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Is that Bayes?

Is that Bayes?

Peyton Manning is not a 51 touchdown per-season quarterback, but that doesn’t mean he won’t average the necessary 2.9 touchdowns per game over his final ten games this season to break Tom Brady’s touchdown record. Before the season, Footballguys.com projected Manning as a 2.38 passing touchdown per game player.  And while he has looked unstoppable thus far, with 22 touchdown throws in six games, Manning has been known to have great spurts before, too.  All quarterbacks have hot and cold streaks, Manning included.  From 2003 to 2012, after removing games where he sat late in the season, Manning averaged 2.17 passing touchdowns per game with a standard deviation of 1.31 touchdowns.1  In the ’04 season, Manning threw at least 20 touchdowns in each of his trailing six game stretches from week 7 all the way through week 15, with a peak of 27 touchdowns in his prior six games in weeks 11 and 12.  Manning also threw 19 touchdowns in his last two full regular season games of 2010 and his first four games of 2011.  White-hot streaks happen, even to the best players, so we shouldn’t just assume that he’s now a 3.67 touchdown per game player.

On the other hand, it would be naive to assume that we should ignore the first six weeks of the season and continue to project Manning as a 2.38 touchdown per game player for the rest of the year.  The question becomes, how much do we base projection over the final 10 games on his preseason projection and how much do we base it on his 2013 results? In Part I, after four games, a regression model produced a projection of 2.56 touchdowns per game the rest of the year. But the problem with a regression analysis is that Manning is an extreme outlier among NFL quarterbacks; to project Manning, it would be best if we could limit ourselves to just quarterbacks named Manning Peyton Manning.

Before continuing, I want to give a special thanks to Danny Tuccitto, without whom this article wouldn’t be possible. Danny provided this great link and also spent a lot of time walking me through the process. To the extent I’ve mucked it up here, you should blame the student, not the teacher. But after walking through some models online, I realized that the best explanation about how to use Bayes Theorem for these purposes was on a sweet site called FootballPerspective.com. And the smartest person on that website had already laid out the blueprint.

In the comments to one of his great posts, Neil explained that we can calculate Manning’s odds using Bayes Theorem if we know four things:

His Bayesian prior mean (i.e., his historical average):

His Bayesian prior variance (the variance surrounding his historical average):

His observed mean:

His observed variance:

Let’s go through each of these:

1) Manning’s Bayesian prior mean: this is simply what we expected out of Manning before the season. I will use 2.38, since Footballguys is the gold standard of football projections in my admittedly biased opinion. But you can use any number you like, as I’ll provide the full formula at the end.

2) Manning’s Bayes prior variance: how much variance should we expect between what Footballguys projected and Manning’s actual results? Well, I have data on Footballguys’ projected number of touchdowns for Manning in every season since 2000. After converting it to a per-game basis, we get the following:

YearProj TDAct TDProj TD/GAct TD/GDiff TD/G

What I think we want to do is measure the variance of the errors, which is the difference between the variance of the actual TD/G column (0.1527), and the variance of the projected TD/G column (0.0541); the difference of those variances is 0.0986, which is Manning’s Bayes prior variance. I say “I think” because there are other ways one could come up with a prior variance.  For example, you could take the variance of the data in the differential column, which is 0.2529.  Or, you could use simply his yearly variance, which is 0.1527.  I’ll invite the readers to comment as to which number (0.0986, 0.2529, or 0.1527) they would prefer to use.

3) Observed mean: 3.667. That one’s pretty easy, as Manning has thrown 22 touchdowns in 6 games.

4) Observed variance of current mean. This one’s a bit trickier. But what I think we need to do here is calculate the number of passing touchdowns per game Manning averaged in the first six games of each season since 2000, along with his average over the rest of the season (in eight, nine, or ten games). Then I took the difference of the variances of each column, as we did in step two.

YearTD/G Thru 6ROY GTD/G ROYDiff

Manning’s variance over the rest of the season is 0.2849 TDs/G, while his variance through six games is 0.1740; the differential there is 0.1109, which is the variance of our current mean.  Although again, reasonable minds could differ — you could take simply the variance of the differential (which is 0.2088), or just his variance through six games (0.1740) or perhaps you’d prefer to compare his variance through six games to his variance over the course of the season. This is intended to be an inclusive post, so add your thoughts in the comments.

Once you have your number for these four variables, then you substitute those numbers into this equation:

Result_mean = [(prior_mean/prior_variance)+(observed_mean/observed_variance)]/[(1/prior_variance)+(1/observed_variance)]

Or, using our numbers:

[(2.38 /0.0986) + (3.667 / 0.1109)] / [(1/0.0986) + (1/0.1109]

which becomes

[24.13 + 33.07] / (19.16) = 2.99

In other words, this formula projects Manning to average 2.99 passing touchdowns per game the rest of the year…. which would have him break Brady’s record by two touchdowns.2 We can also calculate the standard deviation around that average, which is just the square root of 1/ 19.16, which is 0.23.  That’s a pretty decent-sized range, of course, but this says that we can be reasonably confident that the best going forward projection for Manning would be between 2.76 and 3.21. Manning needs to average 2.8 passing touchdowns per game to tie the record, which would means he only needs to do better than -0.83 standard deviations below average.3

There are lots of things that Bayes theorem doesn’t consider, of course.  Strength of schedule is one, and Manning’s schedule is about to get a lot harder.  Risk of injury is another, and Manning has been fully healthy this year (although he did miss parts of the Eagles game once the score got out of hand). There are lots of subjective factors that we have to consider, but the main issue at hand is figuring out where on the spectrum of 2.38 to 3.667 we should project Manning. This basically says that the two are about equivalent in value.

If you projected Manning to average 2.00 passing touchdowns per game in the preseason, this model would bump the project to 27.8 touchdowns the rest of the season. If you were pretty clairvoyant — say, you projected him at 45 touchdowns this year, a 2.8 TD/G average — this would say you should revise your projections to 32.1 touchdowns over his final ten games.   Pre-season projections still matter, but you’re going to have to amp up your projections based on how deadly efficient Manning has been to date.

  1. That was after removing week 17 of the ’04, ’05, ’07, ’08, and ’09 seasons, and week 16 of the ’05 and ’09 seasons, when Manning left early. Why did I pick the last ten years? I don’t know, but he won his first MVP in ’03, so that seemed like a useful starting point. []
  2. If you wanted to just use 0.1527 — Manning’s yearly variance — and 0.1740 — his typical variance through six games — you end up with 2.985, so no practical difference there. []
  3. If you permit the making of certain assumptions, that would imply a roughly 80% chance of breaking the record. []
  • Anders

    Im no probabilistic expert and somebody might correct me if im wrong, but for me I think point 4 should just be the variance of the observed 6 games.

    This give me a result_mean =2.42 and a stdev of 0.31

    • James

      I agree.

  • Chad Wittman

    Excellent, excellent work. Thank you for the fine read.

  • James

    This Peyton Manning season is crazy for other reasons too:

    1. What kind of impact will this have on QB aging studies? Imagine how much flatter they’d be if Peyton retired after this season, or how much more pronounced they’ll be if Peyton has a ‘typical’ Peyton season in 2014 followed by an injury plagued 2015 and then retirement.
    2. With this Broncos season combined with the Patriots season, does this propel Welker to the top of the WOWY stats? Is this the biggest WOWY change of all time, combining improvement of new team and decline of old team?
    2a. Will this be enough to cement Welker’s HOF chances? Chase and I talked about it after his All-Pro 2011 season, and Welker followed it up with another monster 118 catch, 1354 receiving yards in 2012. He’s probably going to drop down to around a 90 catch, 900 yard season this year, but barring injury he’s a lock to smash his career high of 9 TDs since he already has 8 in 6 games!
    3. I think Scott Kacsmar has been tweeting this, but it’s remarkable how different Brady and Peyton have used Welker, especially in the end zone.

    • Paul

      Its interesting to look back on old comments and see just how impressive some predictions are. Peyton manning did have a terrific 2013, a still great but less impressive 2014, then an injury plagued 2015 and retirement. Most predictions are never right. Spot on james, spot on.

  • Really interesting article. Using Bayes Theorem like this is something I’m trying to do this year too. See: http://www.bayesff.com/bayesian101/

    There’s a bit of a difference, in particular (for computational reasons) I assume a constant variance (shape parameter) but I think this general approach is the right way to think about it.

  • Ajit

    I like this article a ton. I will say, its a fun exercise, but applying bayesian analysis like this can be very dangerous, especially because its especially sensitive to priors and distributions. Rather than attempt to build a highly ornate(and complicated) bayesian model, I might have tried to create a time series regression model, with different impulse response tests.

    • Chase Stuart

      Thanks, Ajit. Any interest in posting that?

      • Ajit

        hahaha, maybe. Actually, I was thinking of doing something similar to what you did on your NY giants 0-6 article. Essentially, I was curious to see what the bust probabilities are for draft picks, given certain parameters and to see how that probability evolves over draft position. I may do that once fall quarter ends.