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## The final value of a passing touchdown

This week, I've been writing about the value of touchdowns scored on different downs. One of the problems with the data, though, is that the sample size isn't very large for each down. There is some down-to-down variation that exists in the data that doesn't make a lot of sense, and is probably a result of a small sample size. Further, I think the numbers should be consistent -- your odds of getting a 1st and goal from the one should be the same as your odds of getting a 4th and goal at the one (and in fact, the numbers say they pretty much are). Your odds of fumbling should be the same, too.

One of the problems with the results from Wednesday's post is that a first down touchdown is undervalued -- that is, it says the situation of being in 2nd and goal from the one is worth 5.50 points. But according to theory that we're very confident in, 1st and goal from the one is worth 5.55 points.There should be a greater spread than that. So here's what I did.

For every play at the one yard line (designated in the table below as 1st, 2nd, 3rd or 4th and goal), a team has a 55% chance of scoring a touchdown, a 2% chance of a turnover or an interception, a 12% chance of a loss of yards, and a 29% chance of no gain. A touchdown will be worth 6.4 points, a fumble is worth 0.9, and an interception -0.25. A play of "no gain" is worth whatever the next sitaution is worth. So no gain on 2nd and goal at the one is worth 4.746, since that's the value of having 3rd and goal at the one. A loss is worth slightly less than 75% of a play for "no gain". For fourth down, there's considered a 50% chance of a field goal and every other situation is reduced by 50%. The table below sums this up:

1ST + G 2ND + G

TD 0.55 6.4 3.52 0.55 6.4 3.52

FUM 0.02 0.9 0.02 0.02 0.9 0.02

INT 0.02 -0.3 -0.01 0.02 -0.3 -0.01

FG 0.00 2.4 0.00 0.00 2.4 0.00

LOSS 0.12 3.9 0.47 0.12 3.5 0.42

NO GAIN 0.29 5.3 1.55 0.29 4.7 1.38

5.55 5.333RD + G 4TH + G

TD 0.55 6.4 3.52 0.275 6.4 1.76

FUM 0.02 0.9 0.02 0.01 0.9 0.01

INT 0.02 -0.3 -0.01 0.01 -0.3 0.00

FG 0.00 2.4 0.00 0.50 2.4 1.20

LOSS 0.12 2.4 0.28 0.06 1.2 0.07

NO GAIN 0.29 3.2 0.93 0.145 1.2 0.17

4.75 3.20

That 5.55 number reflects the value of 1st and goal from the one yard line -- which is what our theory predicts. So now a touchdown on a long bomb is worth 0.85 extra points (6.4 - 5.55), a touchdown on 1st and goal is worth 1.07 extra points, on 2nd and goal is worth 1.65 extra points, on 3rd and goal is worth 3.20 points and on 4th and goal is worth, still, 4.85 extra points. Using the numbers from Wednesday's post, this means the average passing touchdown is worth 1.325 extra points. If we convert that number to yards, that would mean each passing touchdown, on average, is worth 18.3 extra yards.

However, it's slightly more complicated than that. Sure we know all passing yards aren't equal -- but leaguewide, passing yards aren't evenly distributed. A pass from the 45 to the 50 isn't as valuable as one from the 5 to the end zone; we know that. But it's also true that the former pass happens very, very often, and the latter is relatively rare. In other words, lots of the passing yards that QBs get are of the less than average value variety. And if that's the case, than the average passing yard isn't as valuable as the yards on the field. And if *that's* the case, then a passing touchdown is even more valuable than we thought.

Doug looked at every passing play in 2007 that gained at least one yard. Then he individually looked at each yard it covered, computed the value of that yard, and added that value to a giant running total. If you divide that total by the total number of yards and you should get the league wide value of a typical passing yard. Remember 98 yards is worth 7.1 points, meaning 1 yard is worth 0.724 points. Well, according to Doug, the average passing yard is worth .0653 points and that one point is worth 15.3 "average passing yards". So we should be multiplying 15.3 times the 1.325 points the average passing touchdown is worth. In other words, a touchdown is worth 20.3 passing yards.

Except for one more point. As Vince pointed out on Wednesday, we should also be subtracting one yard from our total. Since we're measuring the point value from the 1 into the end zone, we should then subtract out one yard at the very end. So, for the last time for awhile, I'm going with 19.3 yards as the ***official*** value of a passing touchdown. It's still worth remembering, though, that a generic touchdown on a non-X-and-goal play, is still merely worth only 10.7 yards, or 13.0 "average passing yards".

This entry was posted on Friday, October 3rd, 2008 at 6:53 am and is filed under General. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.

"Remember 98 yards is worth 7.1 points, meaning 1 yard is worth 0.724 points"

.0724 points, not .724

.

I'm not quite getting the math on the value of a TD on 1st and goal. Before you hike the ball, your 1st and goal has a value of 5.55. If you score a TD, the value is 6.4. 6.4-5.55 = 0.85, so shouldn't the value of a TD on 1st and goal be 0.85 and not 1.07? I'd think the value of a TD on 2nd and goal is 1.07, and 1.65 on third and goal, and 3.2 on 4th and goal... I'm not sure where the 4.85 comes from but it feels very wrong. You've assumed they can get 2.4 points every time on 4th and goal, so how can getting 6.4 be worth more than 4 points?

.

I'd think fumbles/interceptions/Pass TDs would be more likely on 4th and goal because there's no incentive to throw the ball away or keep two hands on the ball. I'd think QBs would be more likely to throw it up rather than take the sack or throw it away on 4th and goal....

A goal-to-go play is a high leverage play: if you're successful (TD) you add a lot of value, but if you fail (e.g. no gain) you lose a lot of value. You're giving players credit for those successes (you can count their TDs) but you're not taking anything away for their failures (and you can't, since there's no way to tell from the box score how many failed goal-to-go plays they had). The only way I can see to punish them for those failures is to reduce the value of a TD, assuming that players with more TDs also have more failed goal-to-go attempts.

Using your probabilities, and crediting a player with 20.3 adjusted yards for a 1-yard TD, the average play the opponent's 1 is worth about 9.1 adjusted yards, which is much more than the typical play. One of the arguments for a TD bonus is that we don't want to reward or punish players for the situation (field position, down & distance) that they run plays in - we'd like every play to have about the same expected value in terms of adjusted yards. The 19.3 bonus is too big for that.

Mattie, I'll respond to your point in a bit.

Just wanted to write down here for archive purposes: Doug has the "average rushing yard" in 2007 being worth 0.0669 points, or 14.95 points per "average rushing yard". The average rushing touchdown in '07 was worth 1.407 points, based on the numbers and logic in the past three points. This means the average rushing touchdown was worth 21.03 yards, minus one yard. So the ***official*** value of a rushing touchdown is an even 20 yards. So the average rushing TD is about a yard more valuable than the average passing TD, since the average rushing TD is more likely to come on a X-and-goal down.

On the other hand, you could also argue that the 19.3 bonus is too small. The yards that are gained on a TD-scoring play are among the most valuable yards on the field. The average value of a yard on a non-TD pass must be a bit less than .0653 points, and the average value of a yard on a TD pass has to be considerably more. That implies that you should give a larger TD bonus in order to also credit players for the higher value of the other yards on the TD pass (instead of just crediting them for the higher value of that last yard).

Vince, one of us must be reading the other's mind. I literally just posed that question to Doug about the failure to account for the misses.

This is the part I don't understand...

.

"That 5.55 number reflects the value of 1st and goal from the one yard line โ which is what our theory predicts. So now a touchdown on a long bomb is worth 0.85 extra points (6.4 - 5.55), a touchdown on 1st and goal is worth 1.07 extra points, on 2nd and goal is worth 1.65 extra points, on 3rd and goal is worth 3.20 points and on 4th and goal is worth, still, 4.85 extra points."

.

I'm with you on the long bomb part. Looking at the numbers, the average result of a first and goal play is +5.55. So if we score a TD on 1st and goal, we exceeded the value by 0.85, so that's the value of the play, yes? No gain would be worth 5.33, so you lost 0.22 points by getting no gain. A TD on second and goal would be worth 1.07 (6.4 - 5.33). And so on. Scoring a TD is still 6.4 regardless of how you score it, so we're calculating the value of the play as 6.4 - initial value. The initial value of 4th and goal is 3.2, so the value of a TD on 4th and goal is 3.2 (6.4-3.2). And the value of a field goal would be -0.8 because you're going from 3.2 to 2.4. At least that's how I'm interpreting the chart. So I don't understand why you're saying a TD on 1st and goal is worth 1.07...

Hey Mattie,

What you have to do is not compare the value of the touchdown to the value of that down, but rather to the value of the *next* down. That is, we're using as theory the comparison of a touchdown or a play down to the one.

Think of a fourth down play. It's 4th and goal at the one. QB passes, incomplete. His team is now in a +1.55 sitution. Why? Because for the *offense*, being at your own 1 yard line on 1st and 10 is worth -1.55 points. So the QB's fourth down incompletion puts his team at +1.55; a fourth down TD would have put his team at +6.4. So the value of his fourth down

touchdownis 4.85.Keep in mind that the starting field position on a play is irrelevant. Consider fourth and goal from the 10. A play to the 1 puts the team in a +1.55 situation; a play to the end zone puts the team in a +6.4 situation. The value of the fourth down touchdown is still 4.85.

So you're always comparing the value of a touchdown on down X to the value of being in down X+1. By definition, a long TD must be worth more than a TD on first and goal. Why? Because we're comparing to a play that gets you down to the one yard line. In the former example, you're in 1st-and-G from the one; in the latter, it's 2nd-and-G from the one.

Okay, now I understand where the 4.85 comes from but not why you're comparing a TD to an incomplete rather than the average result of a 4th and goal play. What happened to interceptions in the end zone, or fumbles, or sacks? By that logic, you'd compare made extra points to a missed extra point. If you're going to do that, an extra point attempt is worth right around 1.0 and the kicker is the most valuable player on the team since they don't miss extra points.

In reality, every kicker makes what, 98% of his extra points? More? So we're saying 98% of that value is already there in the "and goal" situations, and a made extra point is really worth the value 1.0 minus the average value of an extra point attempt (0.98, 0.99?) so it's worth 0.01 or 0.02 points and a missed XP would be worth -0.99 or -0.98. So why arent' we doing this for passing plays? To my mind, a TD on 4th and goal is worth 3.2 and an incompletion would be worth -1.65. The difference between those two results is still 4.85 but now it's not all weighted on the positive end.

.

Lets say 1st and 10 from the 20 is worth 3.0 points. (arbitrary)

If we throw a TD, we just incresed our value from 3.0 to 6.4, so the value of that play is 3.4. That's partly for the TD and partly for the 20 yards. Your analysis suggests 2.55 for the 19 yards, then 0.85 for the TD+1 yard.

If we throw a 19 yard pass, then throw a TD on 1st and 1, we've "scored" 3.4 across two plays instead of 1, so they still need to add up to 3.4, right?

2.55 + 0.85 (if it was 1.07, a 19 yard and 1 yard pass would be worth more than a 20 yard pass)

If we throw a 19 yard pass, three incompletions, and a TD pass, it's still worth exactly 3.4 split up over 4 plays:

2.55 - 0.22 - 0.58 - 1.55 + 3.20

If a TD on 4th and goal was worth 4.85, that series of plays would be more valuable than either of the above, which makes no sense to me at all.

Hey Mattie,

Those are two different points so let me address them separately.

Your first point is correct. It's something I've spent a little bit of time thinking about, and I'm not sure what to do. Your XP example is a good one -- the problem with my system is it doesn't address difficulty level. Value and difficulty level often coincide, but not always, and this is probably a good example of that.

On the other hand, I think it's not a serious problem as long as our comparisons are intraposition. If we're comparing kickers to each other, it's okay if they all get 1.00 points for XPs instead of .02 points, as long as we're looking at rate numbers and not totals.

For QBs, the value of a touchdown, on average, is worth about 19.3 yards. Sometimes those touchdowns will be easy, sometimes hard. Sometimes a 10 yard pass is easy, and sometimes it will be hard. I think the only way to do it given the limited data available on older players is to look just at value creaeted. Keep in mind that when we do the best QB ever rankings, we are comparing all QBs to all other QBs, and that handles a lot of these concerns. At least, I think.

Mattie,

Your other point is also a fair one. Yes, a 4th-and-G TD is going to be more worth more than a regular TD. I understand that my system says 19 yard completion, INC, INC, INC, TD is worth more than 20 yard completion for a TD. But it's actually not. Why?

Because I'm giving 19.3 for all TDs. So a 20 yard TD on 1 attempt is worth 39.3 yards/attempt. A 19 yard completion, INC, INC, INC, TD is worth 39.3 yards, but on five attempts. That's a significant difference.

I guess I was thinking about it on a smaller scale, like having an algorithm to dissect QB performance on a play-by-play basis across entire seasons or careers to rank QB's. Or once one knows the value of each play, one could devise a system to distribute those points among the players involved to come up with something along the lines of win shares for all positions. But you're right, that would throw out most of the players since the per-play data isn't available except on relatively recent games, and I don't know if even that has substitutions listed unless they're directly involved in the play.

But if you're calculating the value of a TD and the values of pass TDs on "and-goal" are all inflated, then the average will be inflated too, wouldn't it? Perhaps 19.3 yards/td is maybe a bit over-generous, which could give too much advantage to QBs who threw more TD's per attempt. The difference would probably be lost in the noise after all the other issues with quantifying QB performance are taken into account (the quality of the opponents, defense, WRs, running game, blah blah), and good QB's probably throw a lot more TD's/attempt anyway, but...

Hmm, so are you going to recalculate the average value of an INT now? ๐

What fascinates me about this conclusion is that QBs with lots of TD passes such as Marino and Favre are actually underrated by the official passer rating formula despite their reputation of padding their TD totals (and hence accumulating 10 yard bonuses) with short TDs inside the red zone. In other words, this analysis argues that those short-yardage TDs are actually more valuable than your average TD once you've accounted for the yardage.

On the other hand, QBs with high TD totals often lack a good red zone/short yardage RB, thus giving them more opportunities to throw from point-blank range. One can still wonder how many more TD passes Troy Aikman would have had without Emmitt Smith running it in on the ground.

You made a critical mistake. While it is true that a passing touchdown may be worth 19.3 yards, an average team will score a one yard touchdown 55% of the time anyway (according to your stats). Suppose a hypothetical Tom Brady, instead of passing for 4806 yards and 50 touchdowns in 2007, passed for 4756 yards and zero touchdowns because his receivers took a knee at the one yard line every time. An average team would still be able to score 27.5 touchdowns anyway on one yard plays. Brady's touchdowns are worth 965 yards (50*19.3) in all, but his added value compared to what an average team would have scored is 965-(27.5*19.3), or 442.5 yards.

To account for this difference, we must multiply the value of a passing touchdown (19.3) by 55% to find how much better scoring a touchdown was compared to . This equals 10.6 yards, which is the actual value of a touchdown.

I'm unconvinced that the 20 yards adjustment is a good thing, taken by itself. There's two problems with it:

1) The increase is based on the contextual difference between TD passes and yardage in non-TD passes. That there is such an increase is unsurprising, since goal-to-go can be a high leverage situation. However, a corresponding hit for *non-TD* passes in goal-to-go situations (whether incomplete or not) is not assessed against the QB. The result is that quarterbacks who pass more often in goal-to-go situations will systematically be overvalued relative to other quarterbacks, even if their success rate in those situations is exactly the same.

2) The same contextual adjustments are not applied to the other weights. +10 for TDs and -45 for interceptions are appropriate when compared to changes in first down position for an offense as a whole. But for the actual interception itself, I think you'll find that the value of that play works out to average something like -45 for first down interceptions, -40 for second down interceptions, and -35 for third down interceptions -- interceptions are less costly as a play when the likelihood of making the first down goes down. On 4th down, the cost of an interception is usually no worse than an incompletion, and sometimes better. If we're willing to give an extra bonus to the touchdowns for context, shouldn't the interception context reduce that penalty? And perhaps the sacks should be examined to see if negative sack yardage deserves a negative or positive adjustment, given context?

Aside from all the other problems with Chase's metric, the thing I find most annoying with Adjusted Yards/Attempt is that everybody overlooks the obvious fact that AY/A is a measure of a team's efficiency, not the QB's efficiency. The PFR web site tends to use this metric for interchangable purposes---to prove the value of a QB through Chase's "points-value" system or to use it to demonstrate the relative strenght of a team through Doug's "strenght of schedule" configerations. Well, which way does the PFR authors want us to regard this metric---as a team metric or as an individual QB metric? Yes, the QB has a role to play, but he is only one actor upon the stage. He should not be rewarded or penalized above or below everyone else if a play succeeds or fails. A QB is ambly rewarded for throwing a TD pass and is ambly penalized for throwing an INT by the NFL's passer rating formula. I'm not saying that the NFL's system is right either---as I would do away with all weighted formulas--- but it is senseless to go through all of these convoluted machinations so as to create an artificially contrived stat which only serves to confirm that which we are already aware of. Afterall, "Why shoot for the moon, when we already possess the stars!"

This is my first comment here. I've been interested in the statistical analysis of football (NFL) since I first discovered The Hidden Game of Football. As flawed as that wonderful work was, it still generated a lot of thinking on my part.

I am now trying to reproduce PFR's AV methodology in order to take a view on its usefulness (first rule of scientific method is that a model must be such as to be reproduced by other observers). I've created a spreadsheet and, as a starter, am trying to reproduce 2009 and some "test" QB's.

I have successfully reproduced the 2009 AV for Brady (16.7 vs. PFR Actual of 17), P. Manning (16.9 vs. Actual of 17), Rivers (18.9 vs. 19) and Roethlisberger (13.6 vs. 14), but I can't get what seems to be my properly functioning model to produce the same results as PFR for Drew Brees.

Can someone tell me where I am going wrong or whether PFR is wrong? I have taken the following inputs from PFR and made the listed calculations:

League 2009

Rushing TD's 429

Passing TD's 710

Field Goals Made 756

Field Goal Attempts 930

Total Turnovers 872

Interceptions thrown 525

Fumbles Lost 347

Punts 2451

Rushing Yards 59739

Passing Yards 111851

Total Yards 171590

Saints 2009

Rushing TD's 21

Passing TD's 34

Field Goals Made 22

Field Goal Attempts 28

Total Turnovers 28

Interceptions thrown 12

Fumbles Lost 16

Punts 58

Rushing Yards 2106

Passing Yards 4355

Total Yards 6461

Brees

Passing Yards 4388

Passing TD's 34

Pass Attempts 514

Interceptions thrown 10

Sacks 20

Sack Yards 35

I am using the following factors as given in the "AV Methodology" section:

OLine: .455

Rusher: .220

Rush/Total Yds Avg since 1970: .370

Passer Factor: .260

TD Multiplier: 20

INT Multiplier: 45

I end up with the following, using the formulas provided:

League Average

Offensive Points per Drive 1.90

Offense Points 100

Team Points for o-line 45.5

Team Points for skill positions 55.5

Team Points for Rushers 11.3

Team Points for Passers 11.2

Saints

Offensive Points per Drive 2.67

Offense Points 140.5

Team Points for o-line 63.9

Team Points for skill positions 76.6

Team Points for Rushers 14.9

Team Points for Passers 16.1

Brees Base Score 16.2

Bonus 1.4

Total 17.6

Actual Brees for 2009 per PFR: "16"

League

Adjusted Net Yards Per Pass Attempt 6.04

Pass TD Multiplier 20

INT Multiplier 45

Brees

Adjusted Net Yards Per Pass Attempt 8.82

Pass TD Multiplier 20

INT Multiplier 45

I appreciate the help. This model is getting some attention and, before I evaluate it, I want to be sure (a) that I know how it works and (b) that it works consistently on the web page.