Pictured: Marc-Andre Fleury makes an amazing save at the end of game 7 to win the 2009 Stanley Cup, moments after giving up a rebound. Did he need to make this dramatic save? Should he be credited for it? Looking at the probability of a rebound on the original shot can help lend context.

A few years ago I was a seasoned collegiate goaltender and a raw undergrad Economics major. This was a dangerous combination. When my save percentage fell from something that was frankly pretty good to below average, I turned to an overly theoretical model to help explain this slip in measured performance, for my own piece of mind and general curiosity. The goal was to measure goaltending performance by controlling for the things out of their control, like team defense. Specifically, this framework would properly account for shot quality (of course) and adjust for rebounds, by not giving goalies credit for saves made on preventable rebounds . The former considered things out of the goalies control, the later considers what is actually in the goalies control. Discussing the model with my professor it was soon clear that I included a lot of components that didn’t have available data, such as pre-shot puck movement and/or some sort of traffic index. However, this hasn’t stopped analysts, including myself, from creating expected goals models with the data available publicly. But a public and comprehensive expected goal model remains elusive.

Despite their imperfections, measuring goaltender performance with expected goals are an improvement over raw save percentage and gaining some traction. However, rebounds as they relate to a comprehensive goaltending metric has garnered less research. Prior rebound work by Robb Pettapiece, Matt Cane, and Michael Schuckers suggests preventing rebounds is not a highly repeatability skill, though focusing on pucks frozen might might contain more signal. Building on some of these concepts I hope to give rebound rates some more context by attempting to predict them and explore their effect on a comprehensive goaltending metric consistent with my 2017 RITHAC presentation.[1]

Unfortunately there is nothing to tell us whether a rebound is a “bad” or preventable rebound, so my solution was to create an expected rebounds model using the same framework used to develop an expected goal models. The goal is the same, compare observed goals and rebounds relative to what we would expect a league average goaltender might surrender controlling for as much as we can.

Defense Independent Performance

One of the first iterations and applications of an expected goals model was Michael Schuckers’ Defense Independent Goalie Rating (DIGR). This framework has been borrowed by other analysts, myself included. The idea being the shots goalies face are largely out of their control, they can’t help if they face 3 breakaways in a period or Ovechkin one-timers from the slot. However, goalies can assert some control over rebounds. How much and if this makes a difference is something we will explore.

Regardless of the outcome of the analysis, logic would suggest we discount credit we give goaltenders for facing shots that they could have or should have prevented. Bad rebounds that turn into great saves should be evaluated from the original shot, rather than taking any follow-up shots as a given.

Rebounds Carry Weight

It’s important to note that rebound shots results in higher observed probability of a goal, which makes sense, and expected goal models generally reflect this. However, this disproportionate amount of an expected goal can be confounding when ‘crediting’ goalie for a rebound opportunity against when it could have been prevented. Looking at my own expected goal model, rebounds account for about 3.2% of all shots, but 13% of total expected goals. This ratio of rebounds being about 4 times as dangerous is supported by observed data as well. Shooting percentage on rebounds is about 27%, while it is 5.8% on original shots.

In the clip above and using hypothetical numbers, Luongo (one of my favorite goalies, so not picking on him here) gives up a bad rebound on a wrist shot from just inside the blueline, with an expected goal (xG) value of ~3%, but the rebound shot, due to the calculated angular velocity of the puck results in a goal historically ~30% of the time. Should this play be scored as Luongo preventing about 1/3 of a goal (~3% + ~30%)[2]?

What if I told you the original shot resulted in a rebound ~2% of the time and that the average rebound is converted to a goal ~25% of the time? Wouldn’t it make more sense to ignore the theatrical rebound save and focus in on the original shot? That’s why I’d rather calculate that Luongo faced a 3.5% chance of a goal, rather than ~33% chance of goal. An xG of 3.5% is based on the 3% of the original shot going in PLUS 0.5% chance of a rebound going in (2% chance of rebound times ~25% chance of goal conditional on rebound), and no goal was scored.

Method xG Saved/Prevented Goals Given Up Total xG Faced xG 1st shot xG 2nd Shot Calculation Method Raw xG Calculation 33.0% 0 33.0% 3% 30% Historical probability of goal *given* rebound occurred Rebound Adjusted 3.5% 0 3.50% 3% 0.50% 𝑃(𝐺𝑜𝑎𝑙)=𝑃(𝐺𝑜𝑎𝑙│𝑅𝑒𝑏𝑜𝑢𝑛𝑑)∗𝑃(𝑅𝑒𝑏𝑜𝑢𝑛𝑑) 0.05% = 25% * 2%

Removing Credit Where Credit Isn’t Due

As to not give goaltenders credit for saves made on ‘bad’ rebound shots we can do the following:

Strip out all xG on shots immediately after a rebound (acknowledging the actual goals that occur on any rebounds, of course) Assign a probability of a rebound to each shot Convert the probability of a rebound to a probability of a goal (xG) by multiplying the expected rebound (xRebound) by the probability of a goal on rebound shots, about 27%. This punishes ‘bad’ or preventable rebounds more than shots more likely to result in rebounds. Using similar logic to an expected goals model, some goalies might face shots more likely to become rebounds than others. By converting expected rebounds (xRebounds) to xG, we still expect the total number of expected goals to equal the total number of actual goals scored even after removing xG from rebounds.

To do this we can create a rebound probability model using logistic regression and a similar set of features as an xG model. My most recent model has an out-of-sample area under the ROC curve of 0.68, where 0.50 is random guessing (or assuming every shot has a 3.2% chance of rebound, which is the historical rate). Compare this the current xG model out-of-sample ROC AUC of 0.78, suggesting rebounds are tougher to reliably predict than goals (and we’re not sure there either). A weak rebound model is fine, reflecting the idea an given shot has some probability of turning into a dangerous rebound, maybe a bad bounce or goaltender mishap or fortunate forward, we just have a tough time knowing when.

This does make some sense though, unlike goals where the target is very clear (put the puck in the net), rebounds are less straight forward, they require the puck to hit the goalie and find a opposing players stick before the defense can knock it away. Some defensemen might be able to generate rebounds from point shots more than random, but despite what they might tell you after the fact, players are generally trying to score on the original shot, not create a rebound specifically.

It is also true that goals are targeted, defined events (the game stops, lights go on, goalie feels shame, and the score keeper records it), whereas rebounds escape an obvious definition. Hockey analytics have generally used shots <= 2 seconds from the shot prior, so let’s explore the data behind that reasoning now.

Quickly: What is a rebound?

It’s important to go back and establish what a rebound actually is, without the benefit of watching every shot from every game. We would expect the average shot off of a rebound to have a higher chance of being a goal than a non-rebound shot (all else being equal) since we know the goalie has less time to be able to get set for the shot. And just hypothesizing, it probably takes the goalie and defenders a couple seconds to recover from a rebound. To test the ‘time since last shot’ hypothesis, we can look in the data to see where the observed probability of a goal begins to normalize.

Shots within 2 seconds or less of the original shot are considerably more likely to result in goals than shots than otherwise. There is some effect at a 3 second lag, and certainly some slow-fingered shot recorders around the league might miss a ‘real’ rebound here and there, but the naive classifier of 0-2 seconds between shots is probably the best we can do with limited public data. At 3 seconds, we have lost about half of the effect.

Model Results

Can your favorite goalie prevent rebound compared to what would be expected? If so great, they will be credited with excess xG (xRebounds multiplied by the observed probability of rebound goals 27%) without having to face a bunch of chaotic and dangerous rebound shots. If they give up more rebounds than average, their xG won’t be inflated by a bunch of juicy rebounds, rather replaced by a more modest xG amount indicative of league average goaltending considering what we know about the shots they’re facing.

Which goalies are best at consistently preventing rebounds according to the model? Looking at expected rebound rates compared to actual rebound rates (below), suggests maybe Pekka Rinne, Petr Mrazek, and Tuukka Rask have a claim at consistently being able to prevent rebounds. Rinne has been well documented to have standout rebound control, so we are at least directionally reaching the same conclusions through prior analyses and observations. However, adding error bars consistent with +/- 2 standard deviations dull this claim a little.

Generally, the number of rebounds given up by a goalie over the season loosely reflect what the model predicts. The ends of the spectrum being Rinne with great rebound control in 2011-12 and Marc-Andre Fleury in giving up almost 40 more rebounds than expected in 2016-17. Interesting, Pittsburgh has some of the worst xGA/60 metrics in the league that year and ended up winning the Cup anyway. High rebound rates by both goalies (Murray’s rebound rate was about 1% higher than expected himself) definitely contributed to the high xGA/60 number, perhaps making their defense look worse than it was.

Goal Probability Assumptions

I’ll admit we’re making a pretty big assumption that if a errant puck is controlled and a rebound shot is taken the probability of a goal will be 27%. Maybe some goalies are better than consistently making rebound saves than other goalies, either through skill or ability to put rebounds in relatively low danger areas. Below plots, with +/- standard deviation error bars observed goal % (1 – save %) on rebound shots for goalies with at least 5 seasons since 2010-11.

Devan Dubnyk and Carey Price have been consistent in conceding fewer than 27% (the average for the entire sample) of rebound shots as goals. However, considering the standard deviation we can expect from this distribution given the sample size, this may not be ‘skill.’ It’s also important to explore if their rebound shots are less dangerous than average, whether due to skill, luck, or team defensive structure. This appears to be the case, when adjusted for the xG model, they perform about as well as the model predicts in some seasons, and exceed it in others. Certainly not by enough to suggest their rebounds should be treated any differently going forward.

Looking at intra-goalie performance correlation supports the idea that making saves on rebounds is a less repeatable skill than the original shots. From 2014-2017, splitting each goalies shots faced into random halves, the correlation between the split 1 performance and split 2 is about 0.43. On rebound shots, this correlation falls to 0.24, suggesting that there is considerably less signal. While there is some repeatable skill, its not enough to treat any goalies differently in our model post-rebound due to remarkable ability (or inability) to make saves on rebounds.

Controlling Rebounds, Summary

To reiterate, the problem:

Expected goal models are valuable in measuring goaltending performance, but rebounds are responsible for a disproportionate share of expected goals, which the goalie has some control over.

My solution:

Remove all expected goals credited to the goalie on rebound shots.

Develop a logistic regression model predicting rebounds, the output of which can be interpreted as each shots probability of a rebound.

Explore goalie-level ability to make saves on rebounds shots, to support the assumption that 27% of rebound shots will result in a goal, regardless of goalie.

Replace ‘raw’ expected goals with an expect goal amount based on the probability of goal PLUS probability of a rebound shot multiplied by the historical observed goal % on rebound shots (27%), considering initial, non-rebound shots only.

Finally it’s important to ask, does this framework help predict future performance? Or it just extra work for nothing?

The answer appears to be yes. My RITHAC work attempted to project future goaltender performance by testing different combinations of metrics (xG raw, xG adjusted for rebounds, xG with a Bayesian application, raw save %) and parameters (age regressors, Bayesian priors, lookback seasons). Back testing past seasons, the metrics adjusted for rebounds performed better than the same metrics using a raw expected goal metric as its foundation.

This supports the idea that rebounds, particularly in expected goals models, can confound goaltender analysis by crediting goaltenders disproportionately for chances that they have some control over. In order to reward goalies for controlling rebounds and limiting subsequent chances, goalies can be measured against the amount of goals AND rebounds a league average goalie would concede – which is truer to the goal of creating a metric that controls for team defense and focuses on goaltender performance independent of team quality. Layering in this rebound adjustment increases the predictive power of expected goal metrics.

The limitations of this analysis include the unsatisfactory definition of a rebound and the need for an expected rebound model (alternatively a naive 3.2% of shot attempts result in rebounds can be used). Another layer of complexity might loose some fans and fanalysts. But initial testing suggest that rebound adjustment adds incremental predictive power enough to justify it inclusion in advanced goaltending analysis where the goal is to measure goaltender performance independent of team defense with the publicly data available.

But ask yourself, your coach, your goalie, whoever: should a goalie get credit for a save he makes on a rebound, if he should have controlled it? Probably not.

[1] Pettapiece converted rebounds prevented to goals prevented, but with respect to rebound rate only and to my knowledge did not expand to build into a comprehensive performance metric. (http://nhlnumbers.com/2013/7/15/can-goalies-control-the-number-of-rebounds-they-allow)

[2] Rebound xG actually can’t be added to the original shot like this since we are basically saying the original shot has a 3% chance of going in, so the rebound will only happen 97% of the time. The probability of the rebound goal in the case is 97% * 30%, or 29.4%. But for simplicity I’ll consider the entire play to be a goal 33.3% of the time. The original work and explainer by Danny Page: (https://medium.com/@dannypage/expected-goals-just-don-t-add-up-they-also-multiply-1dfd9b52c7d0)