# The Puck Stops Here

## Defensive Goals Versus Threshold

In today’s sabermetrics and hockey post, I am looking at the defensive portion of Tom Awad’s goals versus threshold system.  Unlike the offensive portion and the goalies portion, this part is much more likely to be inaccurate.  Defence is far harder to show statistically and as a result this part of the system is quite likely to miss significant defensive achievements and reward less impressive ones.

In the goals versus threshold system, Awad attempts to measure defensive skill by using shots allowed.  The big problem is figuring out which portion of shots allowed to credit (or charge against) each player on the ice.

In an effort to find the right percentage, Awad notes that forwards score about 85% of all goals and get about 75% of all assists in the NHL.  A typical goal scored was worth 0.47 goals in the offensive calculation and a typical assist is worth 0.31 goals.  With 1.7 assists per goal this is 0.53 goals credited to assists on a goal.  Forwards get 0.85*.47+0.75*.53=0.80 of the credit for scoring a goal.  This leaves 20% of the credit for defencemen.  A team typically plays 12 forwards per game, so if we assume all are equal (and they are not) that is 0.067 goals credited to each forward for a goal scored.  A team typically plays six defencemen, again assuming all are equal gives 0.033 goals credited to each defenceman for a goal scored.  Thus offensively, forwards are twice as important as defencemen.  It is then assumed that the reverse is true defensively (but is this actually a valid assumption?).  Thus a goal allowed is credited twice as strongly to a defenceman than a forward.  Thus we are assuming all defencemen share equally in preventing goals with one another and all forwards share equally in preventing goals.  This is clearly not accurate in reality.

We further adjust things by using the fact that typically a defenceman gets 33% more ice time than a forward.  So this makes an individual defenceman 50% more important in preventing goals than forwards (as 1.33 * 1.5 = 2).

We begin the calculation by finding the adjusted minutes played for a team using the formula:

ADMPt= sum (MPf) + sum (1.5*MPd)

Here ADMPt is the adjusted defensive minutes played for a team. MPf is the minutes played by a given forward and MPd is the minutes played by a defenceman.

The first factor of defensive goals versus threshold DGVTa is then calculated using:

SPMteam is the shots per minute of a given team and SPa is the average shots per minute allowed in the league and ADMPp is a given player’s adjusted minutes played.

This calculates the number of shots allowed by a team versus the league average.  A team allowing fewer shots will allow fewer goals and a team allowing more shots allows more goals.  These are distributed evenly among a team’s players, taking into account their ice time.  It is likely incorrect to distribute them equally, but there is no good statistical reason to split it up in any other way.

The second factor of DGVT differentiates players on a given team by their individual performance.  This is attempted using +/- ratings.  +/- is not the greatest measure of defensive ability.  It measures both offence and defence and is heavily influenced by the way a player is used and who he plays with.

+/- ratings are calculated using the formula:

+/- = EGF - EGA

where EGF is the even strength goals for and EGA is the even strength goals against.

For each team totals of EGF and EGA are found (and labelled TEGF and TEGA respectively) i.e.

TEGF = sum (EGF) and TEGA = sum (EGA)

A player’s adjusted +/- (APM) in this system is calculated:

APM = EGF - EGA - (EGF + EGA) * (TEGF - TEGA) / (TEGF + TEGA)

However, this number is the result of ten players on the ice at any given time and must be broken up into individual player’s portions.  We use the fact that we have decided defencemen are 1.5 more valuable than forwards in this system, so we calculate the player’s plus minus factor (PMFf for forwards and PMFd for defencemen) as follows:

PMFf = (1/10) * (5/6) = 0.083   and PMFd = (1/10) * (5/6) *1.5 = 0.125

The 1/10 factor comes from the fact ten players are on the ice and 5/6 because there are six weighted players on the ice on a given team (3 forwards and 1.5 * 2 defencemen).

Individual player’s +/- contribution to defensive goals versus threshold (DGVTbf and DGVTbd for forwards and defencemen respectively) using:

DGVTbf = APMf * PMFf   and   DGVTbd = APMd * PMFd

The third factor (DGVTc) is the attribute the arbitrary 0.25 of stopping shots that was not credited to goalies to the players on the ice.  It is given equally to all players on the ice weighted by ice time, although it is likely not equally earned by these players.

Here GGVTraw and GR are as calculated in the goalie’s portion of goals versus threshold.

Finally, the defensive portion of goals versus threshold is calculated using

DGVT = DVGTa + DGVTb + DGVTc

This is the most complex to calculate of the pieces of the goals versus threshold system and also the least accurate part.  I will look at the top players by this system in an effort to show how accurate it actually is.

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## About The Puck Stops Here

The Puck Stops Here was founded during the 2004/05 lockout as a place to rant about hockey. The original site contains over 1000 posts, some of which were also published on FoxSports.com.

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