# The Puck Stops Here

## Calculating Defensive Point Shares

Lately I have been looking at Justin Kubatko`s point shares system published at hockey-reference.com.  Thus far I have looked only at the offensive portions of the system.  It is an attempt to credit the points teams have in the standings to individual players.  The offensive part works reasonably well given the understanding that it can only be approximate at best and that it arbitrarily sets a value for assists that is not based upon observation.

Today I will turn my attention to the defensive part of the system.  This part is far more approximate than the offensive portion because it is harder to give individual credit to defensive play and because many of the numbers we ideally want to use have not been collected throughout the history of the NHL.  There are four different schemes depending upon the statistics gathered in the time frame.  The earlier the time frame the more approximate it becomes because of the rougher numbers used.

The first time frame is from 1998/99 to present.  This is the period that individual player`s ice time has been recorded and is readily available.  The first step is to calculate the proportion of a team`s total ice time goes to each individual player.  This is done by dividing a player`s ice time by the total ice time of all skaters on his team.  Since there are usually five skaters on the ice on a given team, the total ice time for a team is well above the total amount of time the team played in the season.  There is no differentiation based upon the type of ice time (power play, penalty kill, tough opposition, game tied etc.).  As a result, some important information about defensive ability is lost on this step.

Next we want to calculate the proportion of the team`s marginal goals against to assign to skaters.  This is equal to (7 - 2 * (team shots against per minute) / (league average shots against per minute)) / 7.  The seven is because each player is assumed to be contributing offensively and defensively in equal amounts (an assumption that is clearly false), except for the goaltender who contributes only to defence.  This makes seven portions of defensive play with two from the goalie.  Thus any reader who is mathematically inclined should be able to see the logic in the formula.

Next we want to find the position adjustment for each player.  Defencemen are assumed to be twice as involved in defensive play here giving them a 10/7 position adjustment and forwards a 5/7 position adjustment.  Adding up all five players on the ice (2 defencemen and 3 forwards) gives five parts, as would be expected.  This position adjustment is arbitrary and assumed all defencemen have the same contribution to defence and all forwards have the same contribution to defence and it assumes a value between forwards and defencemen arbitrarily.  It also begs the question of what do we do if a player splits his time between forward and defence in a season.

Now we calculate a team`s marginal goals against.  This is calculated using the formula MGA= (1+7/12)* (team games)  * (league goals per game)  - (team goals against).  It is an attempt to see how many goals were prevented by a team`s defensive play relative to that of replacement level players.

Next we want to calculate how far a player is from his team`s level using +/- ratings as a proxy for this.  There are many well-documented problems in using +/- ratings, but it is the best easily available number.  The +/- adjustment is 1/7 * (position adjustment) * (+/-) - (time on ice) * (team +/- for position) / (team time on ice for position).  This treats forwards and defencemen separately for each team.

Next we want to calculate the marginal goals against for each skater.  This is the team proportion added to the +/- adjustment The team proportion is (proportion of team ice time) * (proportion of team`s marginal goals assigned to skaters) * (position adjustment) * (team marginal goals against).  The main problem here is that it meters out the team`s defence more or less equally to all players and that is false.  Some players are more important to defence than others and though this is adjusted by +/- ratings, that is only a moderately good measure of a player`s individual defence.

Next we want to calculate the marginal goals per point.  This is (league goals) / (league points).

Finally, the defensive point shares are (marginal goals against) / (marginal goals per point).

Before 1998/99, we do not have reliable minutes played totals.  Here it is replaced by games played.  This assumes players each play the same amount of ice time in a game, which we know is false, and that makes the calculation more approximate.

Shots against are only available from 1983/84 onward.  Before this we have to get more approximate.  Here we cannot calculate the proportion of the team`s marginal goals against to assign to skaters.  We assume it is a constant 5/7 for each skater and do not have the ability to adjust for the amount of shots.

+/- ratings were only calculated from 1967/68 onward.  Thus before that we must get even more approximate.  Here we drop the +/- adjustment from the equation.  Now there is no adjustment at all for individual players.  Defence is merely the average of the marginal goals against for all skaters.

In the future, I will look at some specific results.  Since it is more approximate the further back in history, we expect to find it preferentially selects more modern players.  Since we give defencemen two times the position adjustment of forwards and defencemen usually have more ice time than forwards, we expect to see it preferentially select defencemen.  This formulation will tend to be biased toward modern defencemen.  We will get an idea how well it works and then add it to the offensive portion to find point shares for individual skaters.

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I’m looking forward to seeing these results.  I’m specifically curious to see how the modern breakdown of centers versus wingers comes out in general, as I feel giving all three forwards equal defensive credit/responsibility will skew this.

That’s not to say that all teams use the center as the more defensively responsible position among the forwards, but it does tend to be very common.

Posted by J.J. from Kansas on 09/03/11 at 04:18 PM ET

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Smileys

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