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Goals Versus Average

A method for measuring player value

Copyright Tom Awad, 2003

Designed by Tom Awad

Written by Tom Awad and Iain Fyffe

OVERVIEW

Hockey statisticians have been known to complain that the treatment of statistics in hockey lags behind that of other sports, especially baseball. One main problem in hockey's statistics is that, while offensive production is relatively easy to measure, the defensive contributions of a player are less so. In addition, it is almost impossible to compare players of different positions, especially goaltenders, due to the very different nature of their functions. This statistical analysis is an attempt to remedy a portion of this problem; we will leave it to the reader to judge if it has succeeded.

The objective of this analysis is to be able to compare hockey players of all positions and over any period of time (there are usually only two time spans over which players are compared; that is, a single season or an entire career). The theory behind this analysis is that hockey, despite being about winning or losing, ultimately comes down to scoring goals and preventing them. Each player's role, no matter his position, is to try and increase the goal differential in favour of his team. Using that standard, all players can be compared by the same yardstick: by how much did they help (or harm) their team's goal differential?

We have divided hockey players' responsibilities into 3 functions: offence, defence, and goaltending. The offensive function is to contribute to goals scored against the opposition. This function has historically been the best measured statistically, since it consists of measuring games played, goals and assists. In fact, it would be rather easy to perform a comparative analysis of offensive contributions back to the NHL's beginnings. We have attempted to follow conventional hockey wisdom and assigned to goals a value of 1.5 assists. Obviously, opinions vary on this point, but we have assumed that this figure is as good as any other. In addition, both goals and assists have less value each on teams that record more assists. As an example, on a team that scores no assists, a goal would have an offensive value of 1.0, whereas on a team that records exactly 1 assist per goal, each goal would be worth 0.6 and each assist worth 0.4.

The second function is defensive: we have defined the defensive responsibility as preventing shots on goal. We are aware of the fact that this measure is not perfect and that not all shots are of equivalent quality; however, until an objective measure of shot quality becomes available, this is the best information we have. The extremes work: a perfect defensive team would allow 0 shots on goal, while a horrible team would allow as many as the opposition will take. A player�s defensive value can be measured by two statistics: the number of shots on goal that his team allows, and his plus-minus compared to his team�s goal differential. Also, as an attempt at mirroring conventional hockey wisdom, we have assumed that each defenseman has twice the defensive responsibilities that a forward has. One of the components of this extra value is that defensemen typically play an average of 20 minutes a game while forwards play an average of 15 minutes a game, so they have 1.33 times more ice time. We also assume that, for each minute of ice time, they have 1.5 times the defensive responsibilities of forwards, so 1.33 * 1.5 = 2. We understand that this value could also be debated indefinitely without answer, so we will leave it at that.

The third function is goaltending: blocking the shots that do get taken on goal. Ultimately, this is a goaltender's only responsibility. All other goaltending statistics such as wins, shutouts, goals-against average, and others are just by-products of blocking shots. And a goaltender never gets the choice of how many shots he faces.

Now that each function has been defined, it becomes possible to determine how many goals were scored or prevented by a player's contribution. But how do we compare them? The technique we use here is to establish the average productivity of a player at each position and to measure each player's contribution relative to this average. This has a few interesting side-effects:

- A player's contribution can be positive or negative. This is a somewhat odd result given that we always assume that a player contributes something, but it makes sense when comparing to the average. Because the average is the only real yardstick that we are capable of arriving at objectively, it is the best reference point that we can use. By allowing negative values, we make sure that the system works versus ANY reference point, not just versus the average; for example, one could choose to use a �replacement level� rather than the average as the reference point: then all players� values would be increased, but some of them could still be negative. Among other things, allowing negative contributions shows that having a player in your lineup who contributed nothing has a cost: it prevents another player from being in the lineup. Note that the concept of positive and negative values is not entirely foreign to hockey statistics: plus-minus works in much the same way.

- The sum of the contributions of the players on a team equals that team's goal differential on the season. This is a useful result because it makes it much easier to measure "how good would this team be replacing player A with player B?"

- Better players stick out more from the pack than they do using other statistics. For example, Bobby Orr's shortened career ensured that he would forever be far down on any list of career scorers or producers using normal statistical measurements. However, because this system only measures delta from the average, he ranks much higher, which fits in better with the �intuitive� ranking of all-time players. In effect, comparing Bobby Orr (who played eight full seasons since 1967) to Paul Coffey (who played 20) is like comparing the sum of Orr's productivity plus that of an average defenceman for the 12 other seasons.

Because every player is compared to others of his position, defencemen get a leg up in terms of offensive production, so a defenceman with similar stats to a forward will be assigned a higher offensive value. This makes sense because it indicates that good offensive defencemen are harder to find than good offensive forwards.

Plus-minus has only been recorded since the 1967-68 season, and shots on goal against goaltenders have only been officially recorded since 1982-83. However, thanks to the diligent work of Edward Yuen, a hockey fan from Regina who compiled shots-on-goal stats for the seasons 1954-55 to 1966-67, as well as a few in between up to 1982-83, we have stats that stretch back all the way to 1954. For the seasons for which we had no shots information, we have had to �guess� the amount of shots on goal given the performance of the various goaltenders on a team. In effect, the rule we used is that, for any goal that seems to be above or below the league average, we split the credit between the goaltenders and the team defence (therefore, team numbers in the defence and goaltending categories will be identical). Obviously, if some goaltenders on a team allow fewer goals than others, they will still have better ratings. Because of the lack of plus-minus for the years prior to 1967, we assumed that all players of a given position on a team had the same defensive contribution per game. Although this is obviously not true, we could do no better with the limited statistics we had. Luckily, for recent seasons, none of these limitations hold, and we can fully analyze the performances of each player.

There are a few things to keep in mind as to what this analysis does and does not do:

- It only analyzes regular-season statistics. We have not found a good electronic database of playoff statistics, and in addition to this the analysis becomes much more open to bias during the playoffs. During the regular season, you can assume that the average of all the opponents you face will approximate the average of the league. During the playoffs that is far from being the case: you may play only seven games, all against the best defensive team in the league! Another problem is that, given the large number of games in the regular season, goal differential is a good proxy for overall success. In the playoffs, this is not true: a team could muscle through 3 rounds and get to the final, only to get trounced, and could finish with an even or negative goal differential, even though they got to the final! Although the analysis could still be interesting (especially for players renowned for being playoff heroes), it will have to wait for a later time.

- It does not measure a player's quality or talent: it is made to measure his contribution to the team's goal differential. A goaltender that faces zero shots will have a value of zero, regardless of whether he is Patrick Roy or Ken Wregget. It also does not take into account environment: a player will score more with better linemates, and we make no attempt to adjust for that. In effect, we are not trying to see what information is �hidden� in the statistics; we are simply trying to better characterize the information that is at hand.

Once the goal differential for a single season is established, to compare players� performances across different seasons, it is necessary to normalize their performances. In this case, we have normalized for 2 things: schedule length and the average number of goals per game. Obviously, in order to correctly compare performances, a player playing a 70-game season must be given more credit per goal than one playing an 80-game season. Luckily, over the last 35 years, the schedule has only varied from 74 to 84 games, with the exception of the 48-game season in 1994-95. We have normalized to the current length, an 82-game schedule.

The other normalization factor is goals per game. Obviously, over the last 35 years, the level of openness of the game has varied, and players deserve credit for playing within the level of their era. A 40-goal scorer is much more impressive in 2002 than it was in 1982. The average number of goals per game per team since 1968 is 3.41, so this is what we have normalized to.

THE METHOD

A player's GVA value is the sum of three things: his Offensive Goals Versus Average (GVA), his Defensive Goals Versus Average (DGVA), and his Goaltending Goals Versus Average (GGVA). Each of these factors are calculated independetnly, as follows.

Offensive GVA

Offensive contributions are well-measured statistically, using goals and assists. But to arrive at a single value for offence we must combine goals and assists in some manner. NHL scoring races are determined by simply adding the two together; since there are about 1.7 assists awarded per goal in the NHL, this assumes that an individual's goal makes up about 37% of the value of a typical team goal scored, while an assist is worth about 37% as well (1 goal plus 1.7 assists times .37 equals 1.0). We feel that this overestimates the value of an assist somewhat. Therefore, the GVA method assumes that a goal is worth 1.5 times the value of an assist. Using this assumptions, we calculate the value of goals and assists like so:

GVt = (Gt x 1.5)/(At + (Gt x 1.5))

AVt = GVt / 1.5

Where GVt is team goal value, AVt is team assist value, Gt is team goals, and At is team assists. This is done seperately for each team. Typically, it yields the following results: GVt = .47, AVt = .31. Note that .47 plus (1.7 times .31) equals 1.

The GVA method is based on comparing players to the average. Therefore we need to determine what the average offensive contribution is, by position. This is simply done, as follows:

AOCf = [sum (Gf x GVf + Af x AVf)] / [sum (GPf)]

AOCd = [sum (Gd x GVd + Ad x AVd)] / [sum (GPd)]

Where AOCf is average offensive contribution by a forward (per game), AOCd is average offensive contribution of a defenceman (per game), Gf is goals by individual forwards, Gd is goals by individual defencemen, Af is assists by individual forwards, Ad is assists by individual defencemen, GPf is games played by individual forwards, GPd is games played by individual defencemen, GVf is goal value (based on team) for individual forwards, GVd is goal value for individual defencemen, AVf is assist value for individual forwards, and AVd is assist value for individual defencemen.

A player's OGVA is therefore:

OGVA = (G x GV) + (A x AV) - (GP x AOC)

Where G is the player's goals, A is his assists, GP is his games played, GV is his goal value, AV is his assist value, and AOC is the average offensive contribution value for his position.

Goaltending GVA

We'll tackle Goaltending GVA next because it is much easier to calculate than DGVA. A goaltender's contribution to his team's efforts to win is attempting to stop the shots that come his way. This is best measured by save percentage, which indicates the proportion of the shots faced by a goaltender that are stopped.

To compare goaltenders to the average, we must compute average save percentage:

SPa = sum (SVg) / sum (SFg)

Where SVg is saves by individual goaltenders and SFg is shots faced by individual goaltenders.

A goaltender's GGVA is therefore calculated as the number of saves he made, less the number of saves that a goalie with an average save percentage would have made:

GGVA = SV - (SF x SPa)

Where SV is the goaltender's saves, SF is his shots against, and SPa is the league average save percentage.

Defensive GVA

Defence is by far the most difficult aspect to assign value to in the game of hockey. It is not particularly well-measured statistically. Therefore, a safe way to proceed is to assume that if a player played on a good defensive team, he is likely good defensively himself. Put another way, the sum of the defensive contributions of all players on a team must equal that team's total defensive output. Since we have defined a goaltender's contribution to winning as stopping the shots he faces, the defence's contribution must be preventing shots from being taken. Therefore we use team shots allowed to assign value to team defence.

The responsility of defence is not equal among all players. Specifically, defencemen are more responsible for defence than are forwards, just as forwards are more responsible for offence than are defencemen. But how much more responsible? We could just assume a number, but let's see if we can arrive a figure logically.

Forwards score, on average, about 85% of all goals and record 75% of all assists. A typical GV is 0.47; 85% of this is .40. A typical AV is 0.31; this, times 1.7, times 75% is also .40. Forwards therefore record .80 offensive value on each goal scored. This leaves .20 for defencemen. Typically, 12 forwards play in each game for each team. This means that an individual forward records .067 offensive value on each goal (.80 divided by 12). Typically six defencemen play each game; therefore, an individual defenceman records .033 offensive value on each goal. This indicates that individual forwards are responsible for twice the offence of individual defencemen.

It follows that individual defencemen are twice as responsible than forwards for defence, since there is no reason to believe forwards are more valuable on the whole. Therefore the sum of adjusted defensive games played for a team is:

ADGPt = sum (GPf) + sum (2 x GPd)

Where GPf are games played by forwards and GPd are games played by defencemen. The first factor of DGVA is thus:

DGVAa = ADGPp / ADGPt

Where ADGPp is the player's adjusted defensive games played (equal to GP for forwards, 2 x GP for defencemen). This essentially divides defensive responsibility evenly among the team's players, with an adjustment for position. From there, we move on to the second factor of DGVA, which differentiates players on a team based on their performance.

For this, we need an indicator of individual defensive performance. Unfortunately, this is the aspect of hockey that has the least useful statistical information. Plus-minus is the best mainstream statistic in this regard, and even it is only half defensive. While it is often called a defensive stat, plus-minus also reflects offence, since a player scoring a goal receives a plus. There are variety of other problems with plus-minus as well, but it is the best statistic available right now for this purpose.

We can siginificantly improve plus-minus by adjusting it to account for team performance. Players on good teams will have better plus-minus ratings, because plus-minus is significantly affected by one's teammates. We therefore adjust a player's plus-minus based on the average plus-minus for his team. Since forwards and defencemen have different amounts of ice time on average, we do this by position.

For forwards, the team's plus-minus per game is as follows:

PMtf = sum (PMf) / sum (GPf)

Where PMf is each forward's plus-minus and GPf is each forward's games played. For defencemen the calculation is this:

PMtd = sum (PMd) / sum (GPd)

Where PMd is each defenceman's plus-minus and GPd is each defenceman's games played. Players' adjusted plus-minus figures are therefore:

APMf = PM - PMtf x GP

APMd = PM - PMtd x GP

Where PM is the player's plus-minus, PMtf and PMtd are team adjusted plus-minus per game for forwards and defencemen respectively, and GP is the player's games played.

But this adjusted plus-minus cannot be applied directly, because at any one time there are five players on the ice (on average): three forwards and two defencemen. So a single player can only do so much to affect any outcome on the ice. We could simply multiply a player's APM by one-fifth, but remember, we have decided that defencemen are twice as responsible for defensive play. Since some of that difference is included in their 33% average additional ice time (and extra ice time is incorporated into plus-minus, since being on the ice longer means more pluses and more minuses), we calculate that plus-minus for defensemen is 1.5 times more important than for forwards.

So the calculation to find a player's plus-minus factor is as follows:

PMFf = (1 / 5) x (5 / 6) = 0.167

PMFd = (1 / 5) x (5 / 6) x 1.5 = 0.250

The (1 / 5) factor represents the player being one of five on the ice at any one time. The (5 / 6) factor represents players on the ice versus weighted players on the ice (3 forwards * 1.0 + 2 defensemen * 1.5 = 6 weighted players). Defencemen get an additional multiplier to represent their additional defensive responsility, as discussed earlier. Note that these factors will always sum to one (three forwards at 0.167 plus two defencemen at 0.250 equals 1.001, the difference being rounding).

The second factor of defensive contribution is therefore (for forwards and defencemen respectively:

DVGAbf = APMf x PMFf

DVGAbd = APMd x PMFd

And a player's total defensive contribution is as follows:

DVGA = DVGAa + DVGAb

And a player's total Goals Versus Average figure is as follows:

GVA = OGVA + DGVA + GGVA

SAMPLE RESULTS

2001/02 NHL GVA Leaders