## Monday, May 30, 2011

### Stanley Cup Finals 1011 Playoff Probabilities and Predictions

It all comes down to this.

The best team in the West against the best team in the East.*

For the 6th straight year, the Western representative appears to be the stronger team. That's not really surprising - the West has had the better interconference record in every season since 1999-00, and often by a large margin.

I think Vancouver is clearly the better team here and that, if anything, the odds I've presented above understate their chances. That said, these two teams are close enough to one another where it should be a good series.

I'll take the Canucks to win in six games.

VAN in 6.

*As per my probability model. It's possible - perhaps even likely in Boston's case - that neither team is the best team in its respective conference. For what it's worth, I'd take a healthy Pittsburgh team over Boston all day every day. But I digress.
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## Sunday, May 29, 2011

### Team Even Strength Shooting Talent

A while back, I received a comment relating to how my playoff probability model accounts for teams that are outliers with respect to shooting percentage, with the 2009-10 Washington Capitals offered as an example of such a team.

The answer is relatively straightforward: I merely regress each team to the league average based on the extent to which the team to team variation can be attributed to luck over the sample in question. As the variation in even strength shooting percentage at the team level is approximately 66% luck over the course of a regular season, each team's even strength shooting percentage is regressed two-thirds of the way to the mean in order to generate theoretical win probabilities.

The application of above method to data from the 2010-11 regular season yields the following even strength shooting talent estimates for each the league's 30 teams.*

This method, however, is actually a shortcut that relies on assumptions that are unlikely to be true in reality. For one, it assumes that all underlying talent distributions are normally distributed, which may or may not be the case. It's also insensitive to the fact that some teams take more shots than others over the course of a season. A more certain shooting talent estimate can be made with respect to a team that takes 2000 shots as compared to a team that takes 1500 shots, although the model fails to reflect that.

The proper - albeit significantly more complicated and involved - approach would be to actually identify the nature of the underlying talent distribution and work one's way forward from there.

The first step is to look at the observed distribution in performance. I did this through randomly selecting half a season's worth of games for each team and looking at how it performed with respect to EV SH% over that sample, and repeated this 2000 times for every season from 2003-04 to 2010-11. I elected to do this as it provided me with 420 000 data points, thereby allowing me to generate a smooth curve. By comparison, using the actual end-of-season frequencies would have provided me with a mere 210 data points.

I came away with the following curve:

The distribution is slightly right-skewed and therefore not quite normal. This becomes meaningful at the tails - there were approximately 26 times more values 3 standard deviations above the mean than there were values 3 standard deviations below it. In other words, there are many more very good teams than very bad ones when it comes to even strength shooting performance.

The next step is finding a curve that best fits the observed data. This curve should have a mean of approximately 0.081, which was the observed league average shooting percentage. It should also have a standard deviation of approximately 0.0048, which is the skill standard deviation in relation to even strength shooting percentage at the team level. Finally, the curve should be slightly positively skewed.

The beta (236, 2977) curve, shown below, satisfies these criteria.

As a check on the correctness of the selection, I used a random number generator to assign each team an artificial EV shooting percentage based on the above curve. I then simulated a sufficiently large number of half seasons based on those artificial numbers and compared the results to the observed data. If the choice is correct, the simulated results should closely match those observed.

The simulated curve is only based on about 30 000 data points, so it's not as smooth as the observed distribution. That said, the fit is pretty good. The observed distribution appears to have a fatter right tail, and so it's possible that a different beta curve might provide a better match. But it's close enough.

The beta ability distribution can be used to estimate each team's true talent underlying shooting percentage, based on the 2010-11 regular season. How do these estimates compare to those produced by the simple regression approach discussed earlier?

The two approaches produce very similar results - the average difference amounting to only 0.0004. The latter approach is both more precise and principled. But the former achieves substantially similar estimates with a fraction of the effort.

* The mean used was 0.0812, this being the league average EV SH% since the lockout, even though the observed shooting percentage in 2010-11 was a bit lower - a touch under 0.08.
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## Friday, May 13, 2011

### 3rd Round 1011 Playoff Probabilities and Predictions

VAN - S.J

Another extremely even matchup. The relevant facts, as I see them:
• S.J probably has the better powerplay - they generate a ridiculous number of shots
• VAN very likely has the better goaltender
• VAN has home ice advantage
• Both teams are about equally good at controlling the play at EV
• VAN is missing a key forward
If those facts give one club a clear advantage, I can't see it. These are probably the two best teams in the league and this should be a great series. I'll take the Canucks in seven games.

Van in 7.

BOS – T.B

At first glance, Boston seems like the obvious pick here. But the (Patrice) Bergeron injury complicates things. The latest reports indicate that he has yet to resume skating since the incident, so from that it seems as though he might not play at all. That would be a huge loss, as he's probably their best forward, at least by my reckoning.

The issue is whether the Bergeron injury is enough to tip the balance in Tampa Bay's favor. I don't think that it is. Based on regular season play, I have the Bruins as a 61% favorite. While the Bergeron injury necessitates a downward adjusted of that figure, I don't think the loss is profound enough to render the Bruins underdogs. This is supported by the fact that the oddsmakers - who certainly take such things into account - still have Boston as about a 56% favorite.

BOS in 7.

## Tuesday, May 10, 2011

### Team Effects and Penalty Kill Save Percentage

In yesterday's post, I looked at the extent to which team effects contribute to the variation in even strength save percentage between individual goaltenders.

The results were somewhat inconclusive. On the one hand, the inter-year correlation for even strength save percentage is no stronger for goalies remaining with the same team when compared to the value for goalies that changed teams. This suggests that team effects are negligible.

On the other hand, there is a statistically significant correlation between the even strength save percentage of starters and backups. Moreover, the magnitude of the correlation is moderate when viewed in light of the fact that even strength save percentage exhibits low reliability over the course of a single season. This suggests that team effects are important.

The purpose of this post is to look at whether -- and if so, to what extent -- team effects play a role with respect to penalty kill save percentage. The same methods used in yesterday's post will be applied here. If readers are interested in the specifics of each method, I'd encourage a reading of the original post, in which the calculation steps are set out in some detail.

Firstly, a comparison of goalies that changed teams to goalies that remained with the same team. Here's a summary of what this method entails:

Goalies that played for more than one team in a single season were excluded. No minimum shots faced cutoff was employed. However, because some of the goalies in the sample faced very few shots in a given season, I used a weighted correlation in which the weight assigned to each season pair was the lower number of shots faced in the two seasons used...[a]dditionally, because the league average [PK SV%] was not uniform over the period in question, I adjusted each goalie's raw [PK SV%] by dividing it by the league average [PK SV%] in that particular season.
The results:

No evidence for team effects here. The correlation for goalies that changed teams is actually larger, although the difference is not statistically significant.

Next, determining the correlation between starters and backups. Again, a refresher as to the specifics of the method involved:

I separated starting goaltenders and backup goaltenders into two groups. A starting goaltender was defined as the goaltender that faced the most shots for his team in a particular season. All other goaltenders were defined as backups, except for goaltenders that played for more than one team in a season, who were excluded from the sample. Just like in the first method, the [PK SV%] for all goaltenders was adjusted by dividing same by the league average [PK SV%] in the particular season. I then determined the weighted correlation between the [PK SV%] of a team's starter with the collective [PK SV%] of his backups. The weight assigned to each data pair was the lower number of shots faced by either the starter or his backups. So, for example, if the starter faced 1000 shots, and his backups collectively faced 1400, the weight would be 1000.

The application of the above steps yields a correlation of 0.07 over 340 data pairs, a value which is not statistically significant - there's a roughly 19% chance that a correlation that large or larger could occur by chance alone. That said, given the low number of shots faced on the penalty kill by the average goaltender over the course of a single season, it is not possible to obtain a statistically significant correlation between starters and backups unless team effects accounted for a substantial percentage of the non-luck seasonal variation in PK SV%. For example, a correlation of 0.10 - which would barely be significant at the 5% level - would imply a very large role for team effects, given that PK SV% for individual goaltenders has a low seasonal reliability (see the next paragraph).

Proceeding on the assumption that the correlation between starters and backups is reflective of a true relationship, the next step is to compute the seasonal reliability co-efficients for each variable. I obtain approximate values of 0.28 for starters and 0.07 for backups. This implies a true correlation of 0.50.

Finally, I have goaltender data at the individual game level for the last three seasons against which the plausibility of the above results can be checked. The penalty kill data I have is inclusive of 4-on-5 situations only, but that shouldn't make a huge difference. The table below displays the split-half reliabilities for starter and backup PK SV%, as well as the split-half correlation between the two variables, both of which have been averaged over 1000 trials.

These values imply a true correlation of 0.54, which is consistent with the results of the second method.

So there you have it - comparing goalies that switched teams to goalies that remained with the same team suggests team effects are unimportant in relation to PK SV%. But there is a positive correlation between the PK performance of starters and backups, which indicates that team effects are relevant. Those who read yesterday's post will be aware that the data for even strength save percentage tells the same story.

Interestingly, the data suggests that team effects may be more important at even strength than on the penalty kill. This is unusual as penalty kill save percentage at the team level is somewhat more reliable than even strength save percentage, once you control for the disparity in sample size - that is, the fact that a team faces many more shots at even strength than it does on the penalty kill.
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### Team Effects and Even Strength Save Percentage

The extent to which a goaltender's team has an impact on his save percentage - and, in particular, his even strength save percentage - has received some attention in the hockey blogging world in the past - see here and here for some good articles.

One way in which team effects on even strength save percentage (EV SV%) can be gauged is to compare goalies that changed teams to goalies that remained with the same team. This can be done through creating two groups of goalies on the basis of the above criterion, and looking at how well EV SV% repeats from one year to the next for each group. If the correlation for the group of goalies that changed teams is significantly smaller than the correlation for the group of goalies that remained with the same team, then that would be evidence of team effects.

Using a spreadsheet kindly supplied by Geoff Detweiler, I performed the above exercise with respect to goaltender data from 1997-98 to 2010-11. Goalies that played for more than one team in a single season were excluded. No minimum shots faced cutoff was employed. However, because some of the goalies in the sample faced very few shots in a given season, I used a weighted correlation in which the weight assigned to each season pair was the lower number of shots faced in the two seasons used. Thus, if a goalie faced 1600 EV shots in one season, and 400 in the next, the weight assigned to the season pair would be 400.

Additionally, because the league average EV SV% was not uniform over the period in question, I adjusted each goalie's raw EV SV% by dividing it by the league average EV SV% in that particular season. Here are the results:

[ n refers to the number of season pairs in each group ]

The correlations are scarcely distinguishable, which implies that team effects aren't important at even strength. This essentially replicates what Vic Ferrari found when performing similar analysis a few years ago.

Of course, that doesn't necessarily settle the issue. For example, another approach would be to look the relationship between the EV SV% of a team's starting netminder and the collective EV SV% of its backups. If the two variables are positively correlated, then that implies the existence of team effects.

The advantage of this method is that it allows for team effects to be measured more directly by examining the relationship between the variables of importance at the within-season level. This is significant as team effects on save percentage - to the extent that they do exist - may not repeat overly well from one season to the next. For example, Tom Awad, in an excellent article written last year, found that while team differences in shot quality over the course of a single season were much larger than what would be predicted from chance alone, the metric exhibited weak season-to-season repeatability.

Using the same goaltender data referred to earlier, I separated starting goaltenders and backup goaltenders into two groups. A starting goaltender was defined as the goaltender that faced the most shots for his team in a particular season. All other goaltenders were defined as backups, except for goaltenders that played for more than one team in a season, who were excluded from the sample. Just like in the first method, the EV SV% for all goaltenders was adjusted by dividing same by the league average EV SV% in the particular season. I then determined the weighted correlation between the EV SV% of a team's starter with the collective EV SV% of his backups. The weight assigned to each data pair was the lower number of shots faced by either the starter or his backups. So, for example, if the starter faced 1000 shots, and his backups collectively faced 1400, the weight would be 1000.

After doing all of that, I obtained a correlation of 0.156. With 340 data pairs, the probability of a correlation that large materializing by chance alone is very small - slightly under 1%, in fact. Moreover, it cannot be accounted for by shot recording bias.* Therefore, it would appear that the EV SV% of individual goaltenders is affected to some degree by team effects. The question that must now be answered is this: how large is the effect?

As discussed before on this blog, the fact that two variables are weakly correlated over a given sample does not in itself mean that there is no strong underlying relationship between those variables. For example, if each of the variables exhibits low reliability over the sample in question, a weak correlation may in fact indicate a close underlying relationship. Thus, ascertaining the reliability values of the two variables is critical in interpreting the significance of the correlation between them.

Applying this to our value of 0.156, it becomes necessary to determine the seasonal reliability co-efficients of both starting goalie EV SV% and backup EV SV%. While it is not possible to perform this calculation directly,** it can be approximated by simulating seasons to match the spread of the averaged observed results and noting the average correlation between such seasons. Using this method, the approximate reliability co-efficients are 0.33 for starter EV SV% and 0.22 for backup EV SV%.

These values imply that the true correlation between the two variables is roughly 0.58. Assuming that both variables are normally distributed***, this means that the variation in one variable would be able to explain 33% of the variation in the other over the long run, suggesting that team effects are important.

As a final note, this post was intended to generate discussion more than anything. Comments demonstrating flaws in my reasoning and/or methodology are welcome, as is the presentation of contrary evidence.

* Shot recording bias causes the save percentages of goalies playing for the same team to be more similar to one another than what would be the case if shots were recorded in the same in every rink. However, because of a) the small number of shots taken over the course of the season, b) the relatively mild nature of the bias, and c) the fact that half of all games are played on the road, the effect is fairly minor. Of the observed correlation of 0.156, only 0.018 can be attributed to shot recording bias.

** Ordinarily, and as I've done in the past, I would calculate the split-half reliability values for each variable and then calculate the split-half correlation between them. This method is superior as no approximation is necessary with respect to determining the reliability co-efficients. Unfortunately, EV SV% data at the individual game level is required in order to do so. As such data is only available for 2007-08 onward, I'm only able to apply this method to the years of 2007-08, 2008-09 and 2009-10 (and even then, for 5-on-5 play rather than all EV situations). Here are the results:

The results imply that team effects have a very important role in relation to 5-on-5 SV% - indeed, that there would be a perfect correlation between the 5-0n-5 SV% of starters and backups in the long run! This being an obvious absurdity, I think it's preferable to ignore this and concern ourselves with the results from the larger 12 year sample instead.

*** This is merely a simplifying assumption. In reality, it is unlikely that either variable is normally distributed.

## Sunday, May 1, 2011

### Loose Ends - Part III C: The Power Play

[EDIT: The table relating to the penalty kill was labeled incorrectly and suggested that I was looking at shorthanded scoring ( I accidentally put 'PKSF/60' and PK 'GF/60' instead of PKSA/60' and PKGA/60, respectively). The table has now been fixed.]

This post is a tad overdue.

It's the second of two follow up posts relating to powerplay performance. While the first post dealt with the relationship between shooting percentage at even strength and shooting percentage on the powerplay, this post relates to predicting future powerplay performance.

The variation in powerplay shooting percentage at the team level, over the course of a single regular season, is approximately 90% luck, 10% skill. Not surprisingly, powerplay shot rate is a stronger predictor of future powerplay performance than raw powerplay performance (provided that the sample size with which one is dealing isn't overly large). This is precisely what Gabriel Desjardins demonstrated in a post published in early April.

What this post is concerned with, however, is whether the inclusion of missed and blocked shots in the sample has residual value with respect to predicting to predicting powerplay efficiency in the future. While such is the case at even strength, special teams may be a different ballgame. What does the data say?

One preliminary issue that must be dealt with is shot recording bias. Recording bias doesn't really present a problem with respect to even strength shot metrics due to the fact that:

A. What were ultimately interested in is shot ratio/percentage or shot differential, and
B. None of the scorers appear to favor one team over the other (i.e. recording bias is largely, if not entirely, symmetrical).

Not so with special teams. With special teams, we're generally interested with rate stats, in which case recording bias becomes relevant. This is especially true when it comes to the recording of missed and blocked shots. Below is a table showing each team's home/road ratio in recorded shots (saved shots + goals), misses and blocks over the last three regular seasons (from 2008-09 to 2010-11). All game situations were included, although empty net goals or shots that resulted in same were not.

As one might notice, the recording of shots that actually make it to the goal isn't that bad. New Jersey and Minnesota appear to undercount, and Colorado appears to overcount. But every other location is reasonably good.

The recording of misses and blocks, by contrast, is generally fucked up. The N.J, CHI, ATL and BOS scorers seem very reluctant to record misses. Conversely, the guys in L.A, CAR, DAL and TOR seem overly eager.

The data for blocks reveals a similar story. The scorers in ANA, BOS, FLA and N.J count too few, whereas the scorers in NYI, MTL, EDM, S.J, TOR and WSH count too many.

It's a god damn nightmare.

Fortunately, there is a solution. Recording bias can be more or less controlled for by dividing the observed number of home missed or blocked shots by the appropriate co-efficient (that being the applicable H/R ratio, as displayed in the above table).

Once this correction is made, one can determine whether including missed and blocked shots adds value with respect to predicting future powerplay performance.

The following experiment was performed:

- I randomly selected 40 games from the 2010-11 season
- I calculated each team's PP GF/60, PP SF/60, PP Fenwick/60, PP Corsi/60 over that selected sample
- PP Fenwick/60 = [(powerplay shots + powerplay missed shots)/PP TOI]*60
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PP Corsi/60 = [(powerplay shots + powerplay missed shots + powerplay blocked shots)/PP TOI]*60
- I then selected an independent 40 game sample, and calculated each team's PP GF/60 in respect thereof
- I then looked at how each of the four above variables ( PP GF/60 , PP SF/60, PP Fenwick/60, PP Corsi/60), as calculated over the 1st sample of games, predicted PP GF/60 over the 2nd sample of games
- I repeated this exercise 1000 times
- I then repeated the entire exercise for the 2008-09 and 2009-10 regular seasons

The results:

Just like Gabe Desjardins found, shot production is a better predictor of future powerplay success relative to raw performance (with respect to 40 game sample sizes). And while missed shots have some informational value, blocked shots do not.

Does the same apply to the penalty skill? Interestingly, no.

Unlike with the powerplay, raw performance on the penalty kill (over a 40 game sample) is a superior predictor of future PK performance than is shot prevention. Part of that can be attributed to the fact that penalty skill save percentage is considerably more reliable than powerplay shooting percentage.

Furthermore, including misses and blocks is of no assistance. It seems as though better penalty kills force their opponents to take a greater proportion of missed and blocked shots.