Sunday, March 23, 2014

Classifying WRs, TEs, and RBs by Where They Catch the Ball

Abstract
Principal Component Analysis (PCA) is a useful tool to simplify complex datasets. The results of the PCA can be then used either to reconstruct the original data or to classify it into different groups. In this post I apply PCA to reception data for a sample of 150+ NFL receivers. I find that PCA generally does a good job of discriminating between wide receivers, tight ends, and running backs. A few tight ends, however – generally ones known more for their use as receivers than blockers – have significant overlap with the wide receivers. This result indicates that PCA may be useful for determining how to designate players, Jimmy Graham for example, for the franchise tag.

Introduction

Alright, so I lied. Well, partially – I am very busy with job applications, but I've also been teaching myself some new machine learning techniques (mostly from this excellent textbook) and they're just so damn cool that it's been hard not to think of ways to apply them to NFL data.

One of these methods is called Principal Component Analysis (PCA for short), and it's designed to reduce a large, complex dataset down into its most important pieces. These pieces (the 'component' part of PCA) can be used as basis functions to reconstruct the original data with minimal information loss, providing a form of data compression. Or, the coefficients for a given component can be compared between all the observations in a dataset, and trends in these coefficients may be used to classify the data into groups.

One of the great things about PCA is that it relies on no assumptions about how the data are distributed. This means that PCA can be used on just about anything. Something that is especially appropriate for a PCA is the distribution of yardage gained by a player every time they touch the ball. Credit where credit is due: the analysis in this post is partly inspired by Brian Burke of Advanced NFL Stats, who looked at the distribution of yards gained (or lost) on rush attempts in an effort to distinguish between power running backs and smaller, faster RBs. Burke (largely visually) compared the raw yardage histograms, and found that there were only small differences between each type of back. Burke suggests using a gamma distribution to parameterize these gains, although given the distinct rush distribution for each player he shows it seems unlikely that every running back will be well-represented by such a (relatively) simple model (to his credit Burke himself is quite upfront about this). PCA allows us to produce accurate representations of such data without choosing a distribution a priori, which means we don't have to worry about limiting or biasing ourselves by such a decision.

For this post I'll apply PCA to reception statistics rather than rush attempts. One reason for my choice is that Burke's analysis (despite the limitations I mentioned earlier) is pretty thorough, and I prefer to break new ground when I can. The other (more interesting) reason is that while most rush attempts come from a single position group (RBs), the target for a pass attempt can be a WR, TE, or RB. So in addition to looking for differences between possession receivers and home run threats it's also possible to see how the different positions are utilized on passing plays.

Data and Model
I queried my copy of the Armchair Analysis database (which spans the 2000-2011 seasons) and grabbed the yardage gained from every reception for each player with 200+ catches in the database. (I impose this reception threshold to ensure that statistical noise doesn't dominate the data.) The final sample consists of 114 wide receivers, 37 tight ends, and 33 running backs. The reception distribution of the total dataset is shown in Figure 1.
Figure 1: Distribution of all receptions in the sample. It has a strong peak at a gain of around 7-10 yards, with a long tail showing big passing plays. 
I next computed the reception distribution for each player, then ran the PCA. The details of exactly how PCA works are beyond the scope of this blog, but I'll give a brief overview of the method here so that at least the general concept is (hopefully) clear.

First off, each player's reception distribution is normalized so receivers with more catches don't bias the analysis, and the mean yardage distribution for the whole dataset is subtracted. From this point the algorithm gets to work, computing a function which minimizes the variation in the residual data. This process is repeated, and each successive iteration accounts for more and more of the fine details of the dataset. Eventually (when the number of iterations approaches the number of players in the sample) the PCA will perfectly reproduce the original data. Of course, that sort of exact duplication isn't the point of PCA; rather since the most variation is explained by the first components, the goal is to truncate the algorithm after only N iterations, where N is much smaller than the number of players in the dataset.

The script I wrote to do this analysis can be found here. It's a fairly long program, but a large chunk of it is just to make the diagnostic plots to show how well the PCA worked – the meat of the PCA happens between lines 107-115.

Results
Figure 2: The first four components of the PCA as a function of reception yards. The first component is the average of all the players, while subsequent components have been computed by the PCA algorithm. Components beyond the second and third are very jagged, signs that they are fitting individual player variation rather than useful information.

I ran the PCA on the reception data to N = 15, but a look at the first four components (Figure 2) indicates that after the first couple of iterations the PCA is mostly fitting differences in the reception distributions for individual players. I can prove (hopefully) this to you via Figure 3.
Figure 3: Sample PCA reconstruction for Anquan Boldin, showing both the original reception distribution (in black) and a reconstruction using the first three PCA components (red). The reconstruction generally does a good job of mimicking the data even with only three components.
In addition to providing the maximal reduction in variance, the PCA also provides a list of coefficients for each component. These coefficients can be used with the components to produce a reconstruction of the original data – in the case of Figure 3, for Anquan Boldin. You can see that just the first three PCA components are required to recover a fairly good representation of Boldin's catch distribution – consistent with what the shape of the components indicated in Figure 2.

Now that we have verified that the PCA is working as intended, we can get to the good stuff – using the PCA to differentiate between players. As I mentioned earlier the data contain WRs, TEs, and RBs. A plot of the coefficients of the first PCA component (PCA1), color-coded by player position, is shown in Figure 4.
Figure 4: The distribution of the first PCA coefficient. Note how WRs are cleanly separated from RBs, while TEs partially overlap with WRs. 
This figure is quite striking – running backs all cluster with (relatively) large coefficients, while nearly every wide receiver has negative values for PCA1. Tight ends tend to fall in the middle, although there is substantial overlap with the wideouts. What this means is that there is something inherently different about where each position grouping tends to catch the ball (and by extension, what routes they run). This is not inherently surprising, given it's fairly easy to see this just by watching how players at the different positions move during a game.

Discussion and Conclusions
What is interesting, however, is the fact that tight ends and wide receivers aren't as cleanly separated from each other as they are from running backs. In fact, while TEs and WRs are clearly not drawn from the same distribution, there is definitely some overlap. This implies that some TEs are being used more like wideouts. Additional evidence for this hypothesis comes from looking at which tight ends are most and least 'wide receiver-like'. Table 1 lists the top and bottom five TEs, sorted by PCA1.

Table 1: Tight Ends with Extreme PCA1 Values
Most WR-likeLeast WR-like
NamePCA1NamePCA1
Owen Daniels-3.8x10-2Steve Heiden6.8x10-2
Antonio Gates-2.9x10-2Donald Lee5.2x10-2
Tony Gonzalez-2.4x10-2Bubba Franks4.6x10-2
Marcedes Lewis-1.4x10-2Eric Johnson3.9x10-2
Tony Scheffler-7.8x10-3Freddie Jones3.7x10-2

The left side of Table 1 generally contains TEs, most notably Antonio Gates and Tony Gonzalez, who are able pass-catchers. The right-hand side, however, consists of players generally not known for their receiving ability. It seems prudent to reiterate here that I'm not claiming that PCA1 is a predictor of skill in any way; rather it merely indicates that some tight ends are being used more like wide receivers than others.

Aside from being a cool result on its own, it also provides a way to classify players based on a statistic that's directly comparable between positions. This is especially relevant right now, as New Orleans Saints TE Jimmy Graham attempts to be treated as a wideout for the purposes of contract negotiation. You can read up on the details for yourself, but the upshot is that if Graham can get himself classified as a WR he can earn himself an extra $5 million over what he would get as a TE. A lot of the discussion has centered around statistics, such as where Graham lines up before the snap or how many receptions he had last year, that aren't directly comparable between wideouts and tight ends.

Unfortunately my data isn't current enough to actually include Graham in this sample (ditto Rob Gronkowski, just FYI), but I would bet that he winds up in the same regime as Gates and Gonzalez. Regardless of whether my intuition is correct, however, PCA provides a way to directly compare players at different positions based only on very basic data, and therefore it could be a very useful tool for position disputes like these.


Monday, March 17, 2014

Hiatus

Hi everyone!

As some of you may know, I'm on the hunt for a new job – outside of academia. Unfortunately, sending out applications takes a lot of time, and despite my best efforts I can't keep doing football analytics and give the job search the attention it needs. So I'm putting PhD Football on hiatus while I figure out what I'm doing next. Once I get my work situation in order I'll get back to the blog – hopefully soon! (And if you happen to know of anyone hiring science PhDs I'd love to hear about it.)

Monday, March 3, 2014

First Down Probability

Abstract
In this post I compute the First Down Probability metric, which predicts how likely a drive will produce at least one more first down for a given down and distance. I find similar overall first down conversion rates to prior studies in the literature, including that third down rushing plays are significantly underutilized. Unlike previous studies, however, I break down these rushing plays by the position of the ballcarrier, and find that a significant portion of this discrepancy comes from rushes by the quarterback, likely from scrambles on broken passing plays. More puzzling is the fact that QB runs on first and second downs don't show this trend, a result that is difficult to convincingly explain.

Introduction
During the course of a football game a fan gets a lot of statistical information. These numbers – QB rating, a running back's average yards per carry, time of possession, etc – generally lack any kind of contextual information about how the game is actually going. At best these statistics are incomplete (showing a WR's average yards per catch after a 99-yard completion, for instance); at worst, they're downright misleading (That QB just had 5 completions in a row...but they were all screens for minimal yardage).

A better statistic is one that takes the game situation into account. For instance, a 5-yard completion should count for more on third and 4 than on third and 16. There are several such statistics already in existence, such as Football Outsider's DVOA metric or Expected Points. These sorts of metrics generally depend on using historical play-by-play data to compute average outcomes for plays at any given down and distance. This approach is (unsurprisingly) more computationally complex, and often can appear opaque to the casual fan. Some of these stats, such as the DVOA, are intricate enough that their creators have decided to keep the full details of their computation private.

A direct and (relatively) simple context-sensitive statistic is Brian Burke's First Down Probability, which I will abbreviate as FDP. That link has more details, but the core insight of this metric is that the average odds of converting the next first down in a series can be estimated for any given down and distance. With this information in hand, it's possible to evaluate the result of a play based on whether it improves or harms the offense's chance of eventually getting a first down.

In this post I'm going to compute the FDP for the plays in the Armchair Analysis database. One may ask why I would recompute this quantity when Burke has already done quite a good job of it. One reason is to ensure the reproducibility of results – while I trust Burke's analysis, everyone makes mistakes. A more basic reason is that while Burke produces a nice visualization of his computed FDP he doesn't provide his data in a tabular form, which makes using his FDP values difficult (at best). I can also extend the FDP calculation to all four downs (Burke only considers second and third downs in his post). Finally, I can (spoiler alert) start using FDP to generate new insights about how teams approach different down-and-distance situations.

Data
As I mentioned before, I'm using the Armchair Analysis database, which covers the 2000-2011 NFL seasons. I grabbed the play-by-play data for all regular season and playoff games, then filtered out plays for several reasons. Plays inside the two minute warnings were discarded because teams play differently in those situations; I removed plays when the game wasn't close (defined as one team being up by more than 16 points) for the same reason. I cut out all punts and field goals as well as penalties (although I keep the results of the penalties in the data: if a team runs for -5 yards on second down but then is the beneficiary of a 15-yard roughing the passer call on third down, the second down play would be considered as ultimately resulting in a first down for the purposes of this analysis). Finally, to avoid biasing the data based on field position I only include plays between the offense's own 10-yard line and the redzone. 

Ultimately this results in a dataset of 262,601 plays, split 56%-44% in favor of passes over runs. I bin these plays as a function of current down and yards to go, eliminating bins with fewer than 200 plays in my dataset. This cut ensures that there are no bins with conversion rates dominated by sampling error. The Python script I used to do this data querying and processing (as well as produce the plots in later sections) can be found here.

Results
Figure 1: FDP as a function of down and distance. The colors denote different downs, while the line styles break down success if the next play in the drive is a run or a pass. In some cases the data for the individual types of plays does not cover the same range of yards to gain. This is due to the minimum play cutoff detailed in the Data section.
Figure 1 shows the raw results, split by down and distance. For the benefit of anyone looking to check my results or to build on them I have also tabulated these results in text files, which can be obtained from my GitHub repository. Feel free to use them as long as you explain where you got them from (and a link back here would be nice as well!). 

Anyway, the first thing to do is to check my results with what Burke obtained. It's a bit difficult to compare directly since I can only eyeball our plots, but in general my results seem to be fairly copacetic with his. The data between downs look fairly similar, with a ~15% shift each down as you go from first and N to second and N, increasing to ~20% from second to third down. There's not much data on fourth down, but I see no reason why it wouldn't resemble the other downs for conversion attempts beyond 2 yards.

More interesting is what happens when you break the conversion percentages down by type of play. Note that when comparing the FDP of runs versus passes at a given down and distance, a higher conversion rate for e.g. a pass doesn't necessarily mean you should always throw the ball in that situation; rather, it implies that currently NFL teams are not playing at the Nash equilibrium. This means that NFL teams should call more passing plays in that situation than they currently do; as defenses adjust to this new reality, there should be more opportunities for successful rushing plays, and eventually the FDP of both types of plays will equalize. Burke has some more detailed discussion of this in his breakdown of first down probability for runs and passes (although he restricts his analysis to third downs).

So again we are treading on old ground, and again it makes sense to compare results. Here we find a bit of a discrepancy, with Burke's rushing FDP on third and short ~5% lower than mine. It's not clear why this would be, although it might be due to the fact that Burke's data only goes through the 2007 season or how he considers sacks (the Armchair Analysis database considers sacks to just be really crappy passes). Regardless, things appear similar enough to proceed.

It's clear that teams aren't passing enough on first and second downs with more than 5 yards to go. Considering teams are already passing a lot in those situations, especially in second and 10+, this would imply that even the occasional rush in such circumstances is too much.

In short yardage, however, things are reversed. On second and 3 or less teams are running less often than they 'should', although the difference is only at about 7% or so. Third down is even more striking: whenever there are fewer than 9 yards to go the data indicate that teams should be running more. This is an even larger discrepancy than Burke finds, and is downright shocking given how unusual 7+ yard runs are under normal circumstances.

But there are two kinds of runs – designed runs and aborted passing plays. Burke considers the latter category to be rare enough to be inconsequential, but I wasn't so certain. So I modified my program to separate out rushes by the position of the ballcarrier – it can't tell if a QB rush was designed that way or if it was improvised, but it's better than nothing.
Figure 2: FDP, corrected for the influence of QB runs (the uncorrected rushing percentages are shown in gray to facilitate direct comparison). 

Figure 2 shows the result, and it turns out that without the QB involved a third down rush becomes a much worse proposition. Indeed, now teams should only be running more on third and 3 or less, consistent with what the data show for second down.

While teams are generally doing better at finding the equilibrium between passes and rushes with RBs, these results indicate that teams are letting their signal-callers run the ball far too infrequently. If you look at the conversion rates just for QB scrambles it's generally 10% or more higher than a rush from a running back in the same situation! Even more interesting is that this offset only applies on third down. On first and second down a QB scramble appears to have similar conversion rates as a regular rush.

Discussion and Conclusions
First of all, the fact that QB rushes are so underused compared to other types of plays is quite interesting. Given the fact that teams generally do not want their prize passers taking hits down the field, most of these successful conversions are likely due to scrambles on passing attempts. But given how high the conversion rate is perhaps coaches should consider running a few more QB draw plays, especially with all the mobile passers entering the league. 

But what's really weird is that QB's rushes aren't more successful than the regular variety on earlier downs. A possible explanation is that defenses are more keyed toward stopping shorter-yardage plays on second down, whereas on third down they sit back and follow the WRs down the field. But in that case you would expect third down rushes to be equally successful, regardless of the runner. I think it's more likely that on second down a QB under pressure isn't concerned with making the sticks, but rather simply looks to get out of trouble. On third down, however, the consequences of playing it safe are much more clear, which encourages passers to scramble for every last yard. 

Of course, I'll be the first to admit that this is just speculation. A definitive analysis of this phenomenon would probably require deep analysis of individual quarterback scrambles, which is way beyond the scope of this work. But it is a cool result from a (relatively) simple metric, and illustrates how deep insights can be gleaned from just a little bit of intelligent digging.

Monday, February 17, 2014

What Positions Do Teams Value in the Draft?

Abstract
Where players are taken in the NFL draft is based not only on their raw skill and potential, but is highly influenced by the perceived value of the position they play as well as the overall supply of players at that position. While quarterback is clearly the most important position on the team, investigating where players at other positions get drafted may provide insights into how NFL teams evaluate the relative importance of those positions. In this post I do a couple of simple analyses of where players get drafted, breaking the data down by position groupings and finding that there are slight variations in where players at different positions can expect to be taken, although more draft data and/or deeper analyses would be necessary to decisively show disparities between the positions.

Introduction
It's pretty clear that quarterback is the most important position on a football team, and their value is reflected in the draft – in the last decade 13 QBs have been taken in the top five picks. But exactly how much are QBs favored by GMs and coaches come May? And what about the perceived value of other NFL position groups?

Data
This one's fairly easy, as Armchair Analysis lists where each player was drafted as well as what position they play in the same table. I downsampled the data to include only players drafted since 2001 (up to 2011, the last year in the database), because the table contained only partial records before that year. You can find the script I used here.

There is also a limit to the granularity of the positions in the database. For instance, no distinction is made between any of the players on the offensive or defensive lines. This does limit how detailed I can make my analyses, and there may be significant difference in the valuations between positions on the O-line (for instance, a left tackle–protecting a passer's blind side–is likely to be more highly sought-after than a right tackle, although things are never that clear-cut).

Results
I first took a look at the raw data, plotting what fraction of picks go to each position grouping in Figure 1. To improve the signal-to-noise of the data (and just make things easier to visualize) I binned the data in groups of 10 picks. 
Figure 1: Percentage of players drafted at each position, as a function of draft position. Kickers added for scale.
While colorful, the stacked nature of this figure makes it somewhat difficult to parse. Figure 2 shows where players of each position get drafted, independent of any other position groups in the sample. While there are still a bunch of overlapping lines, it's now easy to see if and where teams prefer to draft players at each position.
Figure 2: Where players in each position grouping get picked. Most of the positions have flat distributions. The only notable exception to this trend are QBs, which are highly peaked near the first few picks.

What stands out most is the large (and unsurprising) upturn in QB picks in the first bin. Otherwise, however, there don't appear to be any obvious trends. But the eye can deceive, so let's try to be just a bit more rigorous. Toward that end I computed the expectation value for each position. You can get more detail about the expectation value from Wikipedia, but in this case it's basically just the average place players at each position get drafted. You can't make a pretty graph with it, but you can see the results in Table 1.

Table 1: Draft position as a function of player position
PositionQBRBWRTEOLDLLBDBK
Expected Draft Position102±8.1126±5.1117±4.4131±5.8122±4.0114±3.9118±4.2119±3.2164±9.1

Table 1 starts off by confirming what we saw visually in the figures – quarterbacks are by far the most sought-after position, with an expected draft position 12 picks higher than any other position (although the bootstrapped standard deviations are just consistent with defensive linemen being equally valued). Wide receivers are drafted slightly earlier than running backs, a trend that's been picking up steam in recent years as teams realize that RBs tend to have short careers and therefore don't provide as much value for an early pick. Linemen are the highest-drafted of all defensive position groupings, probably driven by teams' desire to press the point of attack – the general wisdom is that QBs have an advantage against the secondary due both to their skill and rules restricting contact on WRs (although the Seattle Seahawks would beg to differ) , and that generating pressure and sacks is seen as the best way to defend the pass. More data would be required to definitively prove that these differences are real, however.

Discussion and Conclusions
While it's unsurprising that QBs are the hottest position in the draft, it's nice to see it confirmed with numbers. More interesting are the expectation values for the other positions, which while much closer to each other could be used as signifiers of broad NFL trends in how talent is evaluated between positions. The analyses presented here are pretty simplistic, but they are indicative of the potential power of draft data. 

Monday, February 3, 2014

Do Defenses Get Tired?

One of the hallmarks of good science is reproducibility – the ability for other researchers to repeat (and thus verify or disprove) your work. While I hope I have laid out enough details in each of my posts for anyone interested to check my analyses, I am happy to report that I will now be uploading my code for each post to GitHub. Check it out!

Abstract
One of NFL announcers' favorite statistics is the time of possession, which is usually discussed in the context of how tired the defense must be when they've been on the field for a long time. But do defenses actually get fatigued over the course of the game? To answer that question I used the raw number of plays a defense is on the field (rather than the less accurate time of possession) and computed the probability that the offense will score as a function of this number. Ultimately, even after 70+ plays there is no increase in the offense's point production – a clear indication that defensive players have plenty of endurance to make it through even the longest games.

Introduction
A common statistic to see quoted during a game is time of possession (lazily referred to as ToP in the rest of this post). Usually referenced between quarters or near the end of the game, commentators generally talk about ToP in the context of noting how long one team's defense has spent on the field. (Offenses generally have more flexibility in keeping their players fresh through skill package substitutions.) The not-very-subtle implication is that the defense is getting worn down by the amount of time they've been playing and will therefore be more likely to allow points.

This is, of course, largely bullshit. Since so much more time is spent between running plays than actually ticks by while the football is in motion, ToP is really only a good indicator of how much standing around the teams are doing. Additionally, since the game clock stops for an incomplete pass (and pass-heavy offenses tend to pick up yards in chunks and have shorter drives as a result) ToP is naturally skewed towards favoring rushing offenses. If ToP was only collected during a play it might have some value, or better yet just strap some pedometers onto the players and figure out how much they're really running around on the field. 

The idea at the core of ToP, however – that a defense spending more energy on the field may eventually show signs of fatigue and therefore allow more points – is not unreasonable. The current ToP statistic is just a terrible way of measuring it. This question is especially interesting because if defenses do get tired over the course of a game it would add more value to a strong rushing attack, a facet of the offense that has come
under significant fire in recent years as being strictly inferior to the passing game. 

While perhaps not quite as good as my earlier pedometer suggestion, the raw number of plays run should be a much better proxy than ToP for investigating whether defenses get tired. By comparing the results of drives as a function of the number of plays run will therefore indicate whether defenses ever become fatigued enough to affect play.

Data
I started with all the play-by-play data in the Armchair Analysis database, and computed the beginning and end of each drive as well as whether any points were scored. By separating this data out between the home and away teams for each game I constructed a running tally of the number of plays run by the offense at the start of each drive.

Before getting into the results it's important to note that for this analysis the devil really is in the details. The data can be biased in many ways, some subtle and some not.  First and perhaps most obvious is that while all games can be expected to start in a similar way, a drive in the 4th quarter of a blowout is going to look much different than one in a close game. To avoid this problem I restricted my sample only to games where the final tally is within one score (8 points). I also throw out special teams plays, as I am most focused on how the defense plays as a unit (although note that on most teams at least some special teams players will see snaps on the offense or defense).

Another issue is penalties. Most infractions are only called after the play is over, and even though (if the penalty is accepted) the original play doesn't count for statistical purposes I still want to count it for this analysis. Some penalties, however, result in the refs immediately blowing the play dead (the most notable examples of this being false starts and encroachment). These penalties I strip out from the final play-counts. Occasionally a penalty occurs after the play is over (e.g. many unsportsmanlike conduct calls). A dead-ball foul should be purged from the data; unfortunately (as far as I can tell) there is no indication in the database whether a penalty is a dead-ball infraction or not, so I choose to leave all of these penalties in my sample. Fortunately these types of penalties are relatively infrequent, and therefore shouldn't significantly affect the results.

Lastly, drives near the end of halves create significant additional bias as well, since many of them are kneel-downs or result in unusual play-calling (Hail Mary passes, record-setting field goals, etc). I cut out the result of any drive that starts within the 2-minute warning of either half, although I include the plays run on those drives in the running totals of plays run during the game.

It is also worth noting that occasionally there are errors in the database, where the down sequence counter I use to determine the length of each drive is not reset between possessions. This issue is most obvious in the existence of some unusually long (20+ play) drives, although it likely affects shorter drives as well. Generally the incidence of these errors is very low (there are only ~10 of these very long drives in the entire sample, for instance), so I do not believe they will bias the results – especially not for shorter drives, where the sheer number of actual drives should drown out the few erroneous ones.

Results
Before diving into the full analysis, I think it's interesting to look at some raw numbers about NFL drives that aren't usually discussed. Take a look at the distribution of drive lengths in Figure 1, and the distribution of drives per game in Figure 2. The plurality of drives take 3 plays, which makes sense as these are 3-and-out possessions. The occurrence of drives longer than ~6 plays is fairly well described by a power law (a straight line on this log-normal plot) with a cutoff at 21 plays (The few plays above this threshold are likely all spurious results as mentioned above).  Note that, assuming a team would punt on any 4th down, the maximum number of plays an NFL drive could take would be 30.
Figure 1: Distribution of drive lengths. After about 5 plays the frequency of drive length decreases quickly, and very few drives take more than ~15 plays.

While Figure 1 has home and away drives lumped together, I've left them separate in Figure 2 – it's pretty clear that there's no significant difference in the number of drives per game between the home team and the visitors. The distributions are well fit by a Gaussian distribution with an offset of almost exactly 10 drives and a standard deviation of a little less than two drives. This indicates that in a normal game a team will have less than 12 chances to score points – not a lot of opportunities! (It also implies that a team scoring 40+ points in a game is reaching the endzone on at least half of their possessions.)

Figure 2: Distribution of drives per game, for both home and away teams. There is very little difference between the home and away histograms. Solid lines show Gaussian fits to the data, which peak around 10 drives.
With the basics out of the way, now let's delve into the good stuff. I have the number of plays already run by the offense at the start of every drive lined up with the result of that drive. From there it's fairly straightforward to calculate the fraction of drives that end in scores as a function of the number of plays that have been run, which is shown in Figure 3.

Figure 3: Fraction of drives resulting in scores as a result of plays run. No trend is observed.
The errors on Figure 3 come from simple counting statistics, and the bin widths are adaptively chosen to have similar errors. If defenses really did fatigue as they spend more time running around on the field, the percentage of drives ending with points should increase as a function of the number of plays, but there is no evidence for this trend. If you look at touchdowns or field goals individually the picture remains the same – even if an offense runs 70+ plays the defense doesn't budge an inch. 

Discussion and Conclusions
It's pretty obvious from Figure 3 that defenses don't get fatigued during games. On a given drive the offense has a ~35% chance of scoring regardless of how much the defense has been on the field. If you look at how rushing averages change over the course of a game you reach the same essential conclusion, which is a good indication that my results are indeed accurate. While on the surface it seems totally reasonable that players would wear down as the game wears on, given the fact that the number of plays a team runs per game is a well known quantity it makes sense that players would have enough conditioning to make it well beyond even the longest of games. (It would be interesting to repeat this analysis for overtime games but my sample size is far too small.) 

So what are the implications of this result? Well, for one it means that announcers should stop talking about how long the defense has been on the field over the course of a game! More importantly, it means that there's one less reason for teams to rely on running the ball – if a coach feels that throwing deep every play best suits the talent on his offense, they should feel free to do so without consideration for their defense.

Monday, January 20, 2014

Playoff Fairness Through Win-Loss Records?

I want to thank Kenny Rudinger for helping me check and correct some of my numbers on this analysis. Kenny is a physics graduate student at UW-Madison, a great Ultimate Frisbee teammate, and a Bills fan. His website is here.

Abstract
A subtle result of the (relatively) short football season is that teams have a very small sample size of games from which to determine how good they are. The fact that the outcomes of individual games can strongly depend on a few crucial plays adds an additional element of chance any given Sunday. Combine these features with a fairly inclusive playoff system and you have a recipe for upsets. In this post I'll talk about a theoretical model which attempts to produce a handicap, giving better teams a point boost to reduce this variance.

Introduction
Generally the ideas for this blog come while I'm watching NFL games; sometimes they come from discussions with friends. But recently I came across an article in Slate by Neil Paine that got me thinking. In case you're not a fan of following links (your loss, Slate is generally awesome), the basic premise is that the current playoff structure of the NFL, with only a minimal nod to regular season performance, doesn't properly reward teams who've played statistically better during the season. Paine's insane (his words, not mine) suggestion is to give the supposedly superior team a starting point advantage.

Putting aside whether such a handicapping system would be good for the game (as Paine himself argues it's probably not, given that it's fun to see unlikely outcomes and boring to watch your team start a game with a 20-point deficit), how would you go about computing how many points to spot the better team? Paine advocates an approach based on assuming that win-loss records and point differentials are governed by normal distributions

If you're interested in the full calculation you can get all the details from the links in his article, but the gist is fairly simple. You start win the raw win-loss records for each team, then adjust them to account for the fact that 16 games isn't enough to properly sample a team's actual talent level. The next step is to convert the adjusted winning percentages into an estimate the expected likelihood for each team to win the game. Finally, this win expectancy can be converted into a point value, where broadly a higher likelihood translates into more points. 


The result is a prediction of how many points the team with the better winning percentage would need to start off with in order to win with the same probability that they are truly the better team – for instance, a team with a 65% probability of being the better squad would be spotted enough points so that statistically they should have a 65% chance of winning the game.


Paine's method relies on several assumptions which, while not unreasonable, have not all been well-tested. Specifically, although the fact that winning percentages and point spreads are normally distributed has been fairly well tested, his analysis is also predicated on the premise that the relative skill of the two teams have no bearing on the outcome of any individual game. That's quite a claim, and one that I think is worth testing.


Data
Before getting into the real meat of the data, I first wanted to test another assumption Paine makes, which is that the variance of team win-loss records is equivalent to that from repeated coin flips. Since NFL scheduling is definitely not random I was skeptical about this assumption, but a quick Monte Carlo analysis of randomly generated standings indicated that the details of how match-ups are determined is not strongly biasing these statistics. 

I grabbed game records from 2002-2011 from my copy of the Armchair Analysis database, then computed running winning percentages for all teams over the course of each season. Using Paine's methodology I computed the pregame win expectancy for the team with the better record. I then converted these percentages into points, using the same values for home-field advantage and scoring variance as Paine does. 

There's no point in having data from the first week of any season, so I remove all of those games from my data set. (I actually ignore games in the first three weeks to minimize statistical fluctuations.) To remove the possibility of bias from teams which have already qualified for the playoffs but are resting their starters, I also do not use data from weeks 16 or 17. I do however include playoff games in my analysis.

Now that I have the win expectancy and how many extra points that translates into, I can compare how often teams win to how often they should (assuming we've accurately measured a team's true skill, of course). To help preserve signal in the data I binned the games up in win expectancy to have a roughly equal number of games in each bin. 
 
Figure 1: Predicted relationship between win expectancy and winning percentage if the two quantities are perfectly correlated. The relationship is not exactly linear due to the nonuniform size of the bins.
First, to provide a point of comparison, in Figure 1 I plot the winning percentage for each bin provided that the win expectancy directly correlates to win percentage. These points are what we should be shooting for with our handicapping model, as they represent the pure translation between win expectancy and actual winning percentage.
 
Figure 2: Same as Figure 1, but now with the raw winning percentages of the games in our sample shown as the blue histogram. The errors come from counting statistics.
Next, let's look solely at how raw winning percentage trends with win expectancy, which is shown in Figure 2. You can see that while there is a positive correlation between the win expectancy of the statistically better team and their likelihood of actually winning, it's not enough to produce a one-to-one relationship. Finally, in Figure 3 I've added the winning percentage that would result if the "better" team was spotted the number of points given by the model.

Figure 3: Same as Figure 1, but now showing what would happen if teams were given the handicap suggested by Paine as the red histogram.
Discussion and Conclusions
It's pretty clear from Figure 3 that Paine's model, while better than nothing at producing game results in line with the pregame win expectancy, will actually give the team with the better win-loss record a slight additional advantage. Considering the data show that individual games aren't pure coin-flips, an aforementioned assumption of the model, this result is unsurprising. 

That's not to say that Paine's scheme has some sort of fundamental problem. It's clear that individual games are not fully representative of which team is better overall, and there's nothing wrong with the idea of using statistical game results to construct a correction (again, ignoring the issue of whether or not to actually apply such a correction). Games aren't truly random events (another example of reality getting in the way of beautiful, pure statistics), and so any putative correction must take this into account.

Monday, October 14, 2013

QB's Don't Have (Passing) Rhythm

While I have generally been able to maintain a semi-weekly update schedule, the next few months are going to be quite busy (that thesis isn't going to write itself). I will try to keep updating regularly, but I'm not making any promises. Feel free to check the blog regularly to see if there are updates, but if you'd rather use a more efficient method to see when I update put your email address into the box on the right (under 'Get Email Updates') and you'll automagically get an email each time I make a new post. If you're more into twitter, following me @PhDfootball will also let you know when I post (and has the added bonus of giving you direct access to all of my delightfully informative thoughts and comments). Regular updates will hopefully resume after the new year, when the title of this blog will be even more accurate!
Abstract
'Rhythm' is an often-used buzzword in football circles, especially pertaining to a quarterback who is known for being inconsistent. To take a quantitative look at this concept I break down each pass as a function of the one thrown before it, looking for evidence that completing a pass can jump-start a passer into completing more. While this analysis is admittedly superficial, it's a good starting point to tackling this subject. Ultimately, there is no evidence that one pass completion begets another, an argument against the idea that QBs can get into a rhythm.

Introduction
Last season the Jets tried to run an offense involving two quarterbacks, with Mark Sanchez running the regular offense and Tim Tebow coming in to run wildcat-style plays. This was an unarguable failure.

A common reason given by announcers and sportswriters for this unconventional scheme's lack of success was that it never allowed one quarterback to "get into rhythm." That certainly seemed true enough; several times Sanchez would complete a couple of nice passes, then Tebow would come in and run for a few yards, then the drive would stall out once Sanchez came back in.

This is, of course, the same reason given for the failure of Tom Landry's plan to let Craig Morton and Roger Staubach alternate snaps for an entire game (a loss) during the 1971 season.

As usual, there's never any attempt by the announcers to explain what 'rhythm' is or how to tell if a quarterback is in it; this wishy-washy term is generally used as a catch-all to explain why a signal-caller is (or isn't) playing well.

But maybe there is some truth to the idea. There is plenty of anecdotal (and some scientific!) evidence for players getting into 'the zone' during a game, which certainly sounds similar to the concept of 'rhythm'. And football commentators have been using the term for as long as I can remember without any pushback or criticism.

Let's take a look at the concept of QB rhythm (I'll drop the pretentious quoting from this point onward), first attempting to define it in a quantitative manner and then looking at data to determine its validity.

Data
For this experiment I need play-by-play data, which (as usual) comes from Armchair Analysis. Next we need to quantify what statistics could be employed to quantify how much a QB is in rhythm. What would be an observable of a quarterback in rhythm?

The obvious choice is completions. Generally a QB who is in rhythm should be completing several passes in a row, while you would expect a passer who is out of rhythm to be very scattershot. It's difficult to look at completion streaks, as drives can be of variable lengths and we could accidentally bias ourselves towards looking only at very good quarterbacks, who are more likely to have long completion streaks in the first place.

Therefore we'll look at the effect a completion has on just the next pass. While not perfect, this will at least minimize the risk of bias. Additionally, to avoid including situations where one team is being blown out and throwing every play, we'll only include data from the first three quarters of games.

Results
First of all, over the entire sample the completion percentage is a healthy but unspectacular 56.8%. If a quarterback can get into rhythm by completing passes, we'd expect the overall completion percentage on passes attempted after a completed pass to be higher than this overall figure.

Interestingly, it turns out that the opposite is true. If you only look at plays directly after a pass, NFL QBs have a completion percentage of 56.2%. If you loosen your restrictions and check the completion rate specifically for the next pass (even if there may have been several runs in between),  the completion percentage is 56.3%.

Now, it might be that our data are somewhat biased to lower completion percentages because we have to throw out the first completion of each drive. Therefore it might be that we should expect a slightly lower completion percentage than the total 56.8% figure.

To check this possibility I did 1000 random resamplings of the data, keeping the drive data constant but shuffling the type of play (and the result). For both scenarios this test produced completion percentages 56.8+/-0.2%, exactly the same as the overall completion percentage. So if anything, completing their previous pass seems to make quarterbacks more likely to misfire on their next.

Discussion and Conclusions
So what gives? While I'll be the first to admit that this analysis is by no means perfect, it seems pretty clear that this line of inquiry doesn't show any evidence for getting into rhythms. At the very least we can now say that just because a QB has completed a couple passes in a row he's not about to keep up the trend.

One important caveat, especially for the Tebow-Sanchez and Morton-Staubach situations, is that this analysis covers drives where, the vast majority of the time, the QB stayed on the field for every play. Even for wildcat plays the quarterback usually lines up at wide receiver rather than going to the bench - in this way the surprise of the playcall is preserved until the offense breaks their huddle.

With the data currently at my disposal, I can't distinguish between plays where the QB is on the field and those where he is not. Even with that information, there are so few instances where the QB does leave the field during a drive that finding any signal amongst the noise would likely be impossible.

Despite these (very reasonable) concerns, the case against QB rhythm seems fairly strong. While I could believe that quarterbacks get into zones over the course of a season, it doesn't appear to happen on a drive-by-drive basis.

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