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# How we model hockey

By Kevin Mongeon and J. Michael (Mike) Boyle - Jul. 2011

We model hockey using a series of models including:

• Stage 1: Predicted Goals For (PGS)
• Stage 2: Predicted Wins (PW)
• Stage 3: Contribution to Winning (CW)
• Stage 4: Player Lifetime Value (PLV)

Generally, PGS is the probability that a shot will result in a goal assuming an average quality shooter took the shot on an average quality goaltender.  There a number of factors that we use to calculate PGS including shot location, shot type, as well as events preceding the shot.

We use PGS in a number of different ways.  The most common uses of PGS are: PGS For (PGSF), PGS Against (PGSA), and Net PGS (NPGS).  PGSF is the aggregate probability that own-team shots result in goals when a player is on the ice.  PGSA is the aggregate probability that opponent-team shots result in goals when the player is on the ice.  Net PGS, PGSF minus PGSA, is the net probability that a goal was scored while the player was on ice. When aggregated to the game level, PGS for the team is the predicted number of goals that each team would score on average goaltenders.

Other common uses of PGS are: Goals For Per PGS, Goals Against Per PGS.  Goals For Per PGS is a measure of a player’s scoring ability, and Goals Against Per PGS is a measure of a goaltender’s saving ability.

PGS improves on traditional statistics in a number of ways.

First, there are very few traditional statistics that account for a player’s defensive contribution to winning.  The traditional statistics that are commonly used to measure a player's performance are goals, assists, and plus-minus.  Of these traditional statistics, only plus-minus contains a defensive component.  However, the plus-minus statistic contains systematic errors.  Said differently, the plus-minus statistic, on average, is an inaccurate measure of a player’s net contribution.   For example, consider a team with goaltenders that are below average quality.  A player on this team will have a negatively biased plus-minus statistic; or less than they contributed.  Similarly, if a player plays on a line with other players that are above average shooting ability, their plus-minus statistic will be positively biased; or greater than they contributed.  PGS is derived from a probability model; therefore, the systematic errors are removed, resulting in, on average, correct measures of a player’s for, against, and net contribution.

Second, because there are fewer of them, relative to other sports (e.g. basketball), the traditional statistics of goals and assists are heavily influenced by randomness; or luck.  Again, because PGS is derived from a probability model, all randomness (or luck) is removed and player receives the appropriate credit.  Further, because PGS is a derived statistic from shots, and there are many more shots taken (both for and against) in a game compared to goals scored, the difference between PGS and traditional statistics becomes evident over a relatively small sample size; or a small number of games.  Further, because PGS is, on average, accurate, it can be used to accurately measure other aspects of the game including a player’s ability to score and a goaltender’s ability to make saves.  Goals For Per PGS is a more accurate measure a player’s ability to score compared to shooting percentage because it accounts for shot quality.  Similarly, Goals Against Per PGS accounts for the quality of the shot that the goaltender faces compared to save percentage that does not account for shot quality.

Overall, we created PGS to be an, on average, accurate measure of a player’s for, against, and net performance.  However, to determine a player’s contribution, PGS contains a number of limitations.

First, PGS does not account for situational play.  For example, probabilistically, a shot taken in a 3-0 game during the first period contributes less to winning than a shot taken in a tie game with 5 minutes remaining in the game.

Second, while goals and assists are only allocated to, at most, two of the players that are on the ice at the time of the goal, PGS is allocated to all of the players that are on ice at the time of the shot.

Third, a player’s PGSF, PGSA, and Net PGS for a game do not account for decreasing marginal returns to time on ice.  For example, a common method used to compare traditional statistics across different players that play different amounts of time on ice is done by the calculation of rates (i.e. points per minute, or goals per minute).  Using rates assumes that players are not influenced by fatigue and perform at the same level regardless of the amount of time that they play.  Our research shows that this is not the case and players are affected by fatigue and demonstrate decreasing returns to time on ice.

Therefore, in the proceeding stages of our model we correct and account for the previously stated limitations.  In the Predicted Win stage we account for situational play and in the Contribution to Winning stage we properly distribute the PGS from all the players on the ice to the appropriate players, as well as account for decreasing returns to time on ice.