A famous article (in the right circles) by Virgil Carter and Robert E. Machol (1978) established empirically what every fan seems to know intuitively: football coaches kick too many field goals.1 They looked at how many points a possession was worth, on average, and how many points teams earned, on average, by kicking a field goal. Logically, if it is fourth and short and the field goal is relatively long, the chances of getting the first down and, subsequently, an easier field goal attempt or a touchdown make it worthwhile to go for it.

But they were speaking in generalities--what the average team should do--but individual teams are not the average teams. So, I've made an attempt here to build on that work thirty years later.

I have used four variables--yards to go, yard line, quality of kicker (yard line from which he will make 10% of his fieldgoals), and offensive yards per play. If if it fourth and long (high yards to go) that will favor a field goal attempt, as will a high quality of kicker. A high offensive yards per play increase the likelihood of converting on fourth down as well as that of scoring a touchdown, and should be correlated with fewer field goal attempts.

Obviously, I am making several simplifying assumptions. For example, passing teams will do better on fourth and long and running teams will have the advantage on fourth and short--I remember BYU taking delay of game penalties on the goal line so they would have more room for the receivers.

The model I will be using after a fourth down conversion is very simple, and it doesn't allow a team to go for it on fourth down again; they must kick field goal if they get to fourth down again.

Finally, I ignore the subsequent possession. An advantage of going for it on the opponent's 5 is that they are then forced to 95 yards if you don't make it. I hope to add in this factor as well as add some variable options (passing/rushing yards per play and standard deviations, etc.) in a future model, but that will take some time.

I have a second problem. I have five variables as it is. To graph the results we need one dimension per variable and, if you haven't noticed, we experience reality in three dimensions, not five. (Technically I could use time as the fourth dimension, but good luck interpreting that on a with blogarific graphics.) Instead, I use a disaggregated form.

First, I modeled data they had on field goal accuracy, adjusting a little for college football (where the average kicker has less distance and is less accurate). To my utter disgust, the line of best fit was straight= P(fg) = (45-yard line)/.475 . This would suggest that a kicker has a zer0 percent chance of hitting a 52 yard field goal, a 42% chance from 42 yards, and an 84% chance from 22 yards. To adjust the model for accuracy and kicker leg strength I have added a few things:

P(fg) = (38+ LS -yard line)/(47.5-AC)

where LS is a LS rating (0=average, +=better than average) and AC is kicker accuracy (0=average, +=better than average).

The probability of getting a first down on first down is:

P(fd) = sqrt((x/4.5)/(YTG^1.75+.6))

where x is the offensive yards per play and YTG is the yards needed. This equation has some real problems (i.e. if a team averages 8 YPP then they have a greater than 100% chance of getting the first down), but with moderate values it seems to fit the data relatively well.

Next, I use the probability that the team will get "y" yards after they have converted the first down. I use the following formula:

P(yards=>y) =1/((y/10)^1.5*(4.5/x)+1)

And the probability that a team will get exactly "y" yards is:

P(yards==y) =1/((y/10)^1.5*(4.5/x)+1) - 1/(((y-1)/10)^1.5*(4.5/x)+1)

To avoid a rather complicated calculus maneuver I have just decided on the discrete values of each yard line, so that the probability of making a field goal later in the drive after the fourth down conversion is the sum of probabilities from goal line to (yard line - ytg).

The probability of a touchdown is simply

P(y=>yard line - ytg) = 1/((y/10)^1.5*(4.5/x)+1)

Then, to calculate the expected points, I just multiply the P(pfg) (probability of a post-conversion field goal) by 3 and of a touchdown by 7.

So, for an average team with an average kicker getting 4.5 yards per play facing a 4th and 5, we get distributions like this:

If the image were a better quality, you would see quite clearly that a team is better off kicking a field goal on 4th and 5 if they are inside the 28 or so (a 45 yard field). The main factor here is the probability of getting that initial first down.

If the situation is 4th and 3, the average team should never kick a field goal.

The effects of increasing the quality of the kicker or the offense have the expected results--they shift the curves up and down relative to one another. A team with a better kicker is not much more likely to kick the field goal, because he would also be kicking any subsequent field goals.

Because my actual numbers are a little suspect this little study is not too conclusive, but we can draw some basic conclusions. Most importantly, on 4th and short (and the definition of short varies on the offensive prowess of the team) teams kick too many field goals, but kicking the field goal is not a bad option on 4th and 5 or more--so don't fire the kicker.

In some future blog, I will try to clean up some problems (for example, allowing teams to punt or to turn the ball over). I think it is also necessary to consider the field position you will be handing over to the opponent.

(To Most Recent)

BPR | A system for ranking teams based only one wins and losses and strength of schedule. See BPR for an explanation. |

EPA (Expected Points Added) | Expected points are the points a team can "expect" to score based on the distance to the end zone and down and distance needed for a first down, with an adjustment for the amount of time remaining in some situations. Expected points for every situation is estimated using seven years of historical data. The expected points considers both the average points the offense scores in each scenario and the average number of points the other team scores on their ensuing possession. The Expected Points Added is the change in expected points before and after a play. |

EP3 (Effective Points Per Possession) | Effective Points Per Possession is based on the same logic as the EPA, except it focuses on the expected points added at the beginning and end of an offensive drive. In other words, the EP3 for a single drive is equal to the sum of the expected points added for every offensive play in a drive (EP3 does not include punts and field goal attempts). We can also think of the EP3 as points scored+expected points from a field goal+the value of field position change on the opponent's next possession. |

Adjusted for Competition | We attempt to adjust some statistics to compensate for differences in strength of schedule. While the exact approach varies some from stat to stat the basic concept is the same. We use an algorithm to estimate scores for all teams on both sides of the ball (e.g., offense and defense) that best predict real results. For example, we give every team an offensive and defensive yards per carry score. Subtracting the offensive score from the defensive score for two opposing teams will estimate the yards per carry if the two teams were to play. Generally, the defensive scores average to zero while offensive scores average to the national average, e.g., yards per carry, so we call the offensive score "adjusted for competition" and roughly reflects what the team would do against average competition |

Impact | see Adjusted for Competition. Impact scores are generally used to evaluate defenses. The value roughly reflects how much better or worse a team can expect to do against this opponent than against the average opponent. |

[-] About this table

Includes the
top 180 QBs by total plays

Total <=0 | Percent of plays that are negative or no gain |

Total >=10 | Percent of plays that gain 10 or more yards |

Total >=25 | Percent of plays that gain 25 or more yards |

10 to 0 | Ratio of Total >=10 to Total <=0 |

Includes the
top 240 RBs by total plays

Total <=0 | Percent of plays that are negative or no gain |

Total >=10 | Percent of plays that gain 10 or more yards |

Total >=25 | Percent of plays that gain 25 or more yards |

10 to 0 | Ratio of Total >=10 to Total <=0 |

Includes the
top 300 Receivers by total plays

Total <=0 | Percent of plays that are negative or no gain |

Total >=10 | Percent of plays that gain 10 or more yards |

Total >=25 | Percent of plays that gain 25 or more yards |

10 to 0 | Ratio of Total >=10 to Total <=0 |

Includes
the
top 180 players by pass attempts)

3rdLComp% |
Completion % on 3rd and long (7+
yards) |

SitComp% |
Standardized completion % for
down and distance. Completion % by down and distance are weighted by
the national average of pass plays by down and distance. |

Pass <=0 | Percent of pass plays that are negative or no gain |

Pass >=10 | Percent of pass plays that gain 10 or more yards |

Pass >=25 | Percent of pass plays that gain 25 or more yards |

10 to 0 | Ratio of Pass >=10 to Pass<=0 |

%Sacks |
Ratio of sacks to pass plays |

Bad INTs |
Interceptions on 1st or 2nd down
early before the last minute of the half |

Includes the top 240 players by carries

YPC1stD |
Yards per carry on 1st down |

CPCs |
Conversions (1st down/TD) per
carry in short yardage situations - the team 3 or fewer yards for a 1st
down or touchdown |

%Team Run |
Player's carries as a percent of team's carries |

%Team RunS |
Player's carries as a percent of team's carries in short
yardage situations |

Run <=0 |
Percent of running plays that
are negative or no gain |

Run >=10 |
Percent of running plays that
gain 10 or more yards |

Run >=25 | Percent of running plays that gain 25 or more yards |

10 to 0 | Ratio of Run >=10 to Run <=0 |

Includes the top 300 players by targets

Conv/T 3rd | Conversions per target on 3rd Downs |

Conv/T PZ | Touchdowns per target inside the 10 yardline |

%Team PZ | Percent of team's targets inside the 10 yardline |

Rec <=0 | Percent of targets that go for negative yards or no net gain |

Rec >=10 | Percent of targets that go for 10+ yards |

Rec >=25 | Percent of targets that go for 25+ yards |

10 to 0 | Ratio of Rec>=0 to Rec<=0 |

Includes the top 300 players by targets

xxxx | xxxx |

...

Includes players with a significant number of attempts

NEPA | "Net Expected Points Added": (expected points after play - expected points before play)-(opponent's expected points after play - opponent's expected points before play). Uses the expected points for the current possession and the opponent's next possession based on down, distance and spot |

NEPA/PP | Average NEPA per play |

Max/Min | Single game high and low |

Includes players with a significant number of attempts

NEPA | "Net Expected Points Added": (expected points after play - expected points before play)-(opponent's expected points after play - opponent's expected points before play). Uses the expected points for the current possession and the opponent's next possession based on down, distance and spot |

NEPA/PP | Average NEPA per play |

Max/Min | Single game high and low |

Adjusted | Reports the per game EPA adjusted for the strength of schedule. |

Defensive Possession Stats

Points/Poss | Offensive points per possession |

EP3 | Effective Points per Possession |

EP3+ | Effective Points per Possession impact |

Plays/Poss | Plays per possession |

Yards/Poss | Yards per possession |

Start Spot | Average starting field position |

Time of Poss | Average time of possession (in seconds) |

TD/Poss | Touchdowns per possession |

TO/Poss | Turnovers per possession |

FGA/Poss | Attempted field goals per possession |

%RZ | Red zone trips per possession |

Points/RZ | Average points per red zone trip. Field Goals are included using expected points, not actual points. |

TD/RZ | Touchdowns per red zone trip |

FGA/RZ | Field goal attempt per red zone trip |

Downs/RZ | Turnover on downs per red zone trip |

Defensive Play-by-Play Stats

EPA/Pass | Expected Points Added per pass attempt |

EPA/Rush | Expected Points Added per rush attempt |

EPA/Pass+ | Expected Points Added per pass attempt impact |

EPA/Rush+ | Expected Points Added per rush attempt impact |

Yards/Pass | Yards per pass |

Yards/Rush | Yards per rush |

Yards/Pass+ | Yards per pass impact |

Yards/Rush+ | Yards per rush impact |

Exp/Pass | Explosive plays (25+ yards) per pass |

Exp/Rush | Explosive plays (25+ yards) per rush |

Exp/Pass+ | Explosive plays (25+ yards) per pass impact |

Exp/Rush+ | Explosive plays (25+ yards) per rush impact |

Comp% | Completion percentage |

Comp%+ | Completion percentage impact |

Yards/Comp | Yards per completion |

Sack/Pass | Sacks per pass |

Sack/Pass+ | Sacks per pass impact |

Sack/Pass* | Sacks per pass on passing downs |

INT/Pass | Interceptions per pass |

Neg/Rush | Negative plays (<=0) per rush |

Neg/Run+ | Negative plays (<=0) per rush impact |

Run Short | % Runs in short yardage situations |

Convert% | 3rd/4th down conversions |

Conv%* | 3rd/4th down conversions versus average by distance |

Conv%+ | 3rd/4th down conversions versus average by distance impact |

Offensive Play-by-Play Stats

Plays | Number of offensive plays |

%Pass | Percent pass plays |

EPA/Pass | Expected Points Added per pass attempt |

EPA/Rush | Expected Points Added per rush attempt |

EPA/Pass+ | Expected Points Added per pass attempt adjusted for competition |

EPA/Rush+ | Expected Points Added per rush attempt adjusted for competition |

Yards/Pass | Yards per pass |

Yards/Rush | Yards per rush |

Yards/Pass+ | Yards per pass adjusted for competition |

Yards/Rush+ | Yards per rush adjusted for competition |

Exp Pass | Explosive plays (25+ yards) per pass |

Exp Run | Explosive plays (25+ yards) per rush |

Exp Pass+ | Explosive plays (25+ yards) per pass adjusted for competition |

Exp Run+ | Explosive plays (25+ yards) per rush adjusted for competition |

Comp% | Completion percentage |

Comp%+ | Completion percentage adjusted for competition |

Sack/Pass | Sacks per pass |

Sack/Pass+ | Sacks per pass adjusted for competition |

Sack/Pass* | Sacks per pass on passing downs |

Int/Pass | Interceptions per pass |

Neg/Run | Negative plays (<=0) per rush |

Neg/Run+ | Negative plays (<=0) per rush adjusted for competition |

Run Short | % Runs in short yardage situations |

Convert% | 3rd/4th down conversions |

Conv%* | 3rd/4th down conversions versus average by distance |

Conv%+ | 3rd/4th down conversions versus average by distance adjusted for competition |

Offensive Possession Stats

Points/Poss | Offensive points per possession |

EP3 | Effective Points per Possession |

EP3+ | Effective Points per Possession adjusted for competition |

Plays/Poss | Plays per possession |

Yards/Poss | Yards per possession |

Start Spot | Average starting field position |

Time of Poss | Average time of possession (in seconds) |

TD/Poss | Touchdowns per possession |

TO/Poss | Turnovers per possession |

FGA/Poss | Attempted field goals per possession |

Poss/Game | Possessions per game |

%RZ | Red zone trips per possession |

Points/RZ | Average points per red zone trip. Field Goals are included using expected points, not actual points. |

TD/RZ | Touchdowns per red zone trip |

FGA/RZ | Field goal attempt per red zone trip |

Downs/RZ | Turnover on downs per red zone trip |

PPP | Points per Possession |

aPPP | Points per Possession allowed |

PPE | Points per Exchange (PPP-aPPP) |

EP3+ | Expected Points per Possession |

aEP3+ | Expected Points per Possession allowed |

EP2E+ | Expected Points per Exchange |

EPA/Pass+ | Expected Points Added per Pass |

EPA/Rush+ | Expected Points Added per Rush |

aEPA/Pass+ | Expected Points Allowed per Pass |

aEPA/Rush+ | Expected Points Allowed per Rush |

Exp/Pass | Explosive Plays per Pass |

Exp/Rush | Explosive Plays per Rush |

aExp/Pass | Explosive Plays per Pass allowed |

aExp/Rush | Explosive Plays per Rush allowed |

BPR | A method for ranking conferences based only on their wins and losses and the strength of schedule. See BPR for an explanation. |

Power | A composite measure that is the best predictor of future game outcomes, averaged across all teams in the conference |

P-Top | The power ranking of the top teams in the conference |

P-Mid | The power ranking of the middling teams in the conference |

P-Bot | The power ranking of the worst teams in the conference |

SOS-Und | Strength of Schedule - Undefeated. Focuses on the difficulty of going undefeated, averaged across teams in the conference |

SOS-BE | Strength of Schedule - Bowl Eligible. Focuses on the difficulty of becoming bowl eligible, averaged across teams in the conference |

Hybrid | A composite measure that quantifies human polls, applied to converences |

Player Game Log

Use the yellow, red and green cells to filter values. Yellow cells filter for exact matches, green cells for greater values and red cells for lesser values. By default, the table is filtered to only the top 200 defense-independent performances (oEPA). The table includes the 5,000 most important performances (positive and negative) by EPA.

Use the yellow, red and green cells to filter values. Yellow cells filter for exact matches, green cells for greater values and red cells for lesser values. By default, the table is filtered to only the top 200 defense-independent performances (oEPA). The table includes the 5,000 most important performances (positive and negative) by EPA.

EPA | Expected points added (see glossary) |

oEPA | Defense-independent performance |

Team Game Log

Use the yellow, red and green cells to filter values. Yellow cells filter for exact matches, green cells for greater values and red cells for lesser values.

Use the yellow, red and green cells to filter values. Yellow cells filter for exact matches, green cells for greater values and red cells for lesser values.

EP3 | Effective points per possession (see glossary) |

oEP3 | Defense-independent offensive performance |

dEP3 | Offense-independent defensive performance |

EPA | Expected points added (see glossary) |

oEPA | Defense-independent offensive performance |

dEPA | Offense-independent defensive performance |

EPAp | Expected points added per play |

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