Photo: Sergio Estrada, 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.

In Week 4, I explained why touchdowns follow yards, (but yards don't follow back), and predicted that the players with the fewest touchdowns per yard gained would outscore the players with the most touchdowns per yard gained going forward.

In Week 5, I talked about how preseason expectations still held as much predictive power as performance through four weeks. No specific prediction was made.

In Week 6, I looked at how much yards per target is influenced by a receiver's role, how some receivers' per-target averages deviated from what we'd expect according to their role, and predicted that the receivers with the fewest yards per target would gain more receiving yards than the receivers with the most yards per target going forward.

In Week 7, I demonstrated how randomness could reign over smaller samples, but regression dominates over larger ones. No specific prediction was made.

In Week 8, I discussed how even something like average career length could be largely determined by regression-prone fluctuations in incoming talent. No specific prediction was made.

In Week 9, I looked at running backs scoring touchdowns at an unsustainable rate and posited that even Todd Gurley must return to earth.

In Week 10, I delved into the purpose of regression alert and the proper takeaways. No specific prediction was made.

In Week 11, I explained an easy way to find statistics that were more prone to regression and picked on yards per carry one more time.

In Week 12, I went into the difference between regression to the mean, (the idea that production will probably improve or decline going forward), and the gambler's fallacy, (the idea that production is "due" to improve or decline going forward). No specific prediction was made.

In Week 13, I badmouthed interception rate for a bit and then predicted that the most interception-prone quarterbacks to that point would throw fewer picks than the least interception-prone quarterbacks going forward.

Another week, another lay-up for yards per carry. Group B continues to receive more carries than Group A, (as expected), but they've also increased their lead in yard per carry average. Over the last three weeks, Group B backs average 5.18 yards per carry vs. just 5.03 for Group A.

On the interception prediction, this week presented a little bit of a dilemma. Mitchell Trubisky, a quarterback in Group B, was unable to play on Sunday. Normally this isn't a problem because normally my predictions are based on per-game averages, so a missed game neither helps nor hurts a group in expectation. But I set up the interceptions prediction based on total interceptions; in theory, if every Group B quarterback was placed on injured reserve tomorrow, they would throw zero interceptions and win by default.

Were I cleverer and blessed with a tiny bit of foresight, I would have made my predictions based on entire teams instead of individual quarterbacks. Since that's really a much better solution, and because I don't want anyone to think I won on a technicality, that's what I'll be doing going forward. As a result, Group B gets penalized for Chase Daniels' two interceptions in relief of Trubisky.

Other than that snag, everything came up roses for Group B last week. The true randomness of interception rates was best exemplified by Cam Newton against the Tampa Bay Buccaneers. Newton threw interceptions at a rate notably below the league average. The Bucs had just one interception in their first ten games before picking Nick Mullens off twice in Week 12. So what happened when the teams met in week 13? Why, Tampa picked off Newton four times and more than doubled their season-long interception total, naturally!

If interception rate was a stable predictor, such a turn would be virtually unthinkable. Instead, interceptions are so noisy that you could remove Newton's 4-interception game from the sample and Group A still would have thrown more interceptions per game than Group B. As it stands, Group A has a seven interception lead with three games to go.

Pay Attention to the Man Behind the Curtain

I want to draw your attention to what I just did in that interception prediction. I saw a complication and then I used my best judgment to resolve it.

Regression to the mean isn't a force so much as it's an observable law of mathematics, (see also: Central Limit Theorem). Performances can deviate from the true mean in the short run but not in the long run. I'm not setting out to prove this, because this is already proven. I'm merely setting out to demonstrate this in action.

To demonstrate regression to the mean, I have to make choices. These choices are made based on what I feel best demonstrates the concept, not based on what's "fair" or "accurate". I focus on statistics that are especially random, like yards per carry or interception rate, because I'm motivated to make predictions that are especially dramatic; the more random a statistic is the bigger the swings will be once regression kicks in.

I make rules for this column, and those rules are again motivated by my desire to make believers out of you, the readers. One rule is I make predictions over a 4-week span and immediately report on the accuracy. Another rule is that I don't get to select who goes into Group A or Group B beyond setting the parameters and letting the chips fall where they may. (In truth, this is a bit of a lie; since I choose the parameters, I can very easily exclude or include certain players just by tweaking what I'm looking at. But again, because I'm motivated to make things as dramatic as possible, I want to get some stars in Group A and some surprises in Group B.)

It's very important not to lose sight of what this column is— a proof of concept, a bold demonstration, a parlor trick, all this and more— and not let it impact your impression of what regression to the mean is. There are statistics that I don't mention much around here because they're too stable, like yards per pass attempt, or passing yards per game, or receptions per game. These statistics don't make for dramatic reversals, which means they aren't ideal for my mandate to be as impressive as possible. But this doesn't mean they don't regress to the mean; there's nothing at all that doesn't regress to the mean.

Similarly, I'm honor-bound to include players in Group A and Group B against my better judgment to show that regression to the mean works its magic even without meddling humans stepping in and putting their thumb on the scales. But this doesn't mean that meddling humans shouldn't feel free to put their thumb on the scales if they don't want to; if regression tells you that five running backs are due to decline, and one of those running backs is Todd Gurley, and you want to just ignore Gurley and focus on the other four... you should feel free to do that. (Though I will note that often when I think I know better than regression, regression is fond of proving me wrong. Remember, Todd Gurley wound up dragging the Group A average down once I finally worked up the courage to bet against him.)

Beyond my bias towards the dramatic, there are hundreds of little mundane decisions I make about what data to present and how to present it. For instance, consider the on-going yards per carry prediction. I mentioned that Group A averaged 5.03 yards per carry and Group B averaged 5.18 yards per carry. This might seem like a basic statement of fact, but it's not, it's an expression of choice.

I'm calculating yards per carry by adding up all of the rushing yards from Group A and dividing by total carries from Group A. Which seems like the way to do it, right? But there are lots of other ways I could have approached the problem.

For instance, one problem with this approach is it gives extra weight to players who play all four games. This is especially apparent in Group A right now because of injuries. There are seven backs in that group, but Nick Chubb and Matt Breida both missed a game to a bye, Kerryon Johnson missed two games to injury (and might miss a third this week), Melvin Gordon III missed one game to injury (and might miss a second), and Matt Breida has already been ruled out for week 14. Because of this, the three backs who have played every game— Aaron Jones, Phillip Lindsay, and Marlon Mack— contribute more to the overall average.

This is potentially a big deal because... well, to quote myself:

One ironclad rule of football is that everyone's yards per carry regresses to the mean except for Phillip Lindsay's, no exceptions.

— Adam Harstad (@AdamHarstad) October 19, 2018

This was a joke, of course. But it has also held true over the small sample size of this season. It's possible that Phillip Lindsay will wind up representing 1/7th of the players in Group A but 1/5th of the games played. At the moment, removing Lindsay from the sample drops Group A's average from 5.03 all the way down to 4.32! (The most influential member of Group B, by contrast, is Lamar Miller. Removing him from the sample drops Group B's average from 5.18 to 4.88, a much smaller decline.)

Is there another way to calculate averages, though? There are several. I could, for instance, calculate a simple average of each member of Group A's ypc. So Kerryon Johnson averages 5.8 ypc since the prediction, Phillip Lindsay averages 7.9, and so on. A drawback of this method is that for someone like Johnson who only played one game, small samples can produce large outliers. (As an extreme example: if a player ran for 50 yards on his first carry and tore his ACL at the end, his 50ypc average would increase the overall group average by more than 7 yards per carry!)

Or there are other, fancier ways to credibly claim you are finding the "average" of a data set that are less susceptible to outliers— the geometric mean, the harmonic mean, the median, etc. If I wanted to, I could easily say that Group A currently averaged 5.03 ypc, or 5.08 ypc, or 4.87 ypc, or 4.68 ypc. I could say that Group B averaged 1% more yards per carry, or I could say that they averaged 6% more yards per carry, or that they averaged 0.06 more yards per carry, or that they averaged 0.29 more yards per carry. And any single one of those claims would be true.

Again, because of my motivation for writing this column, my personal bias is to keep things as simple and as obvious as possible, to make predictions that feature many good players in Group A and many bad players in Group B, and to focus on statistics that lead to dramatic reversals. When there are ambiguities, I resolve them in the way that makes regression's job more difficult, (in order to make it more dramatic if and when the dramatic reversal occurs). At the same time, when I'm interpreting the data after a prediction wraps up, I'm biased towards interpreting them in a way that makes the reversal look as dramatic, (or the lack of a reversal as understandable), as possible.

I think this makes for a good and interesting lesson. Hopefully, if you've stuck with me this far into the season, you agree. But it's vital to remember that these are all theatrics. They're flourishes. They are not central to the concept of regression itself.

Regression is far less flashy than it may seem in these pages. It's just the slow inexorable pull dragging all sorts of interesting short-run statistical deviation back to the boring long-run mean.