Beyond +/-: A Rating System to Compare NHL Players

1. Beyond +/-: A Rating System to Compare NHL Players Dennis F. Lock Michael E. Schuckers St. Lawrence University

2. Understanding Plus/minus • Each time a goal or point is scored every player on the playing surface for the team scoring receives +1, and every player on the surface for the team being scored on receives -1. • We expand and utilize the notion of plus/minus to develop a few unique rating systems to compare NHL players (forwards and defensemen, goalies excluded).

3. Previous Work by Dr. Dan T. Rosenbaum(2004) • Dr. Rosenbaum designed a method of using least squares and basketball plus/minus to evaluate NBA player performance. • His model: where MARGIN = 100*(Points per possession for the home team – points per possession for the away team) 1 if player j is playing at home, Xjis -1 if player j is playing away, 0 if player j is not playing b0= Home court advantage bj= measures the difference between player j and the reference players, holding the other players constant. • Treating each unit of time in a game without a substitution as an observation, he collected more than 60,000 from the 2002/2003, and 2003/2004 seasons.

4. Problem Transferring to the NHL • Ratio of points/substitutions • Far fewer points scored • The NBA averages 197.48 points per game, while the NHL averaged 5.75 goals per game in 2006/2007. • Many more substitutions • Treating each unit of time in a game without a substitution as an observation does not make much sense.

5. Solving the Low Scoring Average Problem • Since the scoring rate is so low in the NHL we decide to include a method to determine a value for other plays which occur throughout each game. • Table 1 shows a list of the plays that the NHL records in each play-by-play file.

6. Determining Play Values • Goal : 1 • Stoppage and Goalie Pulled : 0 • Face-Off, Giveaway, Hit, Missed Shot, Penalty, Takeaway: where Yi = Value of each play, PGS(i,k) = Probability goal scored k seconds after play i, PG0(i,k) = Probability scored on k seconds after play i. • Shot, Blocked Shot:

7. Value for k • Value of k for plays other then goal, stoppage, and goalie pulled: • For penalties the value of k is determined by the length of a penalty, most commonly 120 seconds. • For all other plays we determined the best value for k to be 10 seconds. • We chose this value since after 10 seconds the change in PGS(i,k)-PGO(i,k) appears to fluctuate near zero.

8. Value for k Figure 1 Change in PGS(i,k) – PGO(i,k) kseconds following an event -Shot -Faceoff(Off) -Hit -Faceoff(Neu) -Takeaway -Faceoff(Def) -Giveaway -Missed Shot -Blocked Shot

9. Play Value Example • Ex./ takeaway: • Using our formula for the value of a takeaway, and the play-by-play provided by the NHL we examine all 17,634 takeaways from the 1,230 games of the 2006/2007 season. • From these takeaways we discovered that the team committing a takeaway scored within 10 seconds in 359 occasions, and got scored on in 51. • So Therefore the value for a takeaway is 0.0174.

10. Play Values • * Indicates adjustment for

11. Gathering Data • Using each play-by-play and matching time on ice data provided for the 1,230 games by the NHL we create an on ice matrix X (359,322 plays x 1,053 players). • For Each Play every player will receive a 1 if the player is on the ice for the home team, -1 if the player is on the ice for the away team, 0 if the player is not present on the ice • Also using the play-by-play and using the play values created by our model discussed previously we create a vector Y (359,322 plays) with the value for each play. • Where for the home team the value of the play is its value, but for the away team the value of the play is (-1) times its value. • Ex./ Missed shot home = 0.0094 Missed shot away = -0.0094

12. Expanded Plus/minus • Using our on ice matrix X and our play value vector Y we can create an expanded plus/minus rating vector R for all of the 1,053 players that played in the 2006/2007 season, accounting for all plays. • For traditional plus minus the only plays involved would be goals, and all elements of Y would be plus or minus 1.

13. Expanded Plus/minus Top Ten

14. Least Squares Model (adjusting for other players on ice) • With our play values, matrix X, and vector Y we create a model similar to Rosenbaum’s model for basketball, where: Yi is the value of play i. 1 if the player is on ice at home, Xi,jis -1 if the player is on ice away, 0 if the player is not on ice. bjis the rating for player j. • Treating each play as an observation we have over 359,000 observations from the 2006/2007 season.

15. Least Squares Model Top Ten • *Won the Hart Memorial Trophy for league MVP.

16. Observed - Expected Model • With our play values, matrix X, and vector Y we create a model similar to Rosenbaum’s model for basketball, where: 1 if home team scored within k seconds Ai is an indicator -1 if away team scored within k seconds. 0 if neither team scored within k seconds. Yiis the expected play value for play i. 1 if the player is on ice at home, Xi,jis -1 if the player is on ice away, 0 if the player is not on ice. βjis the rating for player j. • Treating each play as an observation we have 359,322 observations from the 2006/2007 season.

17. Observed - Expected Model Top Ten • *Won the Hart Memorial Trophy for league MVP.

18. Summary • Traditional plus/minus • Only accounts for goals • Expanded plus/minus • Accounts for most plays • Adjusted expanded plus/minus • Least squares model, accounts for other players on ice. • Observed - Expected plus/minus • Compares actual results to expected performance

19. Directions for Future Work • Recognizing special teams situations. • i.e power play, penalty kill, etc. • Zone information for each play. • Off, Neu, Def • Specific shot information for each shot • Type of shot (wrist, slap, etc.), distance of shot. • Include shootout information

20. Summary • Traditional plus/minus • Only accounts for goals • Expanded plus/minus • Accounts for most plays • Adjusted expanded plus/minus • Least squares model, accounts for other players on ice. • Observed - Expected plus/minus • Compares actual results to expected performance