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Improving Plus-Minus

Copyright Iain Fyffe and Rob Vollman, 2002

Note: This article originally appeared in its current form in The Hockey Research Journal, Vol. V, No. I, Fall 2001. I have not reproduced the technical notes from that article, as that information appears elsewhere on this website. It is essentially an amalgamation of two research articles previously published on this site.

Plus-minus has been a much-maligned statistic since the NHL began to record it officially in 1967. Such status is not deserved. There are certainly problems with the statistic as it is currently formulated, such as the effects of teammates and the short-handed bonus and power-play penalty (which we'll come to in a moment). But plus-minus cannot be taken literally; it must be interpreted, more so than other common statistics. It can be a very informative number when considered in an appropriate form, in the appropriate context and over an appropriate time frame.

In this paper, we will do something that has not often been done in the world of hockey statistics, but should be: we will work within a consistent, defined framework. This framework is established by evalutating player statistics based on the player's opportunity to accumulate said statistics. In this case, we wish to evaluate players' plus-minus stats based on their opportunity to accumulate plus-minus. This is the only fair way to evaluate a player's statistics. Totals are nearly meaningless unless they are considered alongside the opportunities to compile said totals. As you will see, the opportunity to accumulate plus-minus arises only at even strength. Therefore players should be evaluated based on even-strength playing time. We will also be making adjustments that compensate for some of the weaknesses of the plus-minus statistic.

The Nature of Plus-Minus

Before we continue, there is one point that needs to be clarified. Plus-minus is often referred to as being a "defensive" statistic, reflecting the defensive abilities of a player. This is not accurate. Plus-minus encompasses both the offensive and defensive aspects of the game. If a player scores a goal at even strength, he gets a "plus". How does this reflect his defence? A player who is poor defensively can still have a good plus-minus by being very good offensively. Conversely, a poor offensive player can have a good plus-minus by being excellent defensively. Plus-minus is not a one-dimensional statistic. It reflects the overall skill of the player. So if we can transform the raw form of plus-minus into a more informative number, we will have a very useful statistic indeed.

There are several problems with plus-minus in its raw form. For example, it is affected by a variety of factors that the individual player has no control over, such as the skill level of his teammates. In addition, there is a problem with the formulation of the statistic itself. If we can compensate for some of these factors, the imformativeness of the statistic will improve greatly.

Reformulating Plus-Minus

The first problem with plus-minus lies in its formulation. As it stands now, plus-minus excludes power-play goals, which is consistent with our framework of opportunity. In odd-man situations, opportunities to score are not equal. Short-handed goals, however, are not excluded from the calculation. This is inconsistent with our framework. It is our belief that the fairest way to evaluate players is baed on his even-strength play. By including short-handed goals in the calculation, bias is created. It has the effect of penalizing players who play on the power-play (who can receive a minus on the power-play, but never a plus; the "power-play penalty"), and rewarding those who kill penalties (who can get a plus but not a minus; the "short-handed bonus"). Shorthanded goals should be excluded, because otherwise the very formulation of plus-minus is contrary to our framework of evaluation based on opportunity.

Adjusting Plus-Minus

In order to properly examine plus-minus, we must transform it into Adjusted Plus-Minus (APM). APM is a necessary tool when examining plus-minus statistics.

Basically, the idea is to compare an individual player's plus-minus to his expected plus-minus (EPM), which is the value we would expect an average player on that team to have. Without this adjustment, plus-minus stats are largely a simple reflection of the team a player plays for. It's no coincidence that plus-minus leaderboards are usually dominated by players from only a few good teams. In order to fairly evaluate a player, we must eliminate the effect of external factors as much as possible. This is due to the fact that if you play with better players, you have more ooprtunity to accumulate a good plus-minus rating. Conversely, if you play with bad players, you have a greater opportunity to accumulate a poor plus-minus rating. To compensate for the effect of the team, we use APM.

A player's EPM is based on his team's plus-minus. Remember that we are now excluding shorthanded goals from the equation, so this is done on the team level as well. All players on the ice receive a plus or minus when a goal is scored in even-strength situations; a player's EPM is therefore simply the team's PM per even-strength minute times the player's even strength minutes, as follows:

EPM = TPM / TMIN x PMIN

Where TPM is the team's plus-minus, excluding shorthanded goals; TMIN if the team's even-strength minutes; and PMIN is the player's even-strength minutes.

This is the plus-minus we wouls expect an average player on that team to have, based upon the player's playing time. Basing EPM on playing time must be done to reflect the fact that more ice time gives more opportunity to collect pluses and minuses. Once again, only even-strength situations are used, so power-play and shorthanded time are not included. By considering all playing time, players who play on special teams will have their APM stats gravitate toward zero. To compensate for this we consider only even-strength time, since this is the only time a player can now receive a plus or a minus.

Adjusted plus-minus is calculated as follows:

APM = PM - EPM

Where PM is the player's plus-minus, excluding shorthanded goals, and EPM is the player's expected plus-minus, calculated as above.

For example, a player may have an EPM of +5 based on his team and even-strength playing time. However, the player may have an actual plus-minus of only +1. This player would have an APM of -4.

APM should be examined on a per-minute basis, not only as a total. Again, this reflects the idea that players should be evaluated based on opportunity. Also, single-season results can be deceptive. Therefore, it is best to examine a player's APM over a period of several years, or even a career. Over a significant period of time, any flukes that occur should even out, leaving only an indication of the player's real value.

Players who play on a team with many good all-around players will tend to have lower APM figures that they would playing for a weaker team. Conversely, players on a team with many bad all-around players will be overrated using APM. Even APM cannot completely eliminate the effects of teammates on one's rating.

Another important point to consider is who a player plays against. Bob Gainey is often maligned for his low plus-minus figures (which remain low even after adjustment). What is not considered here is that players like Gainey spend much of their time checking their opponent's best scoring lines. That is their job. Obviously, playing regularly against superior offensive players will hurt one's plus-minus rating, and APM cannot compensate for this fact. Similarly, poor defensive players like Yvan Cournoyer will have a higher APM because they rarely play against an opponent's top line. Thus, we must bear in mind a player's role when evaluating him using APM.

Regardless of complicating factors, you should get a very good idea about a player's ability using APM. While APM compensates for many of the weaknesses of traditional plus-minus stats, there are still qualitative and other factors to consider, such as those indicated above.

Sample Results

Here are the leaders in APM per minute from the 2000-01 NHL season. For clarity, APM per 1,000 minutes of even-strength time is shown. Since this is a rate stat, we should have some minimum value of even-strength minutes to reduce the chance of flukes making the list. I have arbitrarily set this minimum at 10 minutes per game, or 820 minutes.