# Building a statistical model for field goal kicker accuracy

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my proposed method for evaluating NFL kickers. To get all the results and

some extra info about the data, check it out.

When you have a hammer, everything looks like a nail, right? Well I’m a big

fan of multilevel models and especially the ability of MCMC estimation to

fit these models with many, often sparse, groups. Great software implementations

like Stan and my favorite R interface to it,

`brms`

make doing applied

work pretty straightforward and even fun.

As I spent time lamenting the disappointing season my Chicago Bears have been

having, I tried to think about how seriously I should take the shakiness of

their new kicker, Eddy Pineiro. Just watching casually, it can be hard to really

know whether a kicker has had a very difficult set of kicks or not and what

an acceptable FG% would be. This got me thinking about how I could use what

I know to satisfy my curiosity.

## Previous efforts

Of course, I’m not exactly the first person to want to do something to account

for those differences. Chase Stuart over

at Football Perspectives

used field goal distance to adjust FG% for difficulty (as well as comparing

kickers across eras by factoring in generational differences in kicking success).

Football Outsiders

does something similar — adjusting for distance — when grading kickers.

Generally speaking, the evidence suggests that just dealing with kick distance

gets you very far along the path to identifying the best kickers.

Chris Clement at the

Passes and Patterns blog

provides a nice review of the statistical and theoretical approaches to the

issue. There are several things that are clear from previous efforts, besides

the centrality of kick distance. Statistically, methods based

on the logistic regression model are appropriate and the most popular —

logistic regression is a statistical method designed to predict binary events

(e.g., making/missing a field goal) using multiple sources of information. And

besides kick distance, there are certainly elements of the environment that

matter — wind, temperature, elevation among them — although just how much

and how easily they can be measured is a matter of debate.

There has also been a lot of interest in game situations, especially clutch

kicking. Do kickers perform worse under pressure, like when their kicks will

tie the game or give their team the lead late in the game? Does “icing” the

kicker, by calling a timeout before the kick, make the kicker less likely to

be successful? Do they perform worse in playoff games?

On **icing**, Moskowitz and Wertheim (2011),

Stats, Inc.,

Clark, Johnson, and Stimpson (2013),

and LeDoux (2016)

do not find compelling evidence that icing the kicker is effective. On the other

hand,

Berry and Wood (2004)

Goldschmied, Nankin, and Cafri (2010),

and Carney (2016)

do find some evidence that icing hurts the kicker. All these have some limitations,

including which situations qualify as “icing” and whether we can identify

those situations in archival data. In general, to the extent there may be an

effect, it looks quite small.

Most important in this prior work is the establishment of a few approaches to

quantification. A useful way to think about comparing kickers is to know what

their **expected FG%** (eFG%) is. That is, given the difficulty of their kicks,

how

would some hypothetical average kicker have fared? Once we have an expected FG%,

we can more sensibly look at the kicker’s actual FG%. If we have two kickers

with an actual FG% of 80%, and kicker A had an eFG% of 75% while

kicker B had an eFG% of 80%, we can say kicker A is doing better because he

clearly had a more difficult set of kicks and still made them at the same rate.

Likewise, once we have eFG%, we can compute **points above average** (PAA).

This is fairly straightforward since we’re basically just going to take the

eFG% and FG% and weight them by the number of kicks. This allows us to

appreciate the kickers who accumulate the most impressive (or unimpressive)

kicks over the long haul. And since coaches generally won’t try kicks they

expect to be missed, it rewards kickers who win the trust of their coaches and

get more opportunities to kick.

Extensions of these include **replacement FG%** and **points above replacement**, which use replacement level as a reference point rather than

average. This is useful because if you want to know whether a kicker is playing

badly enough to be fired, you need some idea of who the competition is.

PAA and eFG% are more useful when you’re talking about greatness and who

deserves a pay raise.

### Statistical innovations

The most important — in my view — entries into the “evaluating kickers with

statistical models” literature are papers by

Pasteur and Cunningham-Rhoads (2014) as well as

Osborne and Levine (2017).

Pasteur and Cunningham-Rhoads — I’ll refer to them as PC-R for short —

gathered more data than most predecessors, particularly in terms of auxiliary

environmental info. They have wind, temperature, and presence/absence of

precipitation. They show fairly convincingly that while modeling kick distance

is the most important thing, these other factors are important as well. PC-R

also find the cardinal direction of every NFL stadium (i.e., does it run

north-south, east-west, etc.) and use this information along with wind direction

data to assess the presence of cross-winds, which are perhaps the trickiest for

kickers to deal with. They can’t know about headwinds/tailwinds because as far

as they (and I) can tell, nobody bothers to record which end zone teams defend

at the game’s coin toss, so we don’t know without looking at video which

direction the kick is going. They ultimately combine the total wind and the

cross wind, suggesting they have some meaningful measurement error that makes

them not accurately capture all the cross-winds. Using

their logistic regressions that factor for these several factors, they calculate

an eFG% and use it and its derivatives to rank the kickers.

PC-R

include some predictors that, while empirically justifiable based on their

results, I don’t care to include. These are especially indicators of defense

quality, because I don’t think this should logically effect the success of a

kick and is probably related to the selection bias inherent to the coach’s

decision to try a kick or not. They also include a “kicker fatigue” variable

that appears to show that kickers attempting 5+ kicks in a game are less

successful than expected. I don’t think this makes sense and so I’m not

going to include it for my purposes.

They put some effort into defining a

“replacement-level” kicker which I think is sensible in spite of some limitations

they acknowledge. In my own efforts, I decided to do something fairly similar

by using circumstantial evidence to classify a given kicker in a given

situation as a replacement or not.

PC-R note that their model seems to overestimate the probability of very long

kicks, which is not surprising from a statistical standpoint given that there

are rather few such kicks, they are especially likely to only be taken by those

with an above-average likelihood of making them, and the statistical assumption

of linearity is most likely to break down on the fringes like this. They also

mention it would be nice to be able to account for kickers having different

leg strengths and not just differing in their accuracy.

Osborne and Levine (I’ll call them OL) take an important step in trying to

improve upon some of these limitations. Although they don’t use this phrasing,

they are basically proposing to use multilevel models, which treat each kicker

as his own group and thereby accounting for the possibility — I’d say it’s a

certainty — that kickers differ from one another in skill.

A multilevel

model has several positive attributes, especially that it not only adjusts for

the apparent differences in kickers but also that it looks skeptically upon

small sample sizes. A guy who makes a 55-yard kick in his first career attempt

won’t be dubbed the presumptive best kicker of all time because the model will

by design account for the fact that a single kick isn’t very informative. This

means we can simultaneously improve the prediction accuracy on kicks, but also

use the model’s estimates of kicker ability without over-interpreting

small sample sizes. They also attempt to use a quadratic term for kick

distance, which could better capture the extent to which the marginal

difference of a few extra yards of distance is a lot different when you’re at

30 vs. 40 vs. 50 yards. OL are unsure about whether the model justifies

including the quadratic term but I think on theoretical grounds it makes a lot

of sense.

OL also discuss using a clog-log link rather than the logistic link, showing

that it has better predictive accuracy under some conditions. I am going to

ignore that advice for a few reasons, most importantly because the advantage is

small and also because the clog-log link is computationally intractable with

the software I’m using.

## Model

on Github

My tool is a multilevel logistic regression fit via

MCMC using the wonderful

`brms`

R package.

I actually considered several models for model selection.

In all cases, I have random intercepts for kicker and stadium. I also use

random slopes for both kick distance and wind at the stadium level. Using

random wind slopes at the stadium level will hopefully capture the *prevailing*

winds at that stadium. If they tend to be helpful, it’ll have a larger absolute

slope. Some stadiums may have swirling winds and this helps capture that as

well. The random slope for distance hopefully captures some other things, like

elevation. I also include interaction terms for wind and kick distance as well

as temperature and kick distance, since the elements may only affect longer

kicks.

There are indicators for whether the kick was “clutch” — game-tying or

go-ahead in the 4th quarter — whether the kicker was “iced,” and whether

the kick occurred in the playoffs. There is an interaction term between

clutch kicks and icing to capture the ideal icing situation as well.

I have a binary variable indicating whether the kicker was, at the time, a

replacement. In the main post, I describe the

decision rules involved in that. I have interaction terms for replacement

kickers and kick distance as well as replacement kickers and clutch kicks.

I have two random slopes at the kicker level:

- Kick distance (allowing some kickers to have stronger legs)
- Season (allowing kickers to have a career trajectory)

Season is modeled with a quadratic term so that kickers can decline over

time — it also helps with the over-time ebb and flow of NFL FG%. It would

probably be better to use a GAM for this to be more flexible, but they are a

pain.

All I’ve disclosed so far is enough to have one model. But I also explore

the form of kick distance using polynomials. OL used a quadratic term, but I’m

not sure even that is enough. I compare 2nd, 3rd, and 4th degree polynomials

for kick distance to try to improve the prediction of long kicks in particular.

Of course, going down the road of polynomials can put you on a glide path

towards the land of overfitting.

I fit several models, with combinations of the following:

- 2nd, 3rd, or 4th degree polynomial
`brms`

default improper priors on the fixed and random effects or

weakly informative normal priors on the fixed and random effects- Interactions with kick distance with either all polynomial terms or just

the first and second degree terms

That last category is one that I suspected — and later confirmed — could

cause some weird results. Allowing all these things to interact with a 3rd

and 4th degree polynomial term made for some odd predictions on the fringes,

like replacement-level kickers having a predicted FG% of 70% at 70 yards out.

## Model selection

I looked at several criteria to compare models.

A major one was

approximate leave-one-out cross-validation.

I will show the LOOIC, which is interpreted like AIC/BIC/DIC/WAIC in terms of

lower numbers being better. This produces the same ordering as the ELPD, which

has the opposite interpretation in that higher numbers are better.

Another thing I looked at was generating prediction weights for the models via

Bayesian model stacking.

I also calculated Brier scores,

which are a standard tool for looking at prediction accuracy for binary

outcomes and are simply the mean squared prediction error. Last among the

quantitative measures is the

AUC

(*a*rea *u*nder the *c*urve), which is another standard tool in the evaluation

of binary prediction models.

Beyond these, I also plotted predictions in areas of interest where I’d like

the model to perform well (like on long kicks) and checked certain cases

where external information not shown directly to the model gives me a relatively

strong prior. Chief among these was whether it separated the strong-legged

kickers well.

Below I’ve summarized the model comparison results. I shade the metrics

darker wherever the number is better — sometimes lower numbers are better,

sometimes higher numbers are. The bolded, red-colored row is the model I

used.

Polynomial degree | Interaction degree | Priors | LOOIC | Model weight | Brier score | AUC |
---|---|---|---|---|---|---|

3 | 2 | Proper | 8030.265 | 0.5083 | 0.1165 | 0.7786 |

4 | 2 | Proper | 8032.332 | 0.1533 | 0.1164 | 0.7792 |

3 | 2 | Improper | 8032.726 | 0.0763 | 0.1160 | 0.7811 |

3 | 3 | Proper | 8033.879 | 0.0001 | 0.1165 | 0.7787 |

3 | 3 | Improper | 8035.616 | 0.2042 | 0.1158 | 0.7815 |

4 | 2 | Improper | 8036.614 | 0.0001 | 0.1158 | 0.7816 |

4 | 4 | Proper | 8038.266 | 0.0000 | 0.1163 | 0.7793 |

2 | 2 | Proper | 8043.221 | 0.0000 | 0.1169 | 0.7778 |

2 | 2 | Improper | 8043.450 | 0.0577 | 0.1165 | 0.7798 |

4 | 4 | Improper | 8043.741 | 0.0000 | 0.1156 | 0.7825 |

So why did I choose that model? The approximate LOO-CV procedure picks it as

the third model, although there’s enough uncertainty around those estimates

that it could easily be the best — or not as good as some ranked below it.

It’s not clear that the 4th degree polynomial does a lot of good in the

models that have it and it increases the risk of overfitting. It seems to

reduce the predicted probability of very long kicks, but as I’ve thought about

it more I’m not sure it’s a virtue.

Compared to the top two models, which distinguish themselves from the chosen

model by their use of proper priors, the model I chose does better on the

in-sample prediction accuracy metrics without looking much different on the

approximate out-of-sample ones. It doesn’t get much weight because it doesn’t

have much unique information compared to the slightly-preferred version with

proper priors. But as I looked at the models’ predictions, it appeared to me

that the regularization with the normal priors was a little too aggressive

and wasn’t picking up on the differences among kickers in leg strength.

That being said, the choices among these top few models are not very important

at all when it comes to the basics of who are the top and bottom ranked kickers.

## Notes on predicted success

I initially resisted including things like replacement status and anything else

that is a fixed characteristic of a kicker (at least within a season) or

kicker-specific slopes in the

model because I planned to extract the random intercepts and use that as my

metric. Adding those things would make the random intercepts less interpretable;

if a kicker is bad and there’s no “replacement” variable, then the intercept

will be negative, but with the “replacement” variable the kicker may not have

a negative intercept after the adjustment for replacement status.

Instead, I decided to focus on model predictions. Generating the expected

FG% and replacement FG% was pretty straightforward. For eFG%, take all kicks

attempted and set `replacement = 0`

. For rFG%, take all kicks and set

`replacement = 1`

.

To generate kicker-specific probabilities, though, I had to decide how to

incorporate this information. I’d clearly overrate new, replacement-level

kickers. My solution to this was to, before generating predictions on

hypothetical data, set each kicker’s `replacement`

variable to his career

average.

For `season`

, on the other hand, I could eliminate the kicker-specific

aspect of this by intentionally zeroing these effects out in the

predictions. If I wanted to predict success in a specific season, of course,

I could include this.

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