The plus-minus; what it is
The plus-minus (or +/-) is relatively new in the hockey statistics world. It was first employed by the Montreal Canadiens in the 1950s and started being tracked by the NHL in 1968. That's over 40 years, but most fans had never heard of it until the last 15 or so (if ever). These days you hear it mentioned several times in most broadcasts, but what is it?
As with many sports, the most well known hockey statistics (goals and assists) are geared towards measuring a player's offensive ability. The plus-minus is intended to be a measure used for defensemen and defense-minded forwards. The statistic is calculated by taking the number of non-power play goals scored by the player's team (so even strength and shorthanded goals) while he/she is on the ice and subtracting the number of non-power play goals scored by the other team while he/she is on the ice. For example, suppose Roy was on the ice for 3 goals scored by his team, one of which was a power play goal. This gives him +2 (since the power play goal doesn't count).If the other team also scored 3 goals, none of them on the power play, then his plus-minus for the game is +2 - 3 = -1.
Once you get used to it, this is pretty straightforward. The idea is that a better defensive player will have a higher plus-minus since the other team will score less when he is on the ice and the offensive players on the team will likely score more since they can be confident that the defense is being taken care of.
What's wrong with the plus-minus
The plus-minus has a number of issues, but there two that I want to address here:
The first is that the plus-minus does not take into account the amount of time a player spends on the ice. A player that spends 20 minutes playing and earns a +1 is seen as contributing the same as a player who plays just 1 minute and also happens to earn a +1. This is not equitable.
The second (and, in my opinion, bigger) problem is that there is no baseline. Suppose Roy's team in the example above lost that game 14-3. Roy's -1 is actually quite good. On the other hand, if Roy's team won the game 14-3 then Roy's -1 is really bad. This becomes worse as you look at season or (shudder) career plus-minus. Someone who is very bad but plays for a very good team can have a season plus-minus that looks great, while someone who is very good but plays for a very bad team can have a plus-minus that looks horrible. (Take John Tavares, for example. His season plus-minus in 2011-12 was -6. That's on the minus side, so looks bad, but when you take into account that the Islanders were 26th in a league of 30 and that they were outscored by their opponents by 55 goals (251-196), his -6 is actually pretty decent. It's not great, but he's also a very offensively-minded player and it is 5th place on the team among players who played at least half the season.)
To make matters worse, stats pages and commentators compare plus-minus values between players on different teams as if that means anything. Is Chris Kunitz's +16 (5th among the Pittsburgh Penguins for players playing at least half the games in 2011-12) really better than John Tavares' -6? That's debatable since the Penguins outscored their opponents by 65 goals (273-218).
To make the plus-minus more comparable between teams, we need a baseline to work from. That's where my proposal for the adjusted plus-minus comes in.
Proposed solution: adjusted plus-minus
The goal behind the adjusted plus-minus is to create a statistic with a known baseline which is therefore comparable for players on different teams. The formula is below, but the idea is simple: adjust everyone's plus-minus so that a player that is neutral (in the sense of not directly affecting which team has more goals in the end) will have an adjusted plus-minus of 0. A positive adjusted plus-minus will show that the player is contributing to the team's success and a negative adjusted plus-minus will show that the player is not (or may, in fact, be contributing to the team's failure). The formula takes into account the player's time on the ice (TOI) as well as the overall performance of his team, thus addressing the two issues described above.
First, some notation:
EVTOI = the player's time on the ice when the teams are even strength. This includes 5 on 5, 4 on 4, 3 on 3, as well as times when one team has pulled the goalie in order to gain a one skater advantage. If either team is on a penalty kill then that time is not counted in the ESTOI.
EVTOT = the team's total time playing at even strength. (Obviously, this is the same for both teams in a single game, but not over the course of many games against differing opponents.)
TPM = the player's team's plus-minus. This is a total of all even strength and shorthanded goals scored by the players team minus all even strength and shorthanded goals scored by the opponent. In short, compute the traditional plus-minus for the entire team as if it was one person.
PPM = the player's traditional plus-minus.
Okay, here's the formula:
Once you get used to it, this is pretty straightforward. The idea is that a better defensive player will have a higher plus-minus since the other team will score less when he is on the ice and the offensive players on the team will likely score more since they can be confident that the defense is being taken care of.
What's wrong with the plus-minus
The plus-minus has a number of issues, but there two that I want to address here:
The first is that the plus-minus does not take into account the amount of time a player spends on the ice. A player that spends 20 minutes playing and earns a +1 is seen as contributing the same as a player who plays just 1 minute and also happens to earn a +1. This is not equitable.
The second (and, in my opinion, bigger) problem is that there is no baseline. Suppose Roy's team in the example above lost that game 14-3. Roy's -1 is actually quite good. On the other hand, if Roy's team won the game 14-3 then Roy's -1 is really bad. This becomes worse as you look at season or (shudder) career plus-minus. Someone who is very bad but plays for a very good team can have a season plus-minus that looks great, while someone who is very good but plays for a very bad team can have a plus-minus that looks horrible. (Take John Tavares, for example. His season plus-minus in 2011-12 was -6. That's on the minus side, so looks bad, but when you take into account that the Islanders were 26th in a league of 30 and that they were outscored by their opponents by 55 goals (251-196), his -6 is actually pretty decent. It's not great, but he's also a very offensively-minded player and it is 5th place on the team among players who played at least half the season.)
To make matters worse, stats pages and commentators compare plus-minus values between players on different teams as if that means anything. Is Chris Kunitz's +16 (5th among the Pittsburgh Penguins for players playing at least half the games in 2011-12) really better than John Tavares' -6? That's debatable since the Penguins outscored their opponents by 65 goals (273-218).
To make the plus-minus more comparable between teams, we need a baseline to work from. That's where my proposal for the adjusted plus-minus comes in.
Proposed solution: adjusted plus-minus
The goal behind the adjusted plus-minus is to create a statistic with a known baseline which is therefore comparable for players on different teams. The formula is below, but the idea is simple: adjust everyone's plus-minus so that a player that is neutral (in the sense of not directly affecting which team has more goals in the end) will have an adjusted plus-minus of 0. A positive adjusted plus-minus will show that the player is contributing to the team's success and a negative adjusted plus-minus will show that the player is not (or may, in fact, be contributing to the team's failure). The formula takes into account the player's time on the ice (TOI) as well as the overall performance of his team, thus addressing the two issues described above.
First, some notation:
EVTOI = the player's time on the ice when the teams are even strength. This includes 5 on 5, 4 on 4, 3 on 3, as well as times when one team has pulled the goalie in order to gain a one skater advantage. If either team is on a penalty kill then that time is not counted in the ESTOI.
EVTOT = the team's total time playing at even strength. (Obviously, this is the same for both teams in a single game, but not over the course of many games against differing opponents.)
TPM = the player's team's plus-minus. This is a total of all even strength and shorthanded goals scored by the players team minus all even strength and shorthanded goals scored by the opponent. In short, compute the traditional plus-minus for the entire team as if it was one person.
PPM = the player's traditional plus-minus.
Okay, here's the formula:
Adjusted plus-minus = PPM - (TPM * EVTOI/EVTOT)
The logic is as follows:
EVTOI/EVTOT gives you the fraction of the team's even strength time during which the player was playing. Shorthanded goals for or against wind up acting as a bonus or penalty for the player, but power play goals either way do not affect them, so that time is not included in the ratio. In this way, the time that the player spends on the ice is taken into account in the statistic.
Multiplying EVTOI/EVTOT by TPM gives you the player's "expected" plus-minus. This is the plus-minus that would be earned by a player who is neutral in the sense described above. This is very likely to be a fractional amount; I've been rounding to two decimal places in my personal calculations, but any number of places could be chosen.
Subtracting the expected plus-minus from the actual plus-minus gives a statistic with a baseline of zero. A neutral player would be "even" (an adjusted plus-minus of 0), no matter how good or bad his/her team was.
Example with full statistics available:
Let's get some more information about Roy's game. The actual score for the game was 9-4, Roy's team lost, scoring once on the power play and giving up two power play goals against. The team's total plus-minus, then is TPM = 3 - 7 = -4. (3 because they scored 3 non-power play goals and 7 because they gave up 7 non-power play goals).
During the game, Roy played a total of 17:30 of ice time: 0:45 on the penalty kill, 1:15 on the power play, and the rest at even strength. That makes Roy's EVTOI = 15:30 (17:30 - 0:45 - 1:15).
During the 60:00 game, the team spent 4:30 shorthanded and 5:15 on the power play. Thus the teams even strength time is EVTOT = 50:15 and Roy's EVTOI/EVTOT = 15:30/50:15 = 0.31 (approximately).
Multiply that by the TPM and you get 0.31 * -4 = -1.24. This is Roy's expected plus-minus. His adjusted plus-minus, therefore is -1 - (-1.24) = +0.24. Roy was actually a little bit of a help to his team in this game, in spite of the -1 rating.
How to adjust for rec league:
Of course, most of us don't have that detailed TOI data to refer to. In my rec league, I've simplified the calculation to simply take the approximate fraction of the game I play based on the number of people playing my position. If I'm one of three centers, for example, then I make the fraction 1/3. If I'm one of 5 defensemen, I make it 2/5 (two defensemen on the ice at any time divided by 5 total). Other than that, I compute it just as described above.
Update 9/24/2012: As an example of what a difference it can make, I offer my own stats from the SCHL for summer 2012. That was a rough session for my team; we went winless and were completely blown out multiple times. My plus-minus for the summer was -21. My adjusted plus-minus was -0.70. I was still on the minus side, which doesn't please me, but at least it shows that I wasn't a complete drag on the team, which a -21 might seem to imply at first.
EVTOI/EVTOT gives you the fraction of the team's even strength time during which the player was playing. Shorthanded goals for or against wind up acting as a bonus or penalty for the player, but power play goals either way do not affect them, so that time is not included in the ratio. In this way, the time that the player spends on the ice is taken into account in the statistic.
Multiplying EVTOI/EVTOT by TPM gives you the player's "expected" plus-minus. This is the plus-minus that would be earned by a player who is neutral in the sense described above. This is very likely to be a fractional amount; I've been rounding to two decimal places in my personal calculations, but any number of places could be chosen.
Subtracting the expected plus-minus from the actual plus-minus gives a statistic with a baseline of zero. A neutral player would be "even" (an adjusted plus-minus of 0), no matter how good or bad his/her team was.
Example with full statistics available:
Let's get some more information about Roy's game. The actual score for the game was 9-4, Roy's team lost, scoring once on the power play and giving up two power play goals against. The team's total plus-minus, then is TPM = 3 - 7 = -4. (3 because they scored 3 non-power play goals and 7 because they gave up 7 non-power play goals).
During the game, Roy played a total of 17:30 of ice time: 0:45 on the penalty kill, 1:15 on the power play, and the rest at even strength. That makes Roy's EVTOI = 15:30 (17:30 - 0:45 - 1:15).
During the 60:00 game, the team spent 4:30 shorthanded and 5:15 on the power play. Thus the teams even strength time is EVTOT = 50:15 and Roy's EVTOI/EVTOT = 15:30/50:15 = 0.31 (approximately).
Multiply that by the TPM and you get 0.31 * -4 = -1.24. This is Roy's expected plus-minus. His adjusted plus-minus, therefore is -1 - (-1.24) = +0.24. Roy was actually a little bit of a help to his team in this game, in spite of the -1 rating.
How to adjust for rec league:
Of course, most of us don't have that detailed TOI data to refer to. In my rec league, I've simplified the calculation to simply take the approximate fraction of the game I play based on the number of people playing my position. If I'm one of three centers, for example, then I make the fraction 1/3. If I'm one of 5 defensemen, I make it 2/5 (two defensemen on the ice at any time divided by 5 total). Other than that, I compute it just as described above.
Update 9/24/2012: As an example of what a difference it can make, I offer my own stats from the SCHL for summer 2012. That was a rough session for my team; we went winless and were completely blown out multiple times. My plus-minus for the summer was -21. My adjusted plus-minus was -0.70. I was still on the minus side, which doesn't please me, but at least it shows that I wasn't a complete drag on the team, which a -21 might seem to imply at first.
Please feel free to ask any questions and/or offer any critiques of this proposed statistic. I've been using it for myself for years, but I'd like to know what other people think.
Great blog, Scott.
ReplyDeleteI'd argue that the +/- is not a useful 'short-run' stat. Like a stock
investment, it's most telling after significant time has past. So while
it does have a purpose it's most often used incorrectly. But I'd be
curious to see your formula applied to a full season. Since the NHL is
questionable, maybe an AHL or college team can provide the data.
Good stuff!
Ryan #6
While I agree that the plus-minus is not a useful short-run stat, I disagree that it has much use even in the long run. There are so many variables that are not taken into account in the statistic (teams played for, time on ice, etc) that I just don't see it as very useful at all.
ReplyDeleteIf you can find a team that would be willing to track this for a season, I would also be interested in seeing it. (I might even be convinced to do the calculations if someone provided me with the necessary data.) I've used it for my own personal stats for years, but that's not really the same.
I like it. I'll give it a whirl personally this season. I'm probably not up to tracking everyone on the Cutters though...
ReplyDeletePS "won" of three centers?
All right, Anonymous Cutter, I fixed the typo. I wonder why spell check didn't catch it? ;-)
ReplyDelete