ESPN has unveiled a new passer rating formula (see also here and here, discussion of the ratings here, here, and here), one that is complex and to be plain, not very straightforward to interpret. In the age of stats that purport to give the contribution to winning in terms of wins per season a player contributes above replacement(i.e. WARP), one really has to wonder about the value of an arbitrary 0 to 100 scale. It’s in all honesty as meaningless as the NFL’s original scale, which maxes at something less than 160.

But in order to critique the new scale at all, in anything other than emotional terms, perhaps it’s best to step back and look at some of the previous critiques of the NFL’s old formula. The one we’ll start with is Brian Burke’s 2007 critique, where he points out that TDs are a pretty arbitrary criterion, and removes them from his formula. He finally decides that the best formula he can come up with is:

`QB Wins Added = (Comp% * 0.18) - (Int/Att * 50.5) - (Sack Yds/Att * 1.57) - 8`

This formula has the advantage of being scaled properly. It is also simple, not as sophisticated as other formulas. How well it works is beyond the scope of this survey, but we note it for those digging for more details.

Football Outsiders uses a method called DVOA to rank quarterbacks. Again, the scale is measured in terms of “success points”, and this is abstract. But it attempts to treat the game of football as something of a state machine, using NFL play by plays as the fundamental data source, and therefore is potentially a better stat than stateless formulas. However, DVOA is a rate stat, not a cumulative stat, and there can be times when a rate stat lies to you (i.e. a high performing player who can’t stay on the field can have a very high DVOA and a very low real value to a team). Nonetheless, this is FO’s attempt to improve on the QBR.

The best and most thorough critique is also an old one, the critique of the NFL QBR by Carroll, Palmer and Thorn in the book “The Hidden Game of Football“. They devote the whole of Chapter 11 to the various formulas the NFL has used, why they were busted, and why the NFL went to the formula they do use. They then critique the formula and offer two ranking formulas of their own. We’re going to spend a lot of time on the THGF critique. To be plain, those who really want to understand it should buy the book, as used copies are cheap.

One thing to note about the Carroll et al’s historical introduction to this problem is that a stat a lot of analysts drool over, YPA, was once used as the sole criterion to judge quarterbacks. When in 1957 Tommy O’Connell won the passing trophy, it became pretty obvious that not only a rate criterion was necessary, but also a cumulative statistical component as well. YPA alone isn’t a good way to rate quarterbacks.

Original and refactored NFL ratings formulas

Later in the chapter, Carroll et al give the NFL formula as the NFL gives it to others, and then refactor the formula so that analyzing the components is easier to do. The original formula is:

```RATE = 100 x [( Completion % - 30)/20 + (Average_Gain - 3)/4 + TD%/5 + (9.5 - INT%)/4]/6```

and after some mathematical gyrations, they break the formula down into the form RATE = A x [ (Completion_term + Yards + TD_term – INT_term)/attempts ] + B

and that formula is (results in the same points, but easier to conceptualize)

`RATE = 100/24 * [ (Completions * 20 + yards + Tds * 80 - ints * 100)/attempts] + 50/24`

Once the easier-to-understand formula is established, they begin their critique in earnest.
The critical passage is as follows:

How do you feel about giving a 20 point bonus for each completion? Not sure? Think of this. If one passer throws 2 passes and completes them both for 10 yards each, he’ll have 60 points. Another passer misses his first toss and then hits his second for 40 yards. He also has 60 points. Both passers rate the same even though the second guy moved his team twice as far!

The NFL system favors the high percentage, nickel passer. It always did, but that wasn’t nearly do obvious until lately, when several teams began to use short passes out in the flat as, in effect, running plays. If Joe Montana dumps off to Roger Craig and the play loses 5 yards, Joe still gets 15 points.

Note that the example in the first paragraph of the quote is stateful. If the example has started at the 20 yard line, then the final state of the short passer would have been a first down on the team’s 40 yard line, while the final state of the “long” passer would have been a first down on the opponent’s 40 yard line. The net expected points (see also here) from the improved field position is higher, so the second scenario should be rewarded more thoroughly. But to get that kind of evaluation requires at the least, play by play stats and to the highest level of detail, video of the game itself.

Finally, Carroll et al give two formulas they regard as superior to the NFL formula:

RATE = ( yards + TD x 10 – int X 45) / att

RATE = ( yards – sacks allowed + TD x 10 – int x 45 ) / (att + sacks)

We’re not here to analyze this formula either, but to present it to those who might be looking at ESPN’s QBR and trying to figure out alternatives.

Note: A NFL QBR calculator is here.