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:

Team	W-L	GF	GA
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:

Team	MGF	MGS	MG%
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:

Team	MGF	MGS	MG%
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.

Marginal Goals in History

One of the projects I've been working on for some time now is an application of my old Point Allocation system to hockey leagues throughout history. Point Allocation, developed in 2002, was the first published attempt at a comprehensive player valuation system, so far as I can tell. It's similar in purpose to Tom Awad's Goals Versus Threshold system used at Hockey Prospectus, in that it attempts to place a value on each player's contributions on both offence and defence in a single measure, to see who contributes the most to their team's success.

The basis of Point Allocation, then as now, is Marginal Goals analysis. This is an idea adapted from Bill James' work in his book Win Shares. Marginal Goals represent what a historically bad team is able to produce in competition; no team scores zero goals or allows an infinite number of them. Marginal Goals are therefore what you can accomplish without really trying to ice a good team. Historically, Marginal Goals on offence are equal to about 0.5 of the league average, and Marginal Goals of defence are 1.5 of the league average.

Any goals a team scores in excess of 0.5 times the league average therefore contributes to winning games, and any goals a team saves below 1.5 times the league average does the same. By adding these together, and dividing by twice the league average in goals scored, we get quite an accurate reflection of a team's winning percentage. As an example, let's look at the 1907 Eastern Canada Amateur Hockey Association (ECAHA) season:

Team	W-L	GF	GA
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

Using Marginal Goals analysis, we get the following results, where MGF is the team's Marginal Goals For, MGS is the team's Marginal Goals Saved, and MG% is the resulting winning percentage:

Team	MGF	MGS	MG%
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

Teams from this time in history will tend to have a wider variance between their actual winning percentage and their MG% than modern teams do, because of the low number of games they played. With only a 10-game season, there is less chance for the luck to even out. But even so, the MG% tracks the teams' actual wins very well, with only the lowly Montreal Shamrocks being off by more than a win. The Shamrocks, a historically terrible defensive team, were lucky to win the two matches that they did in 1907, one against each of the Winged Wheelers and Bulldogs, two other rather bad teams.

The great advantage of this systems, of course, as compared to the more conventional Pythagorean Analysis, is that Marginal Goals gives us a very easy way to not only quantify a team's success, but allocate said success between offence and defence. In 1907, for instance, the Montreal Victorias relied on offence for 58% of their success, while Ottawa was 61% defence. This is the first step in allocating points to individual players, which is the purpose of Point Allocation.

Graphically, the relationship that Marginal Goals Analysis assumes between goal differential and winning looks like this:

As goal differential increases, winning percentage increases in a linear fashion. Now, you may be aware that this relationship isn't entirely realistic. Next time, I'll address that.