## How Likely Is Peyton Manning To Break The Single-Season TD Record? Part I

It's been a magical month for the Broncos passing game.

Sometimes, the simplest questions have the most complicated answers. Peyton Manning has thrown 16 touchdowns so far this season, putting him on pace for 64 touchdowns this year. Now, we can be reasonably sure that Manning’s true ability level — even with Wes Welker, Demaryius Thomas, Eric Decker, and Julius Thomas — isn’t four touchdowns per game. But he doesn’t need to keep up that pace to break Tom Brady’s single-season record of 50 touchdown passes: Manning “only” needs to averaged 2.92 touchdowns per game over the final 12 games. But to figure out his odds of averaging nearly three touchdowns per game, we need to figure out his true ability level. So how do we determine that number?

Even for a man who averages four touchdown throws per game over four games, averaging nearly three touchdowns per game going forward is still a tall order. Footballguys.com projected Manning to averaged 2.38 touchdowns per game this year. In 2012, he threw 37 touchdowns, an average of 2.31 touchdowns per game. From 2003 to 2012, excluding games1 he exited early, Manning averaged 2.17 touchdown passes per game. As a Colt, Manning averaged 1.92 touchdowns per game.

It doesn’t take any advanced math skills to figure out that Manning is likely to average somewhere between 2 and 4 touchdowns per game over the rest of the season. But that doesn’t help us very much: we need to be precise, since the threshold he needs to hit is 2.92 touchdowns per game. I’ll get to the more complicated math in Part II. For now, let’s look at some history.

Prior to 2013, 33 quarterbacks threw at least 11 touchdowns in their first four games. They’re all presented in the table below; let’s walk through Kurt Warner to guide you through the table. In 1999, Warner threw 14 touchdowns in his first 4 games, an average of 3.5 touchdowns per game. He played in 12 games the rest of the year, producing 27 touchdowns, an average of 2.25 touchdowns per game over the rest of the season.

QuarterbackYearTeamTDs (4G)TD/G (4G)Games (RoY)TDs (RoY)TD Avg (RoY)
Kurt Warner1999STL143.512272.25
Don Meredith1966DAL143.59101.11
Daunte Culpepper2004MIN133.2512262.17
Dan Marino1984MIA12312363
Brett Favre1996GNB12312272.25
Drew Bledsoe1997NWE12312161.33
Len Dawson1966KAN12310141.4
Steve Young1998SFO12311242.18
Dan Marino1994MIA12312181.5
Aaron Rodgers2011GNB12311333
Len Dawson1963KAN1239141.56
Ryan Fitzpatrick2012BUF12312121
Brett Favre2008NYJ12312100.83
Dan Marino1986MIA112.7512332.75
Jim Kelly1991BUF112.7511222
Charley Johnson1965STL112.75771
Peyton Manning2004IND112.7512383.17
Matthew Stafford2011DET112.7512302.5
Peyton Manning2010IND112.7512221.83
Al Dorow1961NYT112.751080.8
Kurt Warner2001STL112.7512252.08
Matt Ryan2012ATL112.7512211.75
Tony Romo2007DAL112.7512252.08
Len Dawson1962DTX112.7510181.8
Brian Sipe1983CLE112.7511151.36
Donovan McNabb2005PHI112.75551
Mark Malone1985PIT112.75620.33
Tommy Kramer1986MIN112.759131.44
Average--11.72.9210.820.11.79

What does this mean for Manning 2013? One thing we could do is run a regression using two variables: touchdown passes per game through 4 games and league average touchdown passes that season. I ran those numbers and the best-fit formula was:

-1.57 + 0.57*TD(4G) + 1.24*LgAvg

The p-value on both variables was not statistically significant, and the R^2 was just 0.05. In other words, I wouldn’t put much stock in this formula. On the other hand, this is a good way to ballpark Manning’s projections. And even if the variables aren’t statistically significant, the results seem reasonable. Essentially, this formula tells us that the league average variable is 2.18 times as important as the TDs-per-game-through-four games variable, which makes sense: four games is a really tiny sample size.

If we assume 1.5 touchdowns per team game will be the league average in 20132, then plugging in Manning’s four touchdown per game average would give him a projection of 2.56 touchdowns per game the rest of the year. That would put his 2013 year-end total at 47, a pretty reasonable projection as of Sunday morning.

Anyway, this is just the Part I; in Part II, I’m going to use a different formula to come up with a more precise projection.

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. A reasonable projection, since the average was between 1.45 and 1.48 in each of the last three seasons, and the average is at 1.61 right now, but the average is generally a bit higher in September []
• Richie

Here’s where my prob/statistics is a little rusty.

If Manning averaged 2.31 per game last year, and he averaged 2.17 per game over the past 10 years, it seems like his “true” rate is in the 2.1 to 2.3 TD/game range. Possibly the 2.3 rate has more bearing this year, since he is playing in a similar situation (team, teammates, league environment, etc.) than most of the 2.1 rate he has.

So, given a “true” rate of approximately 2.3 TD/game, is it more likely that he throws 2.3 per game the rest of the year, or that he finishes the year with a 2.3 rate? (I think 2004 and 2012 are the only 2 seasons that he averaged 2.3+ for a full season.)

I would think it’s most likely that he averages between 1.9 per game (to put him at 2.3 by season’s end) and 2.3 per game (his “true” rate) and he finishes with between 37 and 44 TD passes for the year.

• James

“So, given a “true” rate of approximately 2.3 TD/game, is it more likely that he throws 2.3 per game the rest of the year, or that he finishes the year with a 2.3 rate? ”

It’s the former – you’d expect Peyton to throw 2.3 TDs/game for the rest of the year. The past is the past, and his future TD rate is (mostly) independent of what has already happened, and it’s not going to decrease so his final actual rate matches his true talent rate; that’s called the Gambler’s fallacy.

Tom Brady’s 2011 is probably the best example of this. Based on his previous four seasons (exluding 2008) you’d expect Brady’s true TD rate to be about 2.2 TDs/game. Four games into 2011 he had thrown 3.25 TDs/game, but going forward you’d still expect him to only throw at his true rate of 2.2 TDs/game, and lo and behold he did. However, because of how well he did the first four games, his final rate was 2.4 TDs/game.

So if you think Peyton’s true talent rating is 2.3 TDs/game then your estimate should be 2.3 TDs/game for the rest of the season, giving Peyton a final rate of 2.7 from when this post was written.

• Richie

It’s the former – you’d expect Peyton to throw 2.3 TDs/game for the rest of the year. The past is the past, and his future TD rate is (mostly) independent of what has already happened, and it’s not going to decrease so his final actual rate matches his true talent rate; that’s called the Gambler’s fallacy.

Yeah, after I posted that I was thinking more about it. I suppose it would be like a coin toss. If I got “heads” my first 5 tosses, and was doing 10 total tosses, I shouldn’t expect 5 “tails” going forward. I should expect 2 or 3 tails (50%).

• Chase Stuart

Pretty much.

But the other thing to keep in mind is that you should still be increasing your projection.

If Manning was at 2.3 TDs/G before season, and at 3.3 TDs/G right now, our going forward projection should be somewhere between 2.3 and 3.3. Once we have new evidence, we need to incorporate that into our projection. The issue is figuring out how much weight should go on the prior knowledge and how much on the new knowledge. This is at the heart of Bayes Theorem.