Find Part 1 - Offense, here.
Offense is easy to quantify compared to defense. This isn't just true in football, but also baseball (where you practically need an advanced math degree to understand the defensive inputs to WAR) and basketball (where player tracking data and plus-minus are the best ways to quantify value). It's no different in football. In an ideal world, players would rotate enough on the field, to get meaningful samples of the team's performance whether they're on the field or not to do a With or Without You (WOWY) analysis for every context--standard downs, passing downs, etc. Most teams don't rotate enough for that, and we don't have play-by-play data on that level anyways. All we can work with is are the standard box score stats:
Assigning Value for Defensive Statistics
The question then becomes, how valuable are each of these acts to a defense? We know from the offensive analysis others have done that interceptions cost offenses an equivalent of 45 yards, so each interception should be worth that much to the defender that intercepts the pass. using this as a baseline, I can determine relatively easily a "yardage value" for PBUs, FFs, and FRs.
In DIII, the ratio of PBUs to INTs is right around 3:1, so it would make sense that a PBU should be worth about 1/3 of an INT, or 15 yards. This makes sense logically, too. The average completion in DIII goes for 12.5 yards--add in the value of first downs and TDs, and a typical PBU actually prevents the offense from gaining around 15 Total Adjusted Yards.
For fumbles, the thought process is similar. If an interception is "worth" an equivalent of 45 yards, a fumble should be worth a similar amount. Unlike an interception, though, where one defender accounts for whole process of the interception (I'm a former D-Lineman, and yes, I'm aware of the faulty logic at play there), a fumble requires someone to force the fumble and one to recover it. Also, only 50% of forced fumbles result in a recovery for the defense. So the cumulative value for two FFs and one FR should be equal to 45 yards, so I valued FFs and FRs at 15 yards apiece.
All that remains are the tackling stats. For tackles, TFLs, and sacks, an assist is worth half as much as a solo. I explained in my OL YARP explanation that the average sack is six yards deep. If you assume that every sack prevents a pass attempt (obviously the QB could scramble and run), sacks should be worth the value of the pass attempt and the lost yardage (plus a little more, because I assume guys who get the sack are also going to have some QBHs unaccounted for in official sacks). The same logic applies to a non-sack TFL, but the average rushing attempt is a little shorter on average, and the average TFL is only 4 yards behind the line of scrimmage, but the alternative result is going to be better than average. Sacks are excluded from the denominator for YPA, but TFLs are included in the denominator for all plays, so the alternative result is going to be the average gain on non-TFL plays.
The last piece of the puzzle, tackles, is the most common statistic for a defender, but probably the hardest to assign value to. On a single play level, what is the value of a tackle? We don't know anything about the play, other than it wasn't a TD, INT, or PBU--even on a FF, the player is still credited with a tackle. For the other non-turnover stats a defender can earn, it's easy to give credit to a sack, TFL, PBU, etc. based on the expected outcome if that didn't occur. No sack? Average pass attempt. No TFL? Average non-TFL rushing/passing play. No PBU? Average completion. What's the average result of a play without a tackle? Well, if nobody makes a tackle, it's either an incompletion of some sort, or a TD, but someone else could have made that tackle. Some tackles are more critical than others--a missed tackle on a deep pass is probably going to be more costly than a missed tackle on ISO. So how do we assign value to tackles? What I decided to do was take the average value of plays that don't end tackles, excluding incomplete passes, and subtract the value of all plays that did end in a tackle. The result is that an average tackle is "worth" about three yards per play.
Re-listing the stats, with their respective yardage values, we now have:
Solo Tackles = 3
Assisted Tackles = 1.5
Solo TFL = 15
Assisted TFL = 7.5
Solo Sacks = 25
Assisted Sacks = 12.5
Pass Breakups = 15
Interceptions = 45
Forced Fumbles = 15
Fumble Recoveries = 15
Add up all of a player's stats, and divide by the number of total plays their defense faced, and you have a player's defensive TAY/P. Just like with offensive players, the determining factor I used for determining All-American teams is TAY/P per game.
Adjusting for Team Quality
I think it goes without saying that it's very difficult to account for all of a player's value with their box score stats alone. A lock-down corner isn't going to have as many INTs or PBUs as would be indicative of his true value, because he's not going to have the ball thrown his way a lot. A block-eating nose tackle isn't going to have as many sacks, TFLs, or tackles as a dynamic 3-tech. As such, I decided to give players credit for one-eleventh of their team's defensive TAY/P relative to the national average.
The thought is to give a player partial credit for their team success, one-eleventh (because there are eleven guys on the field at a time, duh). The average national average TAY/P is 7.5, so the equation for a player's individual contribution to their team's TAY/P is:
[Player Contribution] = ([Avg TAY/P] - [Team TAY/P]) / 11
Combine those two factors to the player's box-score stats, and you have (what I believe is) a better estimate of the player's TAY/P.
Adjusting for Opponents
If you read Part 1 of this series, you already know how I do my opponent adjustments. The national average TAY/P for all plays is 7.5, so the equation to develop Opponent-Adjusted TAY/P (or TAY/P+) is:
[TAY/P+] = [TAY/P] + ([Opponent Off. TAY/P] - [Avg TAY/P])
Setting Replacement Level
The replacement level for defensive players' TAY/P+ isn't exactly the same as for offensive players. For quarterbacks, I decided to set the replacement level so that the 250th-best quarterback or running back was the replacement level, so that in total, 1250 skill players (250 teams, 5 skill players per team) should be above replacement level. For defenders, the replacement level of production should be approximately the 2750th-best defender (250 teams, 11 defenders per team). Also, for offensive players, the rate is determined according to opportunities--routes run for WRs, pass attempts for QBs, and rush attempts for RBs--but for a defender, an opportunity is every play, so the replacement level has to be lower than for offensive players.
Because of those two factors--more opportunities and more players above replacement level--I've decided to change the standard for replacement level in two ways. Because of the increased number of opportunities (essentially double the opportunities of an offensive skill player), and the increased number of players above replacement level (essentially double the number of players as offensive skill players), the replacement level is one standard deviation below average, divided in half, which comes out to 3.0 TAY/P.
Using Subjective Measures to Augment Stats
Even though it pains me to use any subjective measure like All-Conference or All-American teams, for defenders I think it's as good a way to account for skill as any. For getting voted onto an All-Conference team, I gave a boost of 2.5 YARP+/Gm, and an equal boost for being voted onto the D3Football.com All-Region teams (only one player, Grinnell's Ryan Slager was voted All-Region without also being voted All-Conference; Ryan was Second Team All-MWC South Division, which I gave half credit for being All-Conference). The boost for getting on All-whatever teams isn't super large--it's only the equivalent of one or two sacks over a ten game season--but it did make a difference in the cutoff point for a lot of the "Hansen Ratings All-Conference" teams.
Stay tuned for Part 3 of this series, where I detail my methodology for determining TAY/P+ for punters, kickers, and returners.