Approximating the Yardage of an Interception in the Aggregate

So, it is established that an interception is valued at about 60 yards, an increase from the 45 yards it was previously ascribed. The model used to establish that value of 60 yards is one of several based on expected points (EP), this one developed by Brian Burke. EP models provide the average amount of points expected to follow a play on a given down, with a given distance to go, and from a given spot of the ball, based upon the average next-score for all previous plays with a given down, distance, and field position. The next-score may occur on the forthcoming play, the following play, or on a play three possessions later. The next-points could be scored by the offense (TD, FG), the defense (INT or fumble return for TD, safety), or maybe the defense's offense (following TO, turnover on downs, punt or otherwise). Positive EP values indicate offensive success and negative values, failure. 

Estimated points can then be converted to yardage values. For instance, 1st and 10 at the 50-yard line carries +2.0 EP. That indicates that, across many, many 1st and 10 situations at the 50, in many games, and in many seasons, the average next-points equals +2.0 EP. Likewise, 2nd and 8 from the opponent's 48-yard line will carry its own EP.

Anyhow, the value of 60 yards is derived from the expected point difference between interception plays and non-interception passing plays, across teams in many seasons. That is, the average EP for all non-interception passing plays from all down, distance, and field position situations compared to that average for all passing plays resulting in an interception. The EP difference between non-interception and interception pass plays is +3.8. For the offense, the EP value of +3.8 is equitable, roughly, to having the ball at the opponents’ 20 yard line. EP for the offense equal zero when the ball is at an offense’s own 20 yard line. The difference of the two field positions is 60 yards, hence the 60-yard valuation of interceptions.

I used Burke’s work to guide my development of team-specific, aggregate yardage values for interceptions. Of course, I sought a yardage value individualized for each team and, accordingly, EP for given down, distance, and field position situations are known to vary by team for teams' offenses and defenses. However—foreshadowing—computation of within-season team EP values is confounded by a small sample size. Anyways, like EP, an interception yardage value should also vary by team—but how? Well, these were my criteria: INT-yardage should
 

Figure 1.

• increase for teams with productive offenses (as measured by yards per drive and plays per drive), 
• increase for teams with efficient scoring offenses (as measured by % of drives end in a score), 
• decrease for teams that struggle to limit the productivity and scoring efficiency of opposing offenses (as measured by allowed yards per drive and % of drives ending in a score), and
• decrease for teams turnover prone offenses (as measured by % of a defensive team's offensive drives ending in a turnover).

To start, I sketched a football field and partitioned the field according to defensive drive start and opponents yard per drive. We can sum these two values, which I did for all NFL teams from 2010 through 2014. However, we then subtract that value from 100 ensuring a greater value for defenses that allowed fewer yards. This process, I was certain of and we’ll term it the Defensive Component.

I was less certain of how to include team offensive starting position and offensive yards per drive. Rather surreptitiously, I summed those two values with the Defensive Component above and noticed that the mean across team seasons equaled 100.044. I got excited and termed it the Offensive Component.

I simply subtracted 40 from the sum of the Defense and Offensive components. Voila, the average was 60.044.  Normality for the entire data set appears in Figure 1 with the distribution plotted in orange against a normal curve. In Figure 2 we see that the average of 60 yards is stable across all seasons in the data set, as are the centralities of the standard deviations. 

Figure 2.

The greatest yardage value was 73, for the 2012 Denver Broncos and 2011 New Orleans Saints and the least was 45, for 2014 Oakland Raiders and 2011 Indiana Colts. By several metrics (this one and this one),  Denver ranked among the top offenses and defenses, the New Orleans offense ranked top 1 or 2, Oakland amongst the lowest in offense and toward the bottom in defense, and Indiana relegated itself to the lower bounds offensively and defensively, in their respective seasons. 

Analyzed next were the relationships between interception yardage values and various team-offensive and -defensive attributes bulleted above. As seen in Figure 3 at the bottom of the page, most potent were the rate of offensive scoring drives increasing and rate of scoring drives allowed decreasing with increased yardage values.¹ Likewise, the rate of offensive turnovers decreased as interceptions yardage values increased. These associations are ecologically valid because only some interceptions are a pick-six—12.5%—and the defense depends on its offense to capitalize. Absent from Figure 3 is a small r = -.19 (p = .01) for the relationship between Defensive plays per drive and interception yardage.

Also, Pro Football Reference provides EP values for plays so we could use these to compute EP values for, say, the 2012 Denver Broncos.² Following Burke's methods, we compute the average EP for all non-intercepted, non-4th down passes thrown against the ’12 Broncos defense. Separately, we compute the EP for passes intercepted by the Broncos. Because it is such a small sample size, I also computed League-wide average EP on interceptions for 2012. Non-intercepted, non-4th down passes equaled +2.11 EP and intercepted passes equaled -2.73 and -3.02 EP for league and team averages,  yielding differences of +4.83 and +5.14 EP, respectively.

We could plot non-intercepted pass EP values by field position, as I did initially. Using only the ’12 Broncos’ data, EP is first less than zero at the Opponents’ 35 and EP is first greater than 4.31 at the Broncos’ 29 or so. That would make the INT yardage value 36. Meh....the perils of inadequate power. (For example, see this supplementary figure [green = non-INT] and compare it to Burke's graph in this write-up. The INT EP is League-wide 2012 averages. On my figure, that sharp dip in INT EP around the 95 is the result of insufficient.)

Figure 3.

So, as an alternative approach, I computed the average yard-line field positions for EP ≤0, ≥4.83, and ≥5.13 for non-interception pass plays. For non-interception pass plays against the Broncos D, the average EP ≤0 was about the 23-yard line. Using the difference of the League average INT EP, the average field position was 91-yard line (DEN 9). Using the difference of the Broncos team INT EP average, the average field position was the 94 (DEN 6). That puts the differences at between 69 (league) and 71 (DEN) yards. Recall, the value estimated above for the Broncos was 74; so we're fairly accurate, between +2 to +3 yards using the aggregate method I outlined above. But that's only one team.

It's a fairly simple process—obtaining the PFR EP—but I lack the motivation to check for all teams in seasons 2010-2014. I did check the 2011 Indianapolis Colts. Non-intercepted, non-4th down passes against the Indy' D yielded an average +2.19 EP; League INT, -2.5 EP; and Indy team INT, -2.8 EP. Those EP values equate in field position to ~26 (≤0), 82 (League), and 82 (IND). Thus, the difference equals ~56 yards, or, a difference of about 9 yards from the formula I described above.

Given that disparity, I also checked the 2014 Oakland Raiders. Non-intercepted, non-4th down passes against the Raiders defense yielded an average +2.11 EP; League INT, -2.73 EP; and team INT, -3.03 EP. Those EP values equate in field position to ~28 (≤0), ~78 (League and OAK INT). Those both yield yardage differences of about 50. This is only +5 yards than the 45 I mentioned earlier.

In summation I introduced here a method of computing team-specific yardage values of interceptions. It is well-established that that value is about 60 yards but that value is more so a League average, per se. So, I cross-checked the interceptions yardage values for several teams using estimates based on the EP differences (similar to the method used to obtain the 60 yard value). There were discrepancies between the results of my aggregate method and the EP estimates, two of which were |5| and one was +9. These discrepancies may be due to mathematical failures on my behalf or the crude computation of average field position differences for EP values. It could also be that Burke and PFR obtained EP values with different models. Of course, it could be some other issue that I have overlooked.
 





Footnotes:

¹The decimal values in Figure 3 are Pearson Correlations NOT Linear Regression Coefficients. This is because we are simply interested in relationships between variables (i.e., correlations) not explaining variance of the INT yardage values. However, the fuzzy line for each correlation is based on the linear regression equation. I included the line for convenience in visualizing the relationships.

²Some plays in the PFR lack EP data so those were removed from analysis unless it was a pick-6, in which case I made the EPA -7.