Improving Our Score Adjustment

Nov 15, 2014

The formula for this site originated from this article. The remainder of this article assumes the reader knows about Corsi, Fenwick, and most importantly score effects, so if you have not read Eric's article from Broad Street Hockey, it is strongly advised that you do before you continue.

The basic premise of score adjustment is that for each game state we can calculate a team's difference from the league's average in that game state and then weight that difference by how important that particular game state is. Eric's article uses league averaged TOI numbers as those weights but concludes with a note that his metric can be improved by using team specific time on ice (TOI) values. This would change his formula from:

$$F_{SA} = {3.75*(F_{up2}-44\%)+8.46*(F_{up1}-46.1\%)+17.94*(F_{tied}-50\%)+8.46*(F_{down1}-53.9\%)+3.75*(F_{down2}-56\%) \over 42.36} + 50\% $$


$$F_{SA} = {TOI_{up2}(F_{up2}-44\%)+TOI_{up1}(F_{up1}-46.1\%)+TOI_{tied}(F_{tied}-50\%)+TOI_{down1}(F_{down1}-53.9\%)+TOI_{down2}(F_{down2}-56\%) \over \sum \limits_{n=down2}^{up2} TOI_n} + 50\% $$

Given that the percents in the above equation are the league average percents in those score states, we can shrink the equation down to the form:

$$F_{SA} = {\sum \limits_{n=down2}^{up2} {TOI_{n}*(F_{n}-F_{avg_n})} \over \sum \limits_{n=down2}^{up2} TOI_n} + 50\% $$

With that in mind, the first question is whether or not using team TOI weights is more effective than league average TOI. It turns out it is, but not by very much. Correlating score adjusted Fenwick values to the points a team earns in the remainder of the season yields the following graph:

The data used for this chart and the remainder of this article is an average of the past 5 full seasons' even strength numbers - so going back to the 2008-2009 season and excluding the 2012-2013 season (which was only 48 games and has abysmal correlation numbers). Overall, we see an averaged R2 of 0.199 for the Team TOI and 0.196 for League TOI. These numbers are low because they contain areas of the season where Fenwick is weak, such as the very beginning of the season and the majority of the end. However if we look at the 20 game mark where Fenwick is strong we see that League TOI R2 is .299 and Team TOI R2 is .307. For those who remember Eric's numbers and note that they're somewhat higher, the numbers here represent a more recent data set and may be trimmed differently. A handful of play by play logs, from which these numbers are derived, are also unfortunately missing, so the seasons in this case are only 81 games to account for the missing data. Lastly, there's also enough variability between seasons to more than account for this change - for instance, 2013 had a .502 R2 at 20 games, which is above Eric's numbers. Even though the difference between League TOI weights and Team TOI weights is not very substantial, this site will continue to use Team TOI weights as they end up being more reliable. Reliability will be touched on later in this article.

That all being said, the main reason for this article and for the recent changes in this site is that I was curious to see if the same formula would work on Corsi - and it does. Looking at the Team TOI score adjusted numbers and comparing them to the Corsi Close for the same period yields following graph:

The average Corsi Close R2 at all points through these 5 seasons was .207. Applying score adjustment, Corsi R2 jumps to .230 - a significant 11% increase over Corsi Close and a 16% increase over Score Adjusted Fenwick. These gains are significant where it matters most - early in the season. To note the high point, 2013 had an astounding .572 R2 after 13 games - a 40% increase over the same Corsi Close. Of all metrics compared on this site, this makes Score Adjusted Corsi the best correlated statistic to future points.

For those wishing to compare all metrics - score adjusted (both league TOI and team TOI), close, and tied for Corsi, Fenwick, and Shots on Goal - the data is here. One point to note in defense of the Score Adjusted Fenwick: while Corsi Close, on average, outperforms SAF, SAF does outperform Corsi Close early to mid-season, when these statistics are most interesting.

A Note on Reliability

The standard metric for season reliability is Split Half Reliability. Split Half Reliability measures the reliability of a data set to itself by splitting it into two halves and correlating the first half with the second. Split half reliability values for the metrics above are as follows:

CorsiFenwickShots on Goal
Data SetSA TeamCloseTiedSA LeagueSA TeamCloseTiedSA LeagueSA TeamCloseTiedSA League
All Data0.6410.5250.4780.6170.5830.4470.4160.5620.5330.3980.3420.511

It is of note that for all metrics, Team TOI weighted numbers are more reliable than League TOI weighted numbers. And while not shown here, this is the case for each individual season as well. Hence this site will continue to use Team TOIs over League TOIs due to their higher reliability.


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