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## Wednesday 31 December 2014

### GDA-PK, a cleaner powerplay skill measure for NHL hockey

In the Vancouver Canucks' latest game, a 3-1 win against at the Anaheim Ducks on December 29, a paradox happened in the first period:

At the 13:31 mark, Vancouver got a 2-minute minor penalty
At the 14:26 mark, Anaheim got a 2-minute minor penalty
At the 15:18 mark, Vancouver got another 2-minute minor penalty.
All penalties ran their full course without being converted into goals.

So, for
13:31 - 14:25, Anaheim had a 5-4 powerplay (55 seconds)
15:18 - 15:30, Anaheim had a 4-3 powerplay (13 seconds)
16:26 - 17:17, Anaheim had a 5-4 powerplay (52 seconds).

Anaheim enjoyed a total of two minutes of powerplay advantage, which makes sense because Vancouver received one more penalty. However, for the sake of measuring Vancouver's ability to kill penalties or Anaheim's ability to use them, this counts as three distinct powerplays.

To me, this seems counter-intuitive for multiple reasons.
- Anaheim somehow managed to have more powerplays than penalties given in their favour.
- Most powerplays are of two minutes in length, but one of these was only 13 seconds long.
- Had it just been Vancouver's two penalties, this period would be recorded as one long powerplay of 13:31 - 17:17. So by getting a penalty, so it looks like Anaheim got two extra powerplays by getting a penalty.
- Had it just been Vancouver's two penalties and the first ended in a goal, it would be recorded as two powerplays, which would look like Anaheim getting an extra powerplay by scoring a goal.

My general objections to the counting of powerplays, and powerplay-based metrics, PK% and PP%*, include:

- Two-player advantages, although rare, are treated the same as one-player advantages.
- Powerplays from 5-minute major penalties are treated the same as 2-minute minors, even though minor (and double-minor) penalties end when the team with the player advantage scores.
- Shorthanded goals are ignored.
- The output of the statistics, like 82% PK and 24% PP, aren't intuitive to the viewer as to what to expect. We can compare the PK% and PP% teams to say that Anaheim is very good on the powerplay, and that Vancouver is very good shorthanded relative to other teams, but it doesn't give the viewer a good idea of the chances of a powerplay goal.

A better solution already exists; GAA*, as in Goals Against Average, which is normalized by time. I propose to adapt GAA into two alternate measures.

GDA-PK: Goal difference average - penalty killing.
GDA-PP: Goal difference average - power play.

where

(GDA-PK) = (Shorthanded goals scored - Shorthanded goals against) / (minutes shorthanded)

(GDA-PP) = (Powerplay goals scored - Powerplay goals against) / (minutes shorthanded)

- Going through the Vancouver-Anaheim anomaly again, we would record 0 goals against Vancouver in 2 total minutes. the fact that those 120 seconds are interrupted twice doesn't factor into the computation.

- If the powerplay had been cut short by a goal, less powerplay time and 1 goal would be recorded.

- If there had only been the two Vancouver penalties, more powerplay time would have been recorded and the two-player advantage time could be used for something else, such as GDA-2PP.

- The longer time of  five-minute major and four-minute double minor penalties are reflected fairly in GDA-PK and GDA-PP.

- Multiple goals during a major penalty do not result in an absurd measure such as a PP% of more than 100%.

- Shorthanded goals are included and simply treated as goals in the opposite direction.

The results that come out these metrics may look like this (THIS HAS NOT YET BEEN CALCULATED, PLEASE DO NOT QUOTE THESE AS ACTUAL FIGURES) :

----------------------
Vancouver Lumbermen**
- 0.103 Goals/min PK ( 2nd in NHL)

Anaheim Mighty Mallards**
+ 0.138 Goals/min PP ( 4th in NHL)
----------------------

Numbers-keen viewers can get a sense of the average goals any given penalty will cost, and it provides the same comparative sort of ranking as PP% and PK% but with less noise from special cases.  As an added bonus, a fair comparison to even strength play can be made as well by a similar calculation.

Finally, The fact that some Powerplay intervals are cut short by a goal being scored is a non-issue. It has already been addressed (Bartholomew 1957 for the math, Mullet 1977 for the application)***.

Your thoughts? Disagreements welcome in commentary.

========================

* Definitions:
PK% is short for Penalty-Kill Percent. It is a measure of defensive ability in shorthanded situations and calculated by (Goals against when shorthanded / Number of shorthanded situations) X 100%.

PP% is short for Power Play Percent, a measure of offensive ability in player-advantaged situations, is calculated by (Goals scored when player-advantaged / Number of player-advantaged situations) X 100%

GAA, short for Goals Against Average is a metric used to measure goalies and teams' defensive abilities. It is calculated by (Goals against / Hours of play).

** The names of the teams have been changed to emphasize that these numbers are for demonstration only. A lesson to novice statisticians, numbers you throw out, however casually or verbally, have a habit of being quoted as serious analyses. I'm trying my best to avoid that while still giving an example.

*** References:
Bartholomew, D. J. (1957), “A problem in Life Testing,” Journal of the American Statistical Association,
52, 350-355

Mullet, G. M. (1977), “Simeon Poisson and the National Hockey League,” The American Statistician,
31, 8-12