## Wide receivers, Quarterbacks, and consistency

Posted by Doug on July 6, 2006

I recently picked up a copy of a book called *The Wages of Wins*, by David Berri, Martin Schmidt, and Stacey Brook. The authors are economists and they apply their economic outlook to the world of sports. I generally find this sort of thing interesting because the economist worldview has always struck me as very sensible. And because I like sports.

I want to make clear that **this post is not a review of the book**. I may or may not do that in a future post, but right now I have not read enough of it, nor have I read it carefully enough, to construct an adequate review. But I did get an idea while reading through it, and that's a sign of a promising book. The purpose of this post is merely to share the idea.

Most of the book is about baseball and basketball, but there is one chapter devoted to football. Its title is *How are quarterbacks like mutual funds?* It starts with a game-by-game look at Brett Favre's 2004 season, in which he threw 20 touchdowns and 4 interceptions in the odd-numbered games, and 10 touchdowns and 13 picks in the even-numbered games.

The title question is then answered:

The Favre story suggests that NFL quarterbacks are not quite as consistent as NBA players. Like mutual funds, past performance is no guarantee of future returns.

The authors then go on to build a metric of quarterback performance. The discussion will surely get sidetracked if I tell you exactly what it is, so I'm not going to do that. I'll just tell you that it includes passing attempts and yards, rushing attempts and yards, interceptions and fumbles. Using that metric, the authors measure the year-to-year consistency of quarterbacks and show that quarterbacks are, in fact, not very consistent. More precisely, quarterbacks' ratings according to this measurement constructed from yards, attempts, and turnovers are not very consistent. It's not uncommon, for example, to see a quarterback's rating go from the top quintile in the league to the bottom quintile --- or vice versa --- in consecutive years. I had not realized how variable quarterback stats are.

The authors observe that one reason for this inconsistency is that the rating they attach to a quarterback in a given year is dependent on the performance of a lot of players besides the quarterback himself. This point is well-known to all aficianados of football numbers; it is simultaneously the reason football is so fun to watch and the reason it's so hard to analyze. Any measure that penalizes Brett Favre when Donald Driver fails to get open or rewards Favre when Driver steals a sure interception out of the defender's hands is going to make Favre look more inconsistent than he really is. And that's not a criticism of Berri, Schmidt, and Brook's system in particular, because every system does it.

Favre's rating is a function of his performance, his teammates' performance, and random noise (yeah, there's some other stuff in there, too. I'm trying to keep it simple.). A rating of, say, Dwyane Wade, based on his stats would also be a function of his performance, his teammates performance, and random noise. But since Favre has 10 teammates and Wade has only four, it would be reasonable to suspect that Favre's rating is diluted by factors other than Favre's performance to a greater extent than Wade's is. Even if Favre's performance is rock steady from game to game and year to year, his rating, because there is so much other junk mixed into it, might still tend to vary a lot.

To summarize: Berri, Schmidt, and Brook do a very nice job of showing that quarterbacks' *statistics* are inconsistent (compared to those of basketball and baseball players) from year to year. But the open question is whether quarterbacks' *performances* are inconsistent from year to year, or if the variance in the statistics is due to the weak relationship between statistics and performance.

And the issue is not limited to quarterbacks, of course. In fact, quarterbacks might be the least affected by their teammates' performance. This is a wild guess, but I'd say that Donald's Driver's statistics are more influenced by Brett Favre's performance than Favre's are by Driver's performance. If Driver isn't doing his job, Favre can at least try find someone else to throw to. But Driver has no such option if Favre isn't holding up his end of the bargain.

I am going to shift the focus from quarterback to receivers now, and try to come up with a very rough estimate of a first pass at an extremely preliminary vague notion of an idea for how to maybe possibly determine how much of a wide receiver's production is attributable to the receiver himself, the quarterback, and everyone else on the team.

The idea goes like this. Look at all wide receivers who played at least 8 games in two consecutive seasons, and divide those receivers into three groups:

- those who were on the same team both years, and the team had the same starting quarterback both years;
- those who were on the same team both years, but with different starting quarterbacks each season;
- those who were on different teams in the two different years.

It'd be nice to also have a different-team-same-quarterback group, but history just doesn't provide us enough examples of that. Anyway, we press on.

The next step is to compute the year-to-year correlation in receiving yards per game for each of the groups. As many of you already know, a *correlation coefficient* is a number between -1 and 1 that measures the strength and direction of the linear relationship between two quantities. A positive correlation indicates two quantities that vary together (i.e. when one goes up, so does the other) while a negative correlation indicates two quantities that vary inversely. A correlation of 1 (or -1) means that the two quantities are perfectly linearly related. That is, one quantity can be predicted exactly if the other is known. A correlation of 0 means that there is no linear relationship at all between the two, so knowing one is of no use to you in predicting the other.

As this pertains to pairs of consecutive seasons of the same wide receiver, the correlation coefficient tells us roughly how easy it is to predict a receiver's stats this year using only his stats from last year. Here are the numbers:

**Group 1 - same team, same QB**: correlation = .75

**Group 2 - same team, different QB**: correlation = .64

**Group 3 - different team**: correlation = .44

What does this mean? The standard way to interpret these numbers is as follows (from the first statistics book I grabbed off the shelf):

[the square of the correlation coefficient] is the proportion of the total variation in the y's that can be attributed to the linear relationship with x]

Sometimes the square of the correlation coefficient is described in terms of "explanatory power": it's the percentage of the variation in y's that is "explained by" variation in x. The squares of those numbers are: .56, .41, and .19. So roughly speaking, what we have is this:

- For receivers on the same team with the same quarterback, their numbers this year are "explained by" 56% their numbers from last year, and 44% other stuff.
- For receivers on the same team but with a different quarterback, their numbers this year are "explained by" 41% their numbers from last year, and 59% other stuff.
- For receivers on different teams, their numbers this year are "explained by" 19% their numbers from last year, and 81% other stuff.

I really don't know what that means, except in a strict mathematical sense. It's tempting, but not mathematically justifiable, to try to make some conclusions about the role of the quarterback and the rest of the team based on differences between some of the numbers above. I am, for now, just going to say what I know is true: that receiver's stats are definitely more predictable if they stay on the same team, and even more predictable if that team keeps the same quarterback. Not an earth-shattering revelation I realize, but hey, it's just a blog post.

It might be interesting to play this game with other factors too, like coaches for instance. That is, build a same-team / same-quarterback / different-coach group and compare it to a same-team / different-quarterback / same-coach group. I'll put that on the ever growing to-do list.

Nice post. In your post you state that a QB has ten other teamates that dilute his performance. Really there are many more than that. A porous defense can force a team into passing mode making an offense easier to predict and a QB easier to pick. Poor special teams can leave an offense pinned on there own 2 yard line, forcing the QB to throw it away early so he doesn't take a safety.

I would wonder how much those guys fall into the group of other stuff.

Pretty fascinating stuff Doug. If I may make a small suggestion, another thing to consider would be same team/QB but a different #2 reciever (I am assuming your numbers were based primarily on #1s.) Then compare the #2 from the first year to the #2 of the second year to determine which was more effective, and then decide if that made the #1 more or less effective. It's early in the day, hope that wasn't too hard to follow.

I would be curious just to see the correlation coefficients for WR's who play on teams with the same basic offensive lineup from year to year.

Let's say, a comparison of all WR's who have at least 3 of the same QB, other WR, TE, RB on their team from year to year.

[...] Support pro-football-reference.com « Wide receivers, Quarterbacks, and consistency [...]

[...] Inside the group there wasn’t much consistency: only one-third of the RBs averaged within half a yard per carry of their YPC average from Year N. The correlation coefficient, explained here of the YPC for the RBs in Year N and Year N + 1 was just 0.16. This means that the YPC average of the RBs in the second year can be “explained by” 3% their YPC average in the first year, and 97% other stuff. This is a longwinded way of saying a small bit of data (less than 100 carries) just doesn’t tell you very much. What about a bigger piece of data? [...]