Puckerings archive: Goal and Assist Z-Scores (04 Jul 2002)

What follows is a post from my old hockey analysis site puckerings.com (later hockeythink.com). It is reproduced here for posterity; bear in mind this writing is over a decade old and I may not even agree with it myself anymore. This post was originally published on July 4, 2002.

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Goal and Assist Z-scores
Copyright Iain Fyffe, 2002

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Methods have been developed in the past to identify dominant single-season performances. For example, some years ago I developed something I called goal-scoring dominance, which was calculated as the leading player's goals-per-game average divided by the second-leading player's goals-per-game average. A similar calculation was made for assists. I later discovered that Klein and Reif had developed the very same method years before, calling it Quality of Victory.

But this method suffers from a serious flaw. What if two players have outstanding seasons? The Quality of Victory formula will show that no one performed in a dominant manner, because the second-leading player's average is so high. This is not fair, nor is it accurate.

Goal z-scores (GZ) and assist z-scores (AZ) were designed to resolve this problem. It was hoped that they would not create any new problems; unfortunately this is not the case (more on this later). What we do is compare a player's performance to two things: the average individual player performance that year (in terms of goals per game or assists per game), and the degree of variation in individual player performance that year. Standard deviation is a way to measure this variability. For instance, the sets {1,2,3,4,5} and {0,1,3,5,6} have the same mean (5.0), but the second set has more variation, and therefore a higher standard deviation (2.5, compared to 1.6 for the first set). A z-score is simply the number of standard deviations an observation is above the mean (or below the mean in the case of a negative z-score). So, if we have a set of numbers whose mean is 5 and whose standard deviation is 3, then an observation of 8 would have a z-score of 1 ((8 - 5)/3). It's that simple.

In a normal distribution of events, about two-thirds of all observations will fall within one standard deviation of the mean (i.e., have a z-score between -1 and 1). 95% of observations will be within two standard deviations (z-scores between -2 and 2), and almost all will be within three standard deviations (z-scores between -3 and 3). Using z-scores we can determine how outstanding an individual performance was. For instance, only an outsanding season would produce a z-score of 3 or more.

That was the set-up. As it turns out, the results of this study are not that interesting; but what the results indicate may be of interest. The problem with the z-scores is that the top seasons of all time are dominated by recent players. For instance, in the top 40 GZ seasons, we have 5 from the 2000's (in only three years), 17 from the 1990's, 10 from the 1980's, four from the 1970's, and two each from the 1960's (Bobby Hull) and the 1930's (Charlie Conacher). So really is shows only the best of recent seasons. The assist results were predictable; Gretzky has the top 10 almost to himself, with Lemieux following. The top goal results are interesting enough to note (minimum 20 games played):

Rank/Player	Year	GP	GZS
1. Brett Hull	1991	78	5.95
2. Wayne Gretzky	1984	74	5.86
3. Mario Lemieux	1993	60	5.82
4. Cam Neely	1994	49	5.80
T5. Mario Lemieux	1989	76	5.64
T5. Mario Lemieux	1996	70	5.64

So Brett Hull's 1991 campaign, while technically falling short of Gretzky's goal record, is actually more impressive than any of Gretzky's goal-scoring seasons by this analysis. But the real king of the list is Lemieux. In addition to spots 3, 5, and 6, he holds down numbers 11, 17, and 30 on the top 40 (as well as #41). Gretzky has #2, 12, 34 and 38. No contest.

But as I said, the results aren't overly interesting, because they are dominated by recent players. But the fact that recent players dominate is in itself interesting. It indicates that modern players are able to dominate the average players by a larger degree than older players. The cause of this is unclear, as it can be affected by the performance of the top players, as well as what constitutes an "average" player. But it's interesting because it's the exact opposite of what has happened in baseball, where the degree of domination by the top players has decreased over time, rather than increased. Food for thought.

Hall of Fame Standards for the Challenge Era

Today we're going to wrap up our look at the Inductinator, which is a system I devised to determine implicit standards for Hall of Fame player selections. Well, not quite wrap up, since I should make some comment about European and female players, which I will at some point. But let's stick to the things we can mostly explain for now. In Hockey Abstract 2014, I discuss at some length the results from 1930 to the present, both to shed light on history and to make predictions of future inductions. I've already covered the period 1912 to 1929 here on Hockey Historysis. And now we look at the years up to 1911.

This time period, which I'll call the Challenge Era, calls for a somewhat different approach than more recent times. There are no individual awards or All-Star teams to draw information from. Player career statistics are all but useless, in large part because careers were so much shorter in the 19th century, so that comparisons between early professional players in the oughts and senior players before 1900 are not terribly informative.

As it turns out, we don't really need all that, because we have the Stanley Cup. Take note that of all the Hall of Fame players who played before 1911, none of them began their careers before the 1892/93 season, the first that the Stanley Cup was awarded. The best pre-Stanley Cup players such as Tom Paton, Allan Cameron, James Stewart and Jack Campbell have not been honoured. For the 25 Hall of Famers from this era, approximately 60% of their total Inductinator scores is made up of Stanley Cup-related exploits.

Of the 26 Hall of Famers from this era (see below), only five did not win a Stanley Cup championship. So we start by giving skaters 10 points for each Cup championship, and goaltenders 25 per title. Captains of Stanley Cup teams get extra points; players who captained one such team (such as Graham Drinkwater and Tommy Phillips) get 10 points, and each additional Cup captaincy earns a whopping 70 points apiece. Mike Grant, Dickie Boon and Bruce Stuart were captains twice each, while Harvey Pulford was three times, which is enough by itself to get him over the minimum score of 100 for the Inductinator to see the player as being a Hall-of-Famer.

I should say at this point that for this era, the number of Cup championships a player has is not as straightforward as it is for later players. During the challenge era, there were often multiple Cup series played in a single season. The current champion could be called upon by the trustees to defend their title several times in the same season, even sometimes in the middle of a season. In 1908, for example, the Montreal Wanderers had to defend against challenges from the Ottawa Victorias in January, and both the Winnipeg Maple Leafs and Toronto HC in March. For purposes of the Inductinator, we do not count a successful defence of an existing title to be a Stanley Cup championship; it's only when a new champion results from a series that it's counted. The Wanderers don't get credit for three Cup championships for 1908, they get one.

Players winning the Cup with multiple teams get a bonus of 40 points. While this may not seem to sensible, there's not other way to explain how Tom Hooper is in the Hall of Fame. Bruce Stuart, Tommy Phillips and Fred Scanlan also get these points, but they'd have enough points otherwise to still meet the implicit standards. Cecil Blachford also won Cups with two teams, but these points aren't enough to get him to 100. Which is good, since he's not in the Hall of Fame.

Games played, and especially goals scored, in Stanley Cup matches contribute a lot of points the to the Challenge Era Inductinator. Players earn points if they participated in 10 or more Cup matches, and goaltenders earn more per game (since there's so little else to go on for them). A player who did not play in a single Stanley Cup match suffers a penalty of 20 points; otherwise, there would be no way to explain how Herb Jordan is not in the Hall.

But in terms of Challenge Era players being recognized by the Hall of Fame, it seems nothing is as important as scoring goals in Stanley Cup matches. Of the Inductinator scores for the Hall of Famers, a full 26% is earned by Stanley Cup goals alone. Every single player from this era that scored at least 14 goals in Stanley Cup matches is in the Hall of Fame. Fred Whitcroft, who did not play very much top-level hockey but scored 14 goals in eight Cup matches, is in. He gets 100 points for his Stanley Cup goals. He has to, since he did nothing else of note in his hockey career, and we want to explain his induction. Frank McGee scored 41 goals in Cup games, and that explains why he's on the top of the list below.

But wait, you might be aware that Frank "Pud" Glass won a bunch of Stanley Cups with the Wanderers, including one as captain, and scored 13 goals in those games. So how do we explain his exclusion from the Hall? Simple; we consider goals per game as well. Glass took 11 games to score his goals (1.18 per game), while Whitcroft (for example) scored 14 in eight (1.75 per game). Since Glass scored at a subpar rate (for a Hall-of-Famer, anyway), his total goals aren't valued as highly.

Other points are earned for having reasonably lengthy senior careers (important for Hod Stuart and Blair Russel), for playing with one team for at least nine years (again, important for Blair Russel), and for finishing in the top four in goals for a top-level league, or the top two in goals for a lower-level league. Russell Bowie makes out like a bandit in this last category, earning 380 of his 409 points here. He lead a top-tier league in goals five times, was second four times and third once. No one else comes close to that level of production in the Challenge Era.

Finally, we get to the more arbitrary stuff. Tragic deaths are treated favourably by the Hall of Fame. George Richardson was killed in WWI, and although this was after his playing career was done it seems he was more fondly remembered because of it, since otherwise we would not be able explain his induction in this analysis. Hod Stuart's death was all the more noteworthy, as he died in mid-career and as a Stanley Cup champion. Almost all of his Inductinator score (80 out of 102) is derived from this.

All of this so far can be used to explain 22 of the 26 Hall-of-Famers from this era. We're left with Graham Drinkwater, Billy Gilmour, Jack Ruttan and Oliver Seibert.

With Gilmour, one suspects that the true reason he was inducted is that at McGill he played with Frank Patrick, whose brother Lester was of course extremely influential at the time as a member of the selection committee. His Cup wins and goals give him 50 points, so we need another 50. We can attribute that to personal connections and give up, or we can look for something else he might have been famous for. Well, he is one of the very few sets of three brothers who each won a Stanley Cup, and he did it with his brothers (dave and Suddy) on the same team. So, we can give him 40 points for that feat, and an extra 10 for winning the most Cups amongst his set of brothers. It's not the worst thing to recognize such a thing, I suppose, if in fact that's what was being recognized by the committee.

Drinkwater is 40 points short. The only thing I could find to set him apart was the fact that he was one of the three original Allan Cup trustees in 1909, well after his playing career was over. If I were on the committee, I wouldn't assign any player value to this, and maybe they didn't. But it's the only thing I can think of to get him over 100 points. The two other trustees were Dr. H.B. Yates and Sir Edward Clouston. Clouston might also be eligible to collect these points. He never played for the Stanley Cup, of course, since he was 44 years old by the time that mug was first awarded. But Clouston was one of the "Original 18", the 18 men who played in the hockey match at the Victoria Skating Rink in Montreal on March 3, 1875. Clouston played with James Creighton's side, who won that match two goals to one.

Just to make sure we're not being completely arbitrary here, we should also check the original Stanley Cup trustees, to see if they would be put over 100 with this bonus. The two original Stanley Cup trustees were John Sweetland and Philip Ross. Sweetland played no high-level hockey that I'm aware of, but Ross did. He played for McGill in 1879, and later in Ottawa for the famous Rideau Rebels in 1890 and the Ottawa Generals (later the Senators) in 1891. But he never played for the Stanley Cup, so even if we gave him the same 40 points we give Drinkwater, he still wouldn't be over 100 on the Inductinator scale, so we're safe.

The induction of Jack Ruttan, I'll tell you right now, cannot be explained by the Inductinator. He played five seasons of senior hockey in Manitoba in the early 1910s, and won the Allan Cup in 1913. He was very well-regarded as a player in Manitoba, but his accomplishments do not outshine dozens of other players who are nowhere near the Hall. He's a complete and total question mark. I can't explain him, not even close.

Finally, Oliver Siebert. He was certainly a good player. He gains 45 points for leading a lesser league (Western Ontario League) in goals, but loses 20 for never having scored a Stanley Cup goal, for a total of 25. We need another 75 points. Now, there is something that sets Siebert apart from other players, which I suppose we can assign a value of 75 points, although doing so is incredibly silly. That this is this: he has a son (in Earl Seibert) who is a Hall of Fame-calibre player. Oliver was inducted in 1961, and Earl in 1963. We can technically use this to get the elder Seibert over 100 points, though I feel a bit dirty doing so. I suppose such a thing would increase a player's fame, since that's a pretty vague term. I did check other players as well, to make sure such a bonus would not any non-Hall-of-Famers over 100. The closest is goaltender Bert Lindsay, who played after the Challenge Era. Being Ted Lindsay's father is not enough to get him over the threshold, so we can award this bonus to Seibert without producing undesirable results.

Player	Pos	HoF?	Score
Frank McGee	F	yes	416
Russell Bowie	F	yes	409
Bruce Stuart	F	yes	350
Tommy Phillips	F	yes	332
Harvey Pulford	D	yes	272
Marty Walsh	F	yes	250
Harry Westwick	F	yes	231
Alf Smith	F	yes	226
Harry Trihey	F	yes	224
Dan Bain	F	yes	172
Riley Hern	G	yes	167
Fred Scanlan	F	yes	159
Mike Grant	D	yes	122
Tom Hooper	D	yes	105
Graham Drinkwater	D	yes	102
Hod Stuart	D	yes	102
George Richardson	F	yes	101
Bouse Hutton	G	yes	100
Blair Russel	F	yes	100
Dickie Boon	D	yes	100
Billy McGimsie	F	yes	100
Fred Whitcroft	F	yes	100
Art Farrell	F	yes	100
Oliver Seibert	F	yes	100
Billy Gilmour	F	yes	100
Bill Nicholson	G	no	99
Herb Jordan	F	no	99
Pud Glass	F	no	97
Archie Hooper	F	no	96
Billy Breen	F	no	96
Lorne Campbell	F	no	92
Cecil Blachford	F	no	90
Fred Higginbotham	D	no	90
Suddy Gilmour	F	no	89
Rod Flett	D	no	87
Gordon Lewis	G	no	87
Billy Roxburgh	F	no	82
Eddie Geroux	G	no	79
Herb Birmingham	F	no	76
Art Brown	G	no	74
James McKenna	G	no	74
George McKay	F	no	74
Tony Gingras	F	no	73
Robert MacDougall	F	no	73
Oren Frood	F	no	72
Bruce Ridpath	F	no	68
Clare McKerrow	F	no	66
Ezra Dumart	F	no	65
Jack Ruttan	D	yes	0

You can see that, by these standards, there are a number of other players who could just as easily be in the Hall of Fame. Bill Nicholson, Herb Jordan, Pud Glass, Archie Hooper, Billy Breen are all extremely close, and several others are over 90 points as well. Would we view these players any differently today if they had a few more breaks and were elected to the Hall of Fame? Perhaps, but we really shouldn't. The Inductinator analysis reveals that some Hall of Fame selections from this early era seems almost arbitrary, so I cannot recommend putting too much weight on the honour.

Puckerings archive: Point Allocation (09 Apr 2002)

What follows is a post from my old hockey analysis site puckerings.com (later hockeythink.com). It is reproduced here for posterity; bear in mind this writing is over a decade old and I may not even agree with it myself anymore. This post was originally published on April 9, 2002.
 

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Point Allocation
Copyright Iain Fyffe, 2002

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For those who may not know, Bill James is quite a brilliant man. He's known primarily, of course, for his work as a stathead in baseball. It has become fashionable of late (especially amongst younger statheads) to decry James' work. I'll not get into that; I'll just say this: his pure writing about baseball is arguably more impressive and engaging than his statistical work about baseball. Even if he wasn't a brilliant statistician, his work would still be invaluable to any fan who considers himself to be knowledgable about the game.

Fortunately for us hockey folk, some of James' work and ideas can be translated for use in hockey, or can at least be used for inspiration. This paper describes the development of the Point Allocation system, which is a method of evaluating players based on their contributions to their team's success (or lack thereof). It is based on two bits of Bill James; I have adapted his Marginal Runs analysis, which forms the basis for his Win Shares system of player evaluation, and I have extrapolated quite a bit from a fairly casual remark he made in The Politics of Glory.

I'll start with that relatively innocuous comment. James was discussing the common assertion that defence is not reflected well in statistics (at least, the statistics that most people talk about). He pointed out that, to a degree, defence is in fact reflected, through the length of the player's career, and the amount that he plays. For instance, Brooks Robinson was not a particularly great hitter. And yet, he had an exceptionally long career. Why? You know the answer: he was probably the greatest defensive third baseman who ever lived.

Epiphany! Look at this: Guy Carbonneau and Bob Gainey (for example), both with unimpressive offensive totals, both with long careers. Both renowned as defensive players, even if the stats (like plus-minus) "don't show it".

Epiphany again! Maybe we can take this concept down to a team-season level. That is, say we have two players on the same team, who contribute the same offensively, on a per-minute basis. One player plays 15 minutes per game, the other plays 18 minutes per game. Since these players are offensive equals, there can be only one explanation for the discrepancy in playing time: defence. That second player's defence must be sufficiently better than the first's to warrant three extra minutes of playing time per game. More on this later. 

Marginal Goals

Now we move on to the basic ideas behind James' Win Shares system, which I have adapted to create the Point Allocation system. Win Shares is a way of distributing a team's wins amongst its players, based on their relative contributions to the team's success. The building block of Win Shares is Marginal Runs; therefore the building block for Point Allocation is Marginal Goals.
James discovered a new method that predicts team success similarly to his famous Pythagorean analysis (which itself has been adapted to hockey by Marc Foster). I'll explain it in hockey terms. The following formula is an excellent predictor of a team's winning percentage:

E(Pct) = (MGF + MGS) / (2 x AvgG)

Where E(Pct) is expected winning percentage; MGF is Marginal Goals For, calculated as the team's goals for less one-half the league-average goals per team; MGS is Marginal Goals Saved, calculated as one and one-half times the league- average goals per team less the team's goals against; and AvgG is the league-average goals per team.

Marginal Goals is no better at predicting winning percentage than Pythagorean analysis; in fact, it's probably slightly worse. However, what Marginal Goals allows us to do is apportion the team's winning percentage (in the form of points) between a team's offence and defence, as follows:

OP = MGF / TMG x Pts
DP = MGS / TMG x Pts

Where OP is offensive points (points attributable to offence); DP is defensive points (points attributable to defence); TMG is Total Marginal Goals (Marginal Goals For plus Marginal Goals Saved); Pts is the team's points (ties plus two times wins); and MGF and MGS are defined as above.

We simply cannot do this with Pythagorean analysis. Say we have two .500 teams in a league where 300 goals is average. Team A scores and allows 350 goals, while Team B scores and allows 250. Pythagorean analysis will tell us that both of these teams should both be at .500, which they are. But Team A's success is clearly tied to its offence, while Team B relies more on defence. Marginal Goal analysis allows us to determine how much success is attributable to offence and defence (in this example, Team A is 67% offence and 33% defence, and Team B is 33% offence and 67% defence).

Now if you're wondering why 0.5 and 1.5 are used, rather than some other numbers, it is because these produce a result where approximately one-half of all points will be attributed to offence, and one-half to defence. I know this to be true, as I have tested it; I just haven't proven it mathematically.

From here we depart from Mr. James' work. The next step is to allocate the OP and DP amongst the team's players, and the methods used in baseball's Win Shares are not transferable to hockey. So I have devised my own methods of doing so. As an illustrative example, I will go through the Point Allocation calculations for the 2000/01 Detroit Red Wings.

Team Analysis

The Red Wings played 82 games, collecting 107 points (note that I have eliminated points for OT losses, to keep consistency with the entire history of the NHL). In a league in which 226 goals was average, they scored 253 goals and allowed 202 goals. Therefore, their MGF was 140, and their MGS was 137, for a TMG of 277. Thus their 107 points are allocated as follows: 54.1 to offence, and 52.9 to defence.

Offence

Fortunately, hockey stats reflect offensive contribution quite well, through goals and assists. We cannot use scoring points, however, because of the arbitrary way in which they combine goals and assists. There is no reason to think that playmaking is 1.7 times as important as goalscoring (which is what scoring points do in modern times, where there are about 1.7 times as many assists as goals). Since there is no way to determine the relative importance of playmaking and goalscoring, we will assume they are equally important.

Therefore, to allocate offensive points, we need to calculate a new stat, Offensive Contribution (OC), which is simply defined as the player's assists divided by the team average assists per goal, plus the player's goals. For instance, Brendan Shanahan had 31 goals and 45 assists in 2000/01, and the Wings had 1.70 assists per goal. Shanahan's OC is therefore 57 (45/1.7 + 31). Doing this for every Red Wing, we find the team total is 509 (which is twice their goals, with a rounding difference). Shanahan's OC is .112 of the team total; therefore he receives 6.1 Offensive Points (OP), which is .112 of the team's points allocated to offence. Note that in this analysis, goals and assists by goaltenders are ignored. A goaltender's value lies in stopping pucks, not in shooting the puck down the ice into an empty net. Similarly, goalie assists are more a function of team offence, and have little to do with the goalie's skill.

Defence

Defence in hockey is made up of two parts: the skaters who attempt to prevent shots, and the goaltender who attempts to stop those shots that are allowed. Therefore, before we can allocate defensive points amongst a team's players, we need to determine how many go to team defence, and how many go to goaltending.

Since we have defined a defence's job as preventing shots, and a goalie's job as stopping shots, we will use team shots against and goalie save percentage in conjunction with marginal goal analysis to allocate points.

We start with team defence. The defence is responsible for preventing shots. Therefore, we calculate the MGS you would expect for the team based on their actual shots allowed. Detroit allowed 2221 shots in 2000/01, and the NHL average scoring percentage was 9.95%. Therefore we would expect Detroit's defence to have a MGS of 118 ((226 x 1.5) - (2221 x .0995).

Now we move on the goaltending. Since a goalie's job is to stop shots, we evaluate them based on their save percentage. We calculate the MGS we would expect for the goalies based on their save percentage, and then calculate a weighted average for all the team's goaltenders, based on the goalies' playing times. Manny Legace had a .920 save percentage in 2000/01. The NHL average shots per game (excluding empty-net shots) was 2265, and the league average goals (excluding empty-netters) was 219. Legace's MGS is therefore 148 ((219 x 1.5) - (2265 x (1 - .920)). Similarly, Chris Osgood's MGS is 109 based on his .903 save percentage. Legace played 2136 minutes, and Osgood played 2834; the weighted-average MGS for Detroit's goaltending is therefore 126.

We now need to combine these two figures. There are, on average, 4.9 skaters and one goaltender on the ice at any one time. Therefore we will assume that the skaters' value is 4.9 times as important as the goaltending value. Therefore, the DP are distributed as follows:

DPS = DP x MGSS /(4.9 x MGSS + MGSG)
DPG = DP x MGSG /(4.9 x MGSS + MGSG)

Where DPS is Defensive Points allocated to skaters, DPG is Defensive Points allocated to goalies, DP is team Defensive Points, MGSS is the MGS value for skaters, and MGSG is the MGS value for goalies.

For Detroit, this works out to 9.5 for goaltenders and 43.4 for skaters.

Allocating DP to Goaltenders

Allocating the goaltending DP among the team's goaltenders is a simple task. Simply take each goaltender's contribution to the team weighted-average MGS to determine the proportion of DPG he receives. For instance, Detroit's MGSG of 126 was made up of 64 from Manny Legace (148 MGS time his proportion of minutes played) and 62 from Chris Osgood. Therefore, Legace receives 50.8% of the DPG (64/126), or 4.8 points. Osgood receives the remaining 4.7 points.

Allocating DP to Skaters

Skaters' defence is such an ephemeral quality; we all know that the stats don't reflect defence in any meaningful way. But wait! Remember what I was discussing before I got into all of this; a player's defensive value is reflected in his playing time, when his offence and his teammates are taken into consideration. Just like Bill James' breakthrough in fielding analysis, we start at the team level. It's probably easiest to explain by diving right into the illustration.

First, we must assume that a skater's job is made up of equal parts offence and defence, on average. Then we also assume that a player's total value to a team is reflected in his playing time; that is, a team's best players will play the most. I believe these are perfectly logical and safe assumptions. We then compare each player to the average offensive numbers for his position (forward and defence) to find his offensive contribution relative to the team average. Comparing this to his actual playing time, we can estimate his defensive value to the team.

Let's look at some numbers. Detroit's forwards, in total, played 14,250 minutes in 957 games, for an average of 14.89 minutes per game. They had a total OC of 386. The average OC for a Detroit forward was therefore 0.40 per 14.89 minutes.

Now let's look at Brendan Shanahan, Detroit's top scorer in terms of points. He played an average of 18.37 minutes per game, and had an OC per 14.89 minutes of 0.57. His OC was 1.425 times that of an average team forward; if playing time depended only upon offence, we would thus expect him to play 21.22 minutes per game (1.425 x 14.89). But he played only 18.37 minutes per game; 2.85 minutes per game less than the offensive expectation. This difference must be due to his defence, which is obviously not as good as his offence. His defensive minutes per game would be 18.37 minus 2.85, or 15.52; since offence and defence are equally important, this will give us his average playing time of 18.37 per game. If playing time were based solely on defence, Shanahan would probably play about 15.52 minutes per game. He played 81 games, so his total defensive minutes would be 1,257 (15.52 times 81).

We do this for each player in turn. Note that for defencemen, the values for the Red Wings defencemen must be used (9,793 minutes in 512 games, 19.13 minutes per game, 0.24 OC per 19.13 minutes). Also note that it is possible that a player's calculated defensive time would be negative (though usually only for a player playing only a few games). Since we are dealing only with marginal contributions, negative values make no sense. Therefore, any negative value is assumed to be zero in all analysis involving Marginal Goals, and throughout the Point Allocation system. Adding up the minutes, we find Detroit's team total to be 25,209 defensive minutes. We use this total to allocate defensive points to skaters, based on their proportionate contribution to the total. Shanahan had 1,257 of the team's 25,209 defensive minutes, or 0.050 of the total. He therefore receives 0.050 of the 43.4 skater defensive points, or 2.2 points. Adding these to his 6.3 offensive points, we find his total is 8.5 points.

A Final Adjustment

We want this method to be applicable across all years for which the data is available. We don't want any distortion from schedule length or roster size to affect the results. Therefore adjustments are included, to normalize the results to an 80-game schedule, and also to 15 minutes per game for forwards and 20 minutes per game for defencemen. For example, Shanahan played 81 of 82 games; we adjust this to an 80-games schedule, so we give Shanahan 79 GP. He played 18.37 minutes when the average was 14.89; we adjust this for an average of 15, so we credit him with 18.51 minutes per game. So his total minutes are now 1,462 (79 times 18.51), instead of the 1,488 minutes he actually had. We then adjust his OP and DP based upon this adjusted minutes value. These adjustments will eliminate any bias when comparing today's players to players from the days when they played 70 games per year, or when only 17 skaters were allowed to dress. Similarly, goalies' minutes are adjusted to a base of 4800.

Here are the complete team results for the 2000/01 Red Wings. GP is adjusted GP, MIN is adjusted minutes, OP is offensive points (adjusted to MIN), DP is defensive points (adjused to MIN), and TPA is Total Points Allocated (the sum of OP and DP). 

Name	Pos	GP	MIN	OP	DP	TPA
Lidstrom	D	80	2379	5.4	3.7	9.1
Fedorov	F	73	1550	5.9	2.9	8.8
Shanahan	F	79	1462	6.2	2.2	8.4
Lapointe	F	80	1297	4.9	1.9	6.8
Yzerman	F	53	1187	4.1	2.5	6.6
Kozlov	F	70	1038	3.3	1.6	4.9
Legace	G		2063		4.8	4.8
Osgood	G		2737		4.7	4.7
Maltby	F	70	1007	1.8	2.5	4.3
Draper	F	73	987	2.0	2.2	4.2
Gill	D	66	1284	0.9	3.3	4.2
McCarty	F	70	947	2.0	2.1	4.1
Verbeek	F	65	884	2.7	1.4	4.1
Ward	D	71	1261	0.8	3.3	4.1
Holmstrom	F	71	835	3.2	0.5	3.7
Larionov	F	38	699	2.1	1.5	3.6
Murphy	D	56	1112	1.4	1.9	3.3
Dandenault	D	71	1194	2.1	0.9	3.0
Duchesne	D	53	1015	1.9	1.0	2.9
Fischer	D	54	946	0.7	2.2	2.9
Brown	F	59	668	1.9	0.9	2.8
Gilchrist	F	59	693	0.7	1.9	2.6
Devereaux	F	54	550	1.0	1.1	2.1
Chelios	D	23	549	0.2	1.7	1.9
Butsayev	F	15	138	0.2	0.3	0.5
Kuznetsov	D	24	237	0.2	0.3	0.5
Williams	F	5	62	0.2	0.1	0.3
Wallin	D	1	4	0.0	0.0	0.0

So here we have objective evidence that Nicklas Lidstrom is, in fact, Detroit's most valuable player. This surprises no one, I imagine. It is worth noting, however, based on the sampling of team calculations I have thus far made, that it is fairly rare for a defenceman to be a team's MVP (i.e., to have the highest TPA). This may seem to indicate that the system has a bias against defencemen. But I'm not sure this is true. A defenceman's job is primarily defence, and is therefore primarily passive. A defender reacts to an opponent's offense. Therefore, he has less control over his defensive contribution than an attacker has over his offensive contribution. This is reflected in the numbers, where DP tend to be flatter in distribution than OP. So while TPA indicates a team's MVP quite clearly, remember that it is not entirely fair to compare forwards and defencemen directly, since their jobs are so different.

Note how the system also provides objective evidence of the defensive prowess of the Maltby-Draper-McCarty line. Each have a DP total greater than their OP, which is fairly rare for a forward.
For comparison's sake, here are the 1975/76 Montreal Canadiens, one of the greatest teams ever iced. The ice times are estimates calculated using my method for estimating ice time.

Name	Pos	GP	MIN	OP	DP	TPA
Lafleur	F	80	1709	8.7	3.8	12.5
Dryden	G		3580		11.8	11.8
Mahovlich	F	80	1524	6.9	3.3	10.2
Shutt	F	80	1412	5.8	3.2	9.0
Lambert	F	80	1362	4.7	3.5	8.2
Lapointe	D	77	2048	4.2	3.9	8.1
Savard	D	71	1760	2.7	4.4	7.1
Cournoyer	F	71	1097	4.8	1.7	6.5
Risebrough	F	80	1104	3.0	2.9	5.9
Lemaire	F	61	991	3.6	2.0	5.6
Robinson	D	80	1574	2.3	3.1	5.4
Awrey	D	72	1276	0.6	4.6	5.2
Gainey	F	78	1021	2.0	3.1	5.1
Jarvis	F	80	941	2.0	2.6	4.6
Bouchard	D	66	1051	0.7	3.4	4.1
Wilson	F	59	763	2.3	1.7	4.0
Tremblay	F	71	771	1.8	1.9	3.7
Roberts	F	74	783	1.6	2.1	3.7
Larocque	G		1220		3.1	3.1
Van Boxmeer	D	46	672	1.1	0.4	1.5
Nyrop	D	19	326	0.2	1.2	1.4
Chartraw	D	16	233	0.3	0.5	0.8
Goldup	F	3	21	0.0	0.1	0.1
Shanahan	F	4	25	0.0	0.1	0.1
Andruff	F	1	10	0.0	0.0	0.0

Puckerings archive: Harmonic Points (08 Apr 2002)

What follows is a post from my old hockey analysis site puckerings.com (later hockeythink.com). It is reproduced here for posterity; bear in mind this writing is over a decade old and I may not even agree with it myself anymore. This post was originally published on April 8, 2002.

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Harmonic Points
Copyright Iain Fyffe, 2002

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The way it is now, assists are more important than goals in determining scoring championships. Why do I say this? Because for every goal, there are 1.7 assists awarded. Therefore, playmakers have an advantage over goal-scorers, because there are more assists for them to get a piece of. This is not fair. There is absolutely no evidence that playmaking is more important than goal-scoring in terms of scoring goals.

Total Hockey's Adjusted Scoring stats account for this somewhat, by using historic assist rates, which are lower than current rates. But it does not go far enough. Since there is no evidence to indicate which of goal-scoring and playmaking is more important, it is only fair to assume that they are equally important. Thus, when determining a "scoring champion", we should adjust the number of assists to equal the number of goals, on a league-wide basis.

More to the point, I believe we can further refine how we decide who is a "champion" scorer. For instance, say we have three players, all of whom have 80 adjusted scoring points. Player A has 25 goals and 55 assists, Player B has 40 goals and 40 assist, and Player C has 55 goals and 25 assists. I contend that Player B is the superior scorer. Why? Because he is less reliant on other players to produce goals. Player A is a playmaker; if he has no one of talent to pass to, his scoring will suffer. Player C is a goal-scorer; he needs a playmaker to maximize his value. Player B is a more complete player; he is less reliant on teammates, and is therefore a superior individual player.

I do, of course, realize that hockey is a team game, and it takes an entire team to win. But when we are assessing individual players, we should remove the effect of his teammates as much as possible. In this case, we do this with the Harmonic Points system (HP).

HP is based on the mathematical concept of the harmonic mean. The harmonic mean of two numbers is a middle number such that by whatever part of the first term the middle term exceeds the first term, the middle terms exceeds the second term by the same part of the second term. Whew! In other words, if the harmonic mean is 20% (of the lesser term) greater than the lesser term, it will be 20% (of the greater term) lower than the greater term. Still confused? Maybe a numerical example will help.

Take two numbers: 100 and 200. The harmonic mean of these numbers is 133. 133 is 33% (of 100) greater than 100, and 33% (of 200) less than 200. 

I won't keep you in suspense any longer. Here's how to compute HP (which is simply the harmonic mean of goals and assists, times two):

HP = 2 x {(2 x G x A) / (G + A)}

Where HP is Harmonic Points, G is goals, and A is assists. The formula is multiplied by two to retain the "look" of the number of points, since we're taking an average of goals and assists. A player who has an equal number of adjusted goals and adjusted assists will have HP equal to his adjusted points.

In applying HP, I have used Total Hockey's Adjusted Scoring statistics. This is to eliminate much of the bias created by a player's time and place, allowing us to compare players from different eras. In addition, I will be indicating Adjusted Games Played (games played divided by length of schedule times 82), which are not disclosed in Total Hockey, but should be.

But using the idea that playmaking and goal-scoring are equal in importance, we cannot use Adjusted Scoring stats as they are. Adjusted Assists are based on historic assist rates, which, of course, are higher than historic goal rate. So I have adjusted Adjusted Assists to use the same base figure as goals.

Here are the single-season NHL leaders in HP per 82 Adjusted Games Played (minimum 20 AGP), from 1917/18 to 2000/01. There have been 34 100-HP pace seasons in NHL history:

Rank	Name	Club	Year	AGP	HP	Per 82
1.	Howie Morenz	Montreal	1927/28	80	145	149
2.	Mario Lemieux	Pittsburgh	1992/93	59	103	143
	Mario Lemieux	Pittsburgh	1995/96	70	122	143
4.	Wayne Gretzky	Edmonton	1983/84	76	128	138
	Mario Lemieux	Pittsburgh	1988/89	78	131	138
6.	Wayne Gretzky	Edmonton	1981/82	82	129	129
7.	Wayne Gretzky	Edmonton	1984/85	82	127	127
8.	Wayne Gretzky	Edmonton	1982/83	82	122	122
	Mario Lemieux	Pittsburgh	2000/01	43	64	122
10.	Howie Morenz	Montreal	1930/31	73	108	121
11.	Wayne Gretzky	Edmonton	1986/87	81	117	118
12.	Phil Esposito	Boston	1970/71	82	115	115
13.	Mario Lemieux	Pittsburgh	1987/88	79	110	114
14.	Ralph Weiland	Boston	1929/30	82	113	113
15.	Irvin Bailey	Toronto	1928/29	82	112	112
	Jaromir Jagr	Pittsburgh	1995/96	82	112	112
17.	Jaromir Jagr	Pittsburgh	1998/99	81	110	111
18.	Wayne Gretzky	Edmonton	1985/86	82	110	110
19.	Mario Lemieux	Pittsburgh	1989/90	60	80	109
20.	Phil Esposito	Boston	1973/74	82	108	108
	Mario Lemieux	Pittsburgh	1991/92	66	87	108
22.	Mario Lemieux	Pittsburgh	1996/97	76	99	107
23.	Phil Esposito	Boston	1971/72	80	103	106
	Wayne Gretzky	Los Angeles	1988/89	80	103	106
25.	Phil Esposito	Boston	1968/69	80	102	105
26.	Teemu Selanne	Anaheim	1998/99	75	95	104
27.	Aurel Joliat	Montreal	1927/28	82	103	103
28.	Ebbie Goodfellow	Detroit	1930/31	82	102	102
	Gordie Howe	Detroit	1952/53	82	102	102
	Jaromir Jagr	Pittsburgh	2000/01	81	101	102
31.	Mario Lemieux	Pittsburgh	1993/94	22	27	101
	Eric Lindros	Philadelphia	1996/97	52	64	101
33.	Wayne Gretzky	Los Angeles	1990/91	80	98	100
	Steve Yzerman	Detroit	1988/89	82	100	100

It's clear, by this analysis, that Mario Lemieux is the greatest offensive player in NHL history, bar none. His competition is, of course, Wayne Gretzky. Lemieux is on this list nine times to Gretzky's eight, but Lemieux also dominates the top of the list, appearing three times in the top five (to Gretzky's once), and seven times in the top 20 (to Gretzky's six). Lemieux is the only player with multiple 140-HP pace seasons (Gretzky never had one), and the only player with multiple 130-HP pace seasons (three, to Gretzky's one).

In terms of career HP per 82 AGP, there are four distinct classes of players: (1) Mario Lemieux, (2) Wayne Gretzky, (3) current stars in their prime, and (4) everyone else. Lemieux, through the 2000/01 season, has 1077 HP in 799 AGP, for a per-82 game figure of 111. No one else is even remotely close. Gretzky is second with an average of 96 (1805 HPP in 1543 AGP). Following these two are a bunch of players in the 80's, all current players in their prime: Eric Lindros, Jaromir Jagr, Teemu Selanne and Paul Kariya. Their averages will most likely drop over time to put them in the final group. The "everyone else" group is headed by Mike Bossy (75 average), Howie Morenz (73) and Phil Esposito (73). Other high averages belong to Gordie Howe, Jean Beliveau, Steve Yzerman, Joe Sakic, Marcel Dionne, and Bobby Hull.

The degree of separation between these classes of players serve to demonstrate how truly impressive Mario Lemieux's (and, to a lesser extent, Wayne Gretzky's) scoring exploits really are. These are the complete scorers, players who can carry a team's offence on their backs, all by themselves.

Puckerings archive: Does Playoff Experience Matter? (30 Oct 2001)

What follows is a post from my old hockey analysis site puckerings.com (later hockeythink.com). It is reproduced here for posterity; bear in mind this writing is over a decade old and I may not even agree with it myself anymore. This post was originally published on October 30, 2001 and was updated on April 9, 2002.
 

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Does playoff experience matter?
Copyright Iain Fyffe, 2002

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We all have heard that playoff experience is critical for playoff success. It's certainly been said often enough. If a team, or rather the players on a team, don't have enough playoff experience, they don't have a prayer of winning in the post-season. I believe it's time we put this idea to the test.

The assertion is this: teams with more playoff experience will be more successful in the playoffs than teams with less playoff experience. We will define success in the playoffs as the winning of playoff series, not necessarily winning the Stanley Cup. We will test the assertion through head-to-head playoff series matchups. If the assertion is true, then a team's relative playoff experience should be a good predictor of the outcome of the playoff series.

To test this assertion, I used data from the past three NHL seasons: 1998/99, 1999/00, and 2000/01. I defined a team's playoff experience as the total career playoff games played in previous years by all players who played for the team in that playoff year. I then used these total playoff experience figures as the sole factor in predicting the winner of each playoff series. That is, I predicted that the team with more total playoff experience would win each series. Here are the results of these predictions:

Year	Series	Right	Wrong	Pct
1998/99	15	10	5	.667
1999/00	15	11	4	.733
2000/01	15	9	6	.600
Total	45	30	15	.667

There you have it. Playoff experience is a very good predictor of playoff success, being right two-thirds of the time. But not so fast; we need to go deeper than this superficial analysis. The problem with this analysis is that a player's playoff experience is not independent of the quality of his team (defined here as regular season points). That is to say, a player's playoff success depends greatly upon him playing for a good regular-season team; but don't take my word for it.
We start with two simple points: (1) good teams generally stay good from year to year, while bad teams stay bad, and (2) teams retain a majority of the same players from year to year. Before I continue, let me demonstrate that these points are true.

To demonstrate the first point, I will simply use correlation. The following are the correlation coefficients for NHL teams' regular season points between 1998/99 and 1999/00, as well as the correlation for points between 1999/00 and 2000/01.

Years	Correlation
1998/99-1999/00	0.67
1999/00-2000/01	0.77

As demonstrated in the above table, last year's points are an excellent predictor of this year's points. A correlation of 0.60 or more is considered high, and the relationship is therefore very strong.

The second point is also simple to demonstrate. I selected a random sample of five teams to test the stability of their rosters. I compiled the regular season games played in 2000/01 for players on each team at the end of the year who were also on the same team at the end of the previous year (1999/00). I then compared these results to the maximum number of man-games, which is 18 skaters plus one goalie times 82 games, or 1558 man-games. Here are the results:

Team	Games	% of Max
Atlanta	1086	70
Los Angeles	915	59
New Jersey	1300	83
Phoenix	1002	64
Toronto	1124	72
Average	1085	70

As you can see, the team you play for this year is most likely the team you played for last year. On average, 70% of a team's games are played by players who also played on the team at the end of the previous year.

Now that I have established these points, let's move on to this question: how good a predictor of playoff success is regular season success? I again tested playoff series for the past three years, this time using regular season points as the sole predictor of series winners. The 'neithers' in the table below are the result of teams having equal points, and therefore no winner being predicted. The results:

Year	Series	Right	Wrong	Neither	Pct
1998/99	15	10	4	1	.700
1999/00	15	11	4	0	.733
2000/00	15	9	5	1	.633
Total	45	30	13	2	.689

As you can see, regular season success is marginally better than playoff experience at predicting playoff winners. What this really shows is that playoff experience has no apparent effect on the results of the playoff. If playoff experience were important, it would be better at predicting winners than regular season points. However, they're virtually identical as predictors. The reason for this is that playoff experience is accumulated through playing for a good team. I have shown that players generally play for the same team from year to year, that good teams are generally good from year to year, and that good teams are successful in the playoffs. Therefore, players on good teams will accumulate large totals of playoff experience not by "knowing what it takes to win in the playoffs," but by playing for a good team that will tend naturally to win more, both in the regular season and in the playoffs.

The crucial point is this: playoff experience is the result of playing for a successful regular season team. Playoff experience is simply a reflection of playing for a good team. There is absolutely no evidence that having greater playoff experience will affect the result of a playoff series. If playoff experience were important, it would be better than regular-season points in predicting playoff series winners; in fact, it's marginally worse. In reality, it's the quality of the team that matters, not the playoff experience of the players.