## Goalie Points Above Expected (PAX)

Pictured: Dominik Hasek, who made 70 saves in a 1994 playoff game, beating the New Jersey Devils 1-0 in the 4th overtime. Hasek didn’t receive goal support for the equivalent of 2 full regulation games, but he won anyway. What is the probability of Hasek winning this game and what does it tell us about his contribution to winning?

## A Chance to Win

I was lucky enough to attend (and later work at) the summer camps of Ian Clark, who went on to coach Luongo in Vancouver and most recently Bobrovsky in Columbus. Part of the instruction included diving into the mental side of the game. A simple motto that stuck with me was: “just give your team a chance to win.” You couldn’t do it all, and certainly couldn’t do it all at once, it was helpful to focus on the task at hand.

You might give up a bad goal, have a bad period, or two or three, but if you can make the next save to keep things close, a win would absolve all transgressions. Conversely, you might play well, receive no goal support, and lose. Being a goalie leaves little in your control. The goal support a goalie receives is (largely[1]) independent of their ability and outside of rebounds, so are most chances they face[2]. Pucks take improbable bounces (for and against) and 60 minutes is a very short referendum on who deserves to win or lose.

Think of being a hitter in baseball and seeing some mix of fastballs down the middle and absolute junk and the chance to demonstrate marginal ability relative to peers on every 20th pitch.

Smart analysis largely throws away what’s out of the goalies control, focusing on their ability to make saves. This casts wins, whatever they are worth, as only a team stat.

Taking a step back, there’s two problems with this:

• A central purpose of hockey analytics is to figure out and quantify what drives winning, and removing wins from the equation to focus on save efficiency feels like cruising through your math test and handing it in, only to realize you missed the last page. So close, yet so far.
• Goalies, coaches, fans, primarily care about winning, so it’s illuminating to create a metric that reflects that. Aligning what’s measured and what matters can be helpful and interesting, at the very least deserves some more advanced exploration.

## What Matters

Analysis is at its strongest when we can isolate what is in the goaltender’s control, holding external factors constant the best we can. For example, some teams may give up more dangerous chances than others, so it is beneficial to adjust goaltender save metrics by something resembling aggregate shot quality, such as expected goals. Building on this we can evaluate a goaltender’s ability to win games as a function of the quality of chances they face and the goal support they receive.

To do this we can calculate the expected points based on the number of goals a team scores and the number of chances they give up. Because goalies are partially responsible for rebounds, we can strip out rebounds and replace with a less chaotic, more stable expected rebounds. The result is weighing every initial shot as a probability of a goal and a probability of a rebound, converting expected rebounds to expected goals by using the historical shooting % on rebounds, 27%.

$$Expected Goals Against_{n} =\sum\limits_{i=1}^n P(Goal)_{i} + (0.27\times P(Rebound)_{i})$$

A visual representation of the interaction between these factors supports the expectation – scoring more goals and limiting chances (expected goals) against increases expected points gained. Summed to team-level this information could be used to create a Wins Threshold metric, identifying which goalies need to stand on their heads regularly to win games.

## Goalie Points Above Expected Metric (PAX Goaltendana)

The expected points gained based on goal support and chances against will be used to compare to the actual points gained in games started by a goaltender. How does this look in practice? Earlier this season, November 4st, Corey Crawford faced non-rebound shots that totaled 2.4 expected goals against, while Chicago only scored 1 goal in regulation. Simulating this scenario 1,000 times suggests with an average goaltending performance Chicago could expect about 0.5 points (the average of all simulations, see below). However, Crawford pitched a shutout and Chicago won in regulation, earning Chicago 2 points. This suggests this Crawford’s performance was worth about 1.5 points to Chicago, or 1.5 Points Above Expected (PAX).

Tracking each of Crawford’s starts (ignoring relief efforts) game-by-game show he’s delivered a few wins against the odds (dark green), while really only costing Chicago one game, against New Jersey (dark red).

The biggest steal of the 2017-18 season so far using this framework? Curtis McElhinney on December 10th faced Edmonton shots worth about 5 expected goals (!) and received 1 goal in support. A team might expect 0.05 points under usual circumstances, but McElhinney pitched a shutout and Toronto got the 2 points.

Other notable performances this season is a mixed bag of big names and backups.

 Goalie Date Opponent Expected GA Goal Support Expected Points Actual GA Actual Points PAX CURTIS MCELHINNEY 12/10/17 EDM 5.07 1 0.06 0 2 1.94 CORY SCHNEIDER 11/1/17 VAN 3.78 1 0.17 0 2 1.83 AARON DELL 11/11/17 VAN 3.18 1 0.27 0 2 1.73 TRISTAN JARRY 11/2/17 CGY 2.93 1 0.33 0 2 1.67 ANTON KHUDOBIN 10/26/17 S.J 4.05 2 0.37 1 2 1.64 CAREY PRICE 11/27/17 CBJ 4.12 2 0.37 1 2 1.64 MICHAL NEUVIRTH 11/2/17 STL 2.66 1 0.38 0 2 1.62 SERGEI BOBROVSKY 12/9/17 ARI 2.72 1 0.39 0 2 1.61 PEKKA RINNE 12/16/17 CGY 3.72 2 0.42 0 2 1.58 ROBERTO LUONGO 11/16/17 S.J 2.49 1 0.42 0 2 1.58

Summing to a season-level reveals which goalies have won more than expected. Goalies above the diagonal line (where points gained = points expected) had delivered positive PAX, goalies below the line had negative PAX.

## Ground Rules

For simplicity, games that go to overtime will be considered to be gaining 1.5 points for each team, reflecting the less certain nature of the short overtime 3-on-3 and shootout. This removes the higher probability of a goal and quality chances against associated with overtime, which is slightly confounding[3], bringing the focus to regulation time goal support.

This brings up an assumption the analysis originally builds on – that goal support is independent of goaltender performance. We know that score effects suggest a team that is trailing will likely generate more shots and as a result are slightly more likely to score. A bad goal against might create a knock-on effect where the goaltender receives additional goal support. While it is possible that the link between goaltender performance and goal support isn’t completely independent (as we might expect in a complex system like hockey), the effect is likely very marginal. But it this scenario a win would be considered more probable, further discrediting any potential win or loss. Generally, the relationship between goaltender performance and goal support is weak to non-existent.

However, great puckhandling goalies might directly or indirectly help aid their own goal support by helping the transition out of their zone, keeping their defensemen from extra contact, and other actions largely uncaptured by publicly available data. Piecemeal analysis suggests goalies have little ability to help create offense, but absence of evidence does not equal evidence of absence. This will have to be an assumption the analysis will have to live with[4], any boost to goal support would likely be very small.

## Taking the Leap – Icarus?

The goal here is to measure what matters, direct contributions to winning. This framework ties together the accepted notion that the best way from a goaltender to help is team win is to make more saves than expected with the contested idea that some are more likely to make those saves in high leverage situations than others, albeit in an indirect way. To most analysts, being clutch or being a choker are just some random processes with a some narrative applied.

However, once again, absence of evidence does not equal evidence of absence[5]. I imagine advanced biometrics might reveal that some players experience a sharper rise in stress hormones which might effect performance (positively or negatively) during a tie game than if down by a handful of goals. I know I felt it at times, but would have difficulty quantifying its marginal effect on performance, if any. A macro study across all goalies would likely be inconclusive as well. Remember NHL goalies are a sample of the best in the world, those wired weakly very likely didn’t make it (like me).

But winning is important, so it is worth making the jump from puck-stopping ability to game-winning ability. The tradeoff (there’s always tradeoffs) is we lose sample size by a factor of about 30, since the unit of measure is now a game, rather than a shot. This invites less stable results if a game or two have lucky or improbable outcomes. On the other hand, it builds in the possibility some guys are able to raise their level of play based on the situation, rewarding a relatively small number of timely saves, while ignoring goals against when the game was all but decided. I can think of a few games that got out of control where the ‘normal circumstances’ an expected goals model assumes begin to break down.

## Winning DNA?

All hockey followers know goalies can go into brick-wall mode and win games by themselves. The best goalies do it more often, but is it a more distinguishable skill than the raw ability to prevent goals? Remember, we are chasing the enigmatic concept of clutch-ness or ability to win at the expense of sample size, threatening statistically significant measures that give analysis legs.

To test this we can split goalie season into random halves and calculate PAX in each random split, looking at the correlation between each split. For example, goalie A might have 20 of their games with a total PAX of 5 end up in ‘split 1’ and their other 20 games with a PAX of 3 in ‘split 2.’ Doing this for each goalie season we can look at the correlations between the 2 splits.[6]

Using goalie games from 2009 – 2017 we randomly split each goalie season 1,000 times at minimum game cutoffs ranging from 20 to 50,[7] checking the Pearson correlation between each random split. Correlations consistently above 0 suggest the metric has some stability and contains a non-random signal. As a baseline we can compare to the intra-season correlation of a save efficiency metric, goals prevented over expected, which has the advantage of being a shot-level split.

The test reveals that goals prevented per shot carries relatively more signal, which was expected. However, the wins metric also contains stability, losing relative power as sample size drops.

Goalies that contribute points above expected in a random handful of games in any given season are more likely to do the same in their other games. Not only does a wins based metric make sense to the soul, statistical testing suggests it carries some repeatable skill.

## Final Buzzer

Goalie wins as an absolute number are a fairly weak measure of talent, but they do contain valuable information. Like most analyses, if we can provide the necessary context (goal support and chances against) and apply fair statistical testing, we can begin to learn more about what drives wins. While the measure isn’t vastly superior to save efficiency, it does contain some decent signal.

Exploring goaltender win contributions with more advanced methods is important. Wins are the bottom line, they drive franchise decisions, and frame the narrative around teams and athletes. Smart deep dives may be able to identify cases which poor win-loss records are bad luck and which have more serious underlying causes.

A quick look at season-level total goals prevented and PAX (the metrics we compared above) show an additional goal prevented is worth about 0.37 points in the standings, which is supported by the 3-1-1 rule of thumb, or more precisely,  2.73 goals per point calculated in Vollman’s Hockey Abstract. Goal prevention explains about 0.69 of the variance in PAX, so the other 0.31 of the variance may include randomness and (in)ability to win. Saves are still the best way to deliver wins, but there’s more to the story.

## Overtime

When I was a goalie, it was helpful to constantly reaffirm my job: give my team a chance to win. I couldn’t score goals, I couldn’t force teams to take shots favorable to me, so removing that big W from the equation helped me focus on what I could control: maximizing the probability of winning regardless of the circumstances.

This is what matters to goalies, their contribution to wins. Saves are great, but a lot of them could be made by a floating chest protector. While the current iteration of the ‘Goalie Points Above Expected’ metric isn’t perfect, hopefully it is enlightening. Goalies flip game probabilities on their head all the time, creating a metric to capture that information is an important step in figuring out what drives those wins.

Thanks for reading! I hope to make data publicly available and/or host an app for reference.  Any custom requests ping me at @crowdscoutsprts or cole92anderson@gmail.com.

Code for this analysis was built off a scraper built by @36Hobbit which can be found at github.com/HarryShomer/Hockey-Scraper.

I also implement shot location adjustment outlined by Schuckers and Curro and adapted by @OilersNerdAlert. Any implementation issues are my fault.

My code for this and other analyses can be found on my Github, including the feature generation and modeling of current xG and xRebound models and PAX calculations.

[1] I personally averaged 1 point/season, so this assumption doesn’t always hold.

[2] Adequately screaming at defensemen to cover the slot or third forwards to stay high in the offensive zone is also assumed.

[3] If a goalie makes a huge save late in a tie game and subsequently win in overtime, the overtime goal was conditional on the play of the goalie, making the win (with an extra goal in support) look easier than it would have otherwise.

[4] Despite it partially delegitimizing my offensive production in college.

[5] Hockey analysts can look to baseball for how advanced analysis aided by more granular data can begin to lend credence to concepts that had been dismissed as an intangible or randomness explained by a narrative.

[6] Note that the split of PAX is at the game-level, which makes it kind of clunky.  Splitting randomly will mean some splits will have more or less games, possibly making it tougher to find a significant correlation. This isn’t really a concern with thousands of shots.

[7]The ugly truth is that an analyst with a point to prove could easily show a strong result for their metric by finding a friendly combination random split and minimum games threshold. So let’s test and report all combinations.

There may be an interesting paradox developing within hockey. The working theory is that as advanced analysis and data-driven decision-making continue to gain traction within professional team operations and management, the effect of what can be measured as repeatable skill may be shrinking. The Paradox of Skill suggests as absolute skill levels rise, results become more dependent on luck than skill. As team analysts continue (begin) to optimize player deployment, development, and management there should theoretically be fewer inefficiencies and asymmetries within the market. In a hypothetical league of more equitable talent distribution, near perfect information and use of optimal strategies, team results would be driven more by luck than superior management.

Goaltenders Raising the Bar

Certainly forecasting anything, let alone still-evolving hockey analytics, is often a fool’s errand – so why discuss? Well, I believe that the paradox of skill has already manifested itself in hockey and actually provides a loose framework of how advanced analysis will become integrated into the professional game. Consider the rise of modern goaltending.

Absolute NHL goaltender ability has continually increased for the last 30 years. However, differential ability between goaltenders has tightened. It has become increasingly difficult to distinguish long-term, sustainable goaltender ability while variations in results are increasingly owed to random chance. Goalies appear ‘voodoo’ when attempting to measure results (read: ability + luck) using the data currently available – much like the paradox of skill would predict.[1] More advanced ways of measuring goaltending performance will be developed (say, controlling for traffic and angular velocity prior to release), but that will just further isolate and highlight the effect of luck.[2]

Will well-managed teams create a similar paradox amongst competing professional teams in the future? Maybe. Consider such a team would maximize the expected value talent acquired, employ optimal on-ice strategies, and employ tactics to improve player development. Successful strategies could be reverse engineered and replicated, cascading throughout the league – in theory. Professional sports leagues are ‘copycat’ leagues and there is too much at stake not to adopt a superior strategy, despite a perceived coolness to new and challenging ideas.

Dominant Strategies“I don’t care what you do, just stop the puck”

How did goaltending evolve to dominate the game of hockey? And what parallel pathways need to exist in hockey analytics to do the same?

1. Advances in technology – equipment became lighter and more protective.[3] This allowed goaltenders to move better, develop superior blocking tactics (standing up vs butterfly), cover more net, and less worry of catching a painful shot. The growth of hockey analytics has been dependent on web scraping, automation, and increasing processing power and will soon come to rely on data derived from motion-tracking cameras. Barriers to entry and cost of resources are negligible lending all fanalysts the opportunity to contribute to the game.
2. Contributions from independent practitioners – The ubiquitous goaltending coach position is a relatively new one compared to most professional leagues. In the early 2000s, I was lucky enough to cross paths with innovative goaltending instructors who distributed new tactics, strategies, and training methods available to young goaltenders. Between their travel, camps, and clinics (and later their own development centers) they diffused innovative approaches to the position, setting the bar higher and higher for students. A few of these coaches went on become NHL goalie coaches – effectively capturing a position that didn’t exist 30 years prior. Now the existence of goalie coach cascade down to all levels of competitive hockey.[4]  Similarly, the most powerful contributions to the hockey analytics movement have been by bright individuals exposing their ideas and studies to the judicious public. The best ideas were built upon and the rest (generally) discarded. Will hockey analytics evolve (read: become accepted widely among executives) faster than goaltending? I don’t know – a goaltending career takes well over a decade to mature, but they play many games providing feedback on new strategies rather quickly.[5] Comparatively, ideas develop quicker but might take longer to demonstrate their value – not only are humans hard-wired to reject new ideas there are fewer managerial opportunities to prove a heavy data-driven approach to be a dominant strategy.
3. Existence of a naïve acceptance – The art (and science) of goaltending is not especially well understood among many coaches, particularly with relative skill levels converging. However, managers and coaches do understand results. Early in my career, I had a coach who was only comfortable with stand-up goaltenders, his own formative experiences occurring when goaltender predominately remained erect (in order to keep their poorly padded torso and head from constant danger). However, he saw a dominant strategy (more net coverage) and placed faith in my ability without a comprehensive understanding or comfort of modern goaltending. Analytics will have to be accepted the same way – gradual but built on demonstrated effectiveness. Not everyone is comfortable with statistics and probabilities, but like goaltenders, the job of analysts is to produce results. That means rigorous and actionable work that offers a superior strategy to the status quo. This will earn the buy-in from owners and senior management who understand that they can’t be at a competitive disadvantage.

Forecasting Futility

Clearly the arc of the analytics evolution will differ from the goaltender evolution, primary reasons being:

• Any sweeping categorization of two-decade-plus ‘movement’ is prone to simplification and revisionist history.
• While goaltending as a whole has improved substantially, incremental differences in ability still obviously exist between goaltenders. In the same way, not all analysts or teams of analysts will be created equal. A non-zero advantage in managerial ability may compound over time. However, the signal will likely be less significant than variation in luck over that extended timeframe. In both disciplines, that rising ability may give way to a paradox of not being able to decipher their respective skills, muddying the waters around results.
• Goaltending results occur immediately and visibly. Fair or not, an outlier goaltender can be judged after a quarter of a season, managerial results will take longer to come to fruition. Not only that, we only observe the one of many alternative histories for the manager, while we get to observe thousands of shots against a goaltender. Managerial decisions will almost always operation under a fog of uncertainty.

Alternatively, it important to consider the distribution of athlete talent against those of those in the knowledge economy. Goaltenders are bound by normally distributed deviations of size, speed, and strength. Those limitations don’t exist for engineers and analysts, but they do operate in a more complex system, leaving most decisions to be subjected to randomness. This luck is compounded by the negative feedback loops of the draft and salary cap, it is unlikely a masterfully designed team would permanently dominate, but it suggests some teams will hold an analytical advantage and the league won’t turn into some efficient-market-hypothesis-all-teams-50%-corsi-50%-goals-coin-flip game. But if a superstar analyst team could consistently and handily beat a market of 29 other very good analyst teams in a complex system, they should probably take their skills to another more profitable or impactful industry.

Because these are confusing times we live in, I’d be remiss if I didn’t mention two other paradoxes of hockey analytics.

• Thorough, rigorous work is often difficult to understand and not easily understood by senior decision-makers. This is a problem in many data-intensive industries – analytical tools outpace the general understanding of how they work. It seems that (much like the goaltending framework available to us) once data-driven strategies are employed and succeed, all teams will be forced to buy-in and trust that they have hired competent analysts that can deliver actionable insights from a complex question. Hopefully.

• With more and more teams buying into analytics, the some of the best work is taken private. The best work is taken in-house seemingly overnight, sometimes burying a lot of foundational work and data. That said, these issues are widely understood and there is a noble and concerted effort to maintain transparency and openness. We can only hope that these efforts are appreciated, supported, and replicated.

Final Thoughts

The best hockey analysis has borrowed empiricism and data-driven decision-making from the scientific method, creating an expectation that as hockey analytics gain influence at the highest levels, we (collectively) will know more about the game.[7] However, assuming the best hockey analysts end up influencing team behavior, it is possible much of the variation between NHL teams[8] will be random chance – making future predictive discoveries less likely and weakening the relationship of current discoveries.

Additionally, when it feels like the analytical approach to hockey is receiving unjustified push back or skepticism, it is important to remember that the goaltender evolution, initiated by fortuitous circumstance, eventually forced buy-ins from traditionalists by offering a superior approach and results. However, increasing absolute skill in a field can have unintended consequences – relative differences in skill will decrease, possibly causing results to become more dependent on luck than skill. Something to consider next time you try to make sense of the goaltender position.

[1] This is not to say all goalies in 2016 are of equal skill levels, but they are absolutely more talented than their ancestors and fall within a smaller range of abilities. That said, outside of a top 2 or 3 guys, the top 5-10 list of goalies is a game of musical chairs, quarter to quarter, season to season.

[2] Goaltenders don’t get a chance to ‘drive the play,’ so it is very important to control for external factors. This can’t be done comprehensively with current data. Even with complete data, it may be futile.

[3] And cooler, possibly attracting better athletes to the position, your author notwithstanding.

[4] Another feature of the paradox of rising skill levels: to fail to improve is the same as getting worse. Hence, employing a goalie coach is necessary in order to prevent a loss of competitiveness. The result: plenty of goalie coaches of varying ability, but likely without a strong effect on their goaltender’s performance. This likely causes some skepticism toward their necessity. This is probably a result of their own success, they are indirectly represented by an individual whose immediate results might owe more to luck than incremental skill aided by the goalie coach.

[5] For example, a strategy devised at 6 years old of lying across the goal line forcing other 6 year-olds to lift the puck proved to be inferior and was consequently dropped from my repertoire.

[7] Maybe even understanding the link between shot attempts and goals (you can read this sarcastically if you like).

[8] And other leagues that are able to track and provide accurate and useful data.