Sunday, 18 December 2011

Modifying Marginal Goals

Last time, we looked at Marginal Goals analysis and how it can be applied to historical hockey stats as the first step toward using the Point Allocation system. We ended by looking at the relationship that Marginal Goals assumes between goal differential and wins, which is strictly linear:
This, of course, is not completely realistic. While the actual relationship between goals and wins is very similar to this along most of the normal range of team performance, it definitely gets squidgy at the extremes. Very good teams, such as the 1907 Montreal Wanderers, and very bad teams, such as the 1907 Montreal Shamrocks, will be significantly mis-estimated by Marginal Goals. Pythagorean Analysis does a much better job at reflecting the actual relationship between goals and wins at these extremes, which looks something like this:
You can see here the excellent Wanderers being overestimated by Marginal Goals, and the pathetic Shamrocks being underestimated. This is what happens with very good and very bad teams under Marginal Goals analysis.

There's a simple solution to this problem, of course. In developing the historical Point Allocation system, I haven't used this simple Marginal Goals analysis, but a modified version. The theory behind the modification is quite simple - after a certain point, each additional goal you score has less and less value. If you've already scored 10 goals in a game, scoring an eleventh isn't going to increase your chances of winning, since you've already essentially won. Similarly, if you've already allowed 10 goals, the eleventh can't make you lose the game any more than you already are.

So, instead of the basic Marginal Goals calculation discussed last time, we put some limits on it. Every goal scored in excess of 1.33 times the league average counts for only one-quarter of a goal, and every goal against in excess of 1.33 times the league average counts for only one-quarter of a goal against. This distorts the upper and lower ends of possible team performance in order to make Marginal Goals analysis better match the actual relationship between goals and wins.

Going back to our 1907 results:

TeamW-LGFGA
Montreal Wanderers 10-0 105 42
Ottawa Senators 7-3 76 54
Montreal Victorias 6-4 101 70
Montreal Winged Wheelers 3-7 58 83
Quebec Bulldogs 2-8 62 88
Montreal Shamrocks 2-8 52 117

We already know that basic Marginal Goals analysis results in the following, where MGF is the team's Marginal Goals For, MGS is the team's Marginal Goals Saved, and MG% is the resulting winning percentage:

TeamMGFMGSMG%
Montreal Wanderers 66.8 72.8 .913
Ottawa Senators 38.3 59.3 .645
Montreal Victorias 62.8 44.8 .704
Montreal Winged Wheelers 20.4 29.8 .334
Quebec Bulldogs 24.3 25.3 .329
Montreal Shamrocks 14.4 -4.2 .068

Modified Marginal Goals, on the other hand, produces this:

TeamMGFMGSMG%
Montreal Wanderers 64.3 72.8 .897
Ottawa Senators 38.3 59.3 .645
Montreal Victorias 62.8 44.8 .704
Montreal Winged Wheelers 20.4 29.8 .334
Quebec Bulldogs 24.3 25.3 .329
Montreal Shamrocks 14.4 8.5 .152

Notice how much closer the Shamrocks are now to their actual result of two wins for the season. Graphically, the results look like this:

The little kinks in the line at the extreme allow the linear "curve" to match the Pythagorean relationship much closer than basic Marginal Goals analysis. We're now able to use Marginal Goals, with all their advantages, without being fooled by extreme teams.

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