Photo: USA Today Sports

Welcome to Regression Alert, your weekly guide to using regression to predict the future with uncanny accuracy.

For those who are new to the feature, here's the deal: every week, I dive into the topic of regression to the mean. Sometimes I'll explain what it really is, why you hear so much about it, and how you can harness its power for yourself. Sometimes I'll give some practical examples of regression at work.

In weeks where I'm giving practical examples, I will select a metric to focus on. I'll rank all players in the league according to that metric, and separate the top players into Group A and the bottom players into Group B. I will verify that the players in Group A have outscored the players in Group B to that point in the season. And then I will predict that, by the magic of regression, Group B will outscore Group A going forward.

Crucially, I don't get to pick my samples, (other than choosing which metric to focus on). If the metric I'm focusing on is yards per target, and Antonio Brown is one of the high outliers in yards per target, then Antonio Brown goes into Group A and may the fantasy gods show mercy on my predictions.

Most importantly, because predictions mean nothing without accountability, I track the results of my predictions over the course of the season and highlight when they prove correct and also when they prove incorrect. Here's a list of all my predictions from last year and how they fared.

The Scorecard

In Week 2, I laid out our guiding principles for Regression Alert. No specific prediction was made.

In Week 3, I discussed why yards per carry is the least useful statistic and predicted that the rushers with the lowest yard-per-carry average to that point would outrush the rushers with the highest yard-per-carry average going forward.

One week in, we've only seen a little movement in yards per carry averages. Group A averaged 5.45 yards per carry in the first two weeks and 4.92 yards per carry in Week 3. Group B averaged 3.61 yards per carry in the first two weeks and 3.82 yards per carry in Week 3. The big story, however, is one of volume; Group A saw its carry per game average fall from 13.6 to 11.0, while Group B's carry per game average held firm from 16.5 to 16.0. Some of Group A's decline no doubt owed to the early ejection of Phillip Lindsay, but not all-- Lindsay would have needed another 23 carries for Group A to maintain its average workload through two weeks.

Events like Lindsay's ejection are also why we track so many players at once, and why we track for a minimum of four weeks. Fluky things happen over small samples. That noise washes out in large samples. The larger the sample you can accumulate, the more dominant regression to the mean becomes. Anyway, on to Week 4!

Touchdowns Are Stochastic

We're going to open today's analysis with a quick vocabulary lesson.

sto·chas·tic
adjective
randomly determined; having a random probability distribution or pattern that may be analyzed statistically but may not be predicted precisely.

You could just say stochastic is just a fancy word for "random", but there's more to it than that. Since 2013, Antonio Brown has scored 54 touchdowns in 80 games, an average of 0.675. We could say that's his "true production level", and over a long timeline, we'd probably expect him to conform to that, averaging 0.675 touchdowns per game going forward.

Despite that being his true production level, though, guess how many times Antonio Brown has scored 0.675 touchdowns in a game? As far as I can tell, that has never happened. Instead, he either scores zero touchdowns... or he scores one touchdown. (Or sometimes he scores two touchdowns, and twice he's scored three touchdowns.) Because they are binary outcomes, we can analyze Antonio Brown's touchdowns statistically, but we cannot predict them precisely.

Yards don't really behave like that. Since 2013, Antonio Brown averages 100.7 yards per game. But it's not like every week he's either getting you 0 yards or else he's getting you 100 yards. Instead, he's usually getting you somewhere between 50 and 150 yards. His yardage total is *much* more consistent from game to game than his touchdown total.

One way to measure consistency is something called standard deviation, which measures how much something varies around the average. The standard deviation of Brown's receiving yards is 48.3 yards. The standard deviation of Brown's receiving touchdowns is 0.82 touchdowns.

Now, these numbers are not directly comparable. But if you divide a player's standard deviation by that player's average, you get something called the coefficient of variation, or CV. CV is a way to compare how volatile different statistics are. The CV of Brown's yards is 48%, meaning it tends to vary by about 48% of his overall average. The CV of Brown's touchdowns is 121%. Touchdowns are much more random from week to week than yards are.

Not only that, but touchdowns are also much more valuable than yards. In most scoring systems, one extra touchdown is worth the equivalent of 60 extra yards. Which means if Brown catches the high side of variance and scores a few extra touchdowns early in the year, it can dramatically inflate his fantasy production to date. And if he catches the low side of variance and fails to reach the end zone, it can leave him far lower than we'd otherwise expect.

Which gives rise to my favorite statistic for regression: yard-to-touchdown ratios. Some players are really, really good at getting yards and/or not quite as good at scoring touchdowns. The most famous recent example is Julio Jones, who has gained 218 receiving yards in his career for every touchdown he has scored. This is a very high average, but there are plenty of other wide receivers in this general range; Andre Johnson averaged 203 yards for every touchdown, Henry Ellard averaged 212, etc.

Other players are really, really good at getting touchdowns but typically aren't commensurately good at getting yards. For his career, Davante Adams scores a touchdown for every 104 yards he gains receiving. Again, this is a very low average, but not historically implausible; Dez Bryant averaged 102 yards for every touchdown, while Randy Moss was all the way down at 98 yards per touchdown.

Importantly: yard-to-touchdown ratio is not a measure of player quality. Davante Adams has twice scored 10 or more touchdowns with 1,000 or fewer yards. All else being equal, a guy who gains 1500 yards and 10 touchdowns is better than a guy who gains 1000 yards and 10 touchdowns, even if the latter guy has a "better" yard-to-touchdown ratio. Heading into this season, the Pittsburgh receiver with the "best" yard-to-touchdown ratio was not Antonio Brown or Juju Smith-Schuster... it was Justin Hunter.

With that in mind, over the long term, receivers tend to average between 100 and 200 yards per touchdown, with the majority of the league clustered between 120 and 180. Any rate that falls in that range is plausibly sustainable and perhaps a true representation of a player's relative skill at scoring touchdowns. We should not expect Julio Jones to score double-digit touchdowns this year.

But because touchdowns are stochastic, in the short run we see yard-to-touchdown ratios that are wildly outside of that "sustainable" zone. And because touchdowns count for so many points in fantasy football, this gives us a ton of targets for regression.

Here's a list of every wide receiver through three weeks with at least 150 yards from scrimmage, along with their yard-to-touchdown ratio. (If the numbers are sometimes off, it's because I'm including both rushing and receiving yards and touchdowns in the calculation, but only displaying receiving stats here.)

Cooper Kupp, T.Y. Hilton, Marvin Jones, Ted Ginn Jr, Albert Wilson, Davante Adams, Tyler Lockett, Stefon Diggs, Mike Williams, Kenny Stills, Chris Godwin, Calvin Ridley, and A.J. Green all average fewer than 100 yards for every touchdown scored to this point in the season. Collectively, they have averaged 12.0 points per game in standard scoring. This is Group A.

Brandin Cooks, Julio Jones, Jarvis Landry, Odell Beckham Jr, Allen Robinson, Terrelle Pryor, Corey Davis, Amari Cooper, Juju Smith-Schuster, Adam Thielen, DeAndre Hopkins, and Golden Tate all average more than 250 yards for every touchdown scored to this point in the season. Collectively, they have averaged 9.8 points per game in standard scoring. This is Group B.

Thanks to the magic of touchdowns, Group A has scored 23% more fantasy points per game in standard scoring to this point in the season. Despite this, thanks to the magic of regression to the mean, I predict Group B will crush Group A going forward.