passing rating a higher utilization is recommended for fixed rushing ability, while better running game reduces the optimal passing utilization.  Note that a rushing EPA higher than 0.3 per play per game is rather unrealistic, and so is having $u=0$.  For the average rushing EPA (marked with the vertical line), the optimal fraction of passing plays is 0.3, 0.47 and 0.63, for a bad, average and great passing offense respectively.

I’d like to note here that the results of the analysis are not and should not be treated as causal, that is, running more does not necessarily cause passing to be more efficient.  It might as well be the case that teams that are trailing in the score turn to more passing and this bias the results. In some of my past analysis I have explored the possibility of similar reverse causality and there are not strong indicators for it. Furthermore, we have treated rushing as being constant regardless of its utilization.  While rushing skill curves are weaker as compared to passing the final results will quantitatively (not qualitatively) change. However, it should be evident that there is a clear interaction between passing efficiency and utilization that makes rushing still a piece of the puzzle in the NFL.