Jake Thompson
October 09, 2015
180

# Statistical Analysis in Sports

Talk given to the Research, Evaluation, Measurement, and Statistics graduate seminar at the University of Kansas

October 09, 2015

## Transcript

1. Statistical
Analysis
in Sports
Jake Thompson

2. October 9, 2015 Statistical Analysis in Sports 2
Creating a Rating System
§ Possessions
§ Scoring efﬁciency
§ Turning statistics into a rating system
§ Expected winning percentages
§ Using the ratings
§ Predicting games
§ Alternative rating methodologies
§ Elo
§ Correlated-Gaussian

3. Measuring Success

4. Some background
§ Wins and losses!
§ Points scored and points allowed.
§ Seems trivial, but points and margin of victory are way more
informative than wins and losses.
§ Points in a game are inﬂuenced by the quality of the
two teams and how fast the game is played.
§ Points per possession, or 100 possession (Oliver,
2004).
§ Tempo-free statistics (Pomeroy, 2012).
October 9, 2015 Statistical Analysis in Sports 4

5. The Impact of Tempo-Free Statistics
§ Kansas and Duke both score and average of 1.10
points per possession.
§ Kansas plays slow, whereas Duke likes to play fast.
§ Both teams play Missouri, which averages 0.95 points
per possession.
October 9, 2015 Statistical Analysis in Sports 5
Team
Team
Efﬁciency
Missouri’s
Efﬁciency
Possessions
Expected
Score
Kansas 1.10 0.95 60 66-57
Duke 1.10 0.95 74 81-70

6. Estimating Possessions
§ From Oliver (2004):
§ Points per possession can then be calculated by:
§ Off. PPP = Points Scored / Total Possessions
§ Def. PPP = Points Allowed / Total Possessions
§ Commonly multiplied by 100 to give us points per 100
possessions.
October 9, 2015 Statistical Analysis in Sports 6

7. Example: Kansas vs. Iowa State
October 9, 2015 Statistical Analysis in Sports 7
Team Points FGM FGA OREB DREB TOV FTA
Kansas 89 32 63 10 25 15 23
Iowa State 76 30 72 15 22 14 14

8. Adjusting for Strength of Schedule
§ Generalized Least Squares (gls in R).
§ PPP = OffenseT
+ DefenseO
§ KU PPP = OffenseKU
+ DefenseISU
§ PPP = β
0
+ β
Off
+ β
Def
+ β
Off_HC
+ β
Def_HC
§ 117.43 = β
0
+ β
KU_Off
+ β
ISU_Def
+ β
Off_HC
§ 100.28 = β
0
+ β
ISU_Off
+ β
KU_Def
+ β
Def_HC
§ The gls() function in R allows us to correlate errors
within a grouping variable.
§ Scores are nested within games.
October 9, 2015 Statistical Analysis in Sports 8

9. Specifying the Model
§ The parameters:
§ One offensive parameter per team (350)
§ One defensive parameter per team (350)
§ One intercept
§ Two home court parameters
§ Selecting the reference teams
§ One reference on offense and defense
§ Selected iteratively
§ Originally the last team alphabetically
§ Model rerun with reference team set to the team with the
average offensive/defensive efﬁciency.
October 9, 2015 Statistical Analysis in Sports 9

10. Specifying the Model
§ After each iteration, calculate each team’s offensive
and defensive efﬁciency.
§ Offensive Efﬁciency = β
0
+ β
Team_Off
§ Calculate the mean offensive and defensive efﬁciency.
§ Determine which team is closest to the mean of each
efﬁciency.
§ These are the new reference teams.
§ Estimate the model again with the updated reference
teams.
§ Continue until the same teams are selected as the
reference teams in consecutive runs.
October 9, 2015 Statistical Analysis in Sports 10

11. Results
October 9, 2015 Statistical Analysis in Sports 11
School Conference Offense Defense Net
Kentucky SEC 120.21 78.51 41.70
Duke ACC 124.26 88.00 36.26
Wisconsin Big Ten 127.09 89.92 37.17
Arizona Pac-12 118.44 84.05 34.39
Villanova Big East 120.73 86.99 33.75
Virginia ACC 114.90 80.94 33.96
Gonzaga WCC 120.19 89.22 30.97
Utah Pac-12 116.14 85.55 30.59
North Carolina ACC 118.48 90.31 28.17
Ohio State Big Ten 115.50 89.39 26.11
Notre Dame ACC 123.85 96.89 26.96
Oklahoma Big 12 110.34 84.68 25.66
Kansas Big 12 114.19 88.57 25.62
Louisville ACC 108.44 83.98 24.47
Iowa State Big 12 117.07 92.66 24.41

12. Turning Statistics
Into a Rating
System

13. Expected Winning Percentage
§ Pythagorean Win Expectation (James, 1983)
§ If teams win in proportion to their “quality”, n = 2.
§ n varies by sport, and reﬂects the role that chance
plays in the outcome of games.
§ MLB: n = 1.83
§ NHL: n = 2.15
§ NFL: n = 2.37
§ NBA: n = 16.5
October 9, 2015 Statistical Analysis in Sports 13

14. Expected Winning Percentage
§ Pythagenpat Win Percentage (Smyth & Heipp, 2009)
§ An adaptation of the Pythagorean rating where each
team has their own exponent.
§ Based on the idea that points are more important in
low-scoring games.
§ The more points that are scored, the higher ni
will be
(less chance).
October 9, 2015 Statistical Analysis in Sports 14

15. Expected Winning Percentage
§ Linear/Logistic Combination Model (Kubatko, 2013)
§ Incorporates average margin of victory into the logit
model.
§ Provides better predictions for extreme seasons and
tends to be more stable over time.
October 9, 2015 Statistical Analysis in Sports 15

16. Choosing the Optimal Exponents
§ Maximum Likelihood Estimation using the optim()
function in R.
§ Calculate the pre-game adjusted efﬁciencies for each
game from the 2002-03 season to the 2014-15 season.
§ Use optim to ﬁnd the exponents that minimize the
binomial deviance, or log loss.
October 9, 2015 Statistical Analysis in Sports 16

17. Exponent Results
§ Use these exponents to calculate the three ratings for
each team.
§ To get a composite rating, I take the mean of the three
ratings, weighted by Log Loss.
October 9, 2015 Statistical Analysis in Sports 17
Method Exponent Log Loss
Pythagorean 9.972 0.532
Pythagenpat 1.208 0.531
Linear/Logistic Combo -0.010 0.531

18. Team Rating Results
October 9, 2015 Statistical Analysis in Sports 18
School Conf. Pythagorean Pythagenpat
Linear/
Logistic
Composite
Kentucky SEC 0.9859 0.9850 0.9846 0.9851
Wisconsin Big Ten 0.9692 0.9776 0.9760 0.9743
Duke ACC 0.9690 0.9751 0.9738 0.9726
Arizona Pac-12 0.9683 0.9691 0.9686 0.9687
Virginia ACC 0.9705 0.9670 0.9672 0.9682
Villanova Big East 0.9634 0.9676 0.9665 0.9658
Gonzaga WCC 0.9513 0.9576 0.9563 0.9551
Utah Pac-12 0.9547 0.9550 0.9547 0.9548
North Carolina ACC 0.9375 0.9442 0.9431 0.9416
Notre Dame ACC 0.9204 0.9391 0.9362 0.9319
Ohio State Big Ten 0.9279 0.9315 0.9310 0.9302
Oklahoma Big 12 0.9334 0.9269 0.9281 0.9294
Kansas Big 12 0.9265 0.9280 0.9278 0.9274
Baylor Big 12 0.9237 0.9253 0.9251 0.9247
Louisville ACC 0.9276 0.9179 0.9197 0.9217

19. Using the
Ratings

20. Predicting Game Winners
§ We can calculate a team’s probability of beating their
opponent by using the Log5 formula (James, 1981).
§ This model generalizes to include the Bradley-Terry-Luce
model commonly used in psychology, and the Rasch model in
psychometrics (Long, 2013).
§ Kansas (0.9274) vs. Ohio State (0.9302):
October 9, 2015 Statistical Analysis in Sports 20

§ From the GLS model:
§ PS = (β
0
+ β
T1_Off
+ β
T2_Def
) - (β
0
+ β
T2_Off
+ β
T1_Def
)
§ Directly from the adjusted efﬁciencies:
§ Team 1 Points = (T1Off
/ υ
Off
) × (T2Def
/ υ
Def
) × υ
All
§ Team 2 Points = (T2Off
/ υ
Off
) × (T1Def
/ υ
Def
) × υ
All
§ PS = Team 1 Points – Team 2 Points
§ From net ratings:
§ PS = (T1Off
– T1Def
) – (T2Off
– T2Def
)
October 9, 2015 Statistical Analysis in Sports 21

22. Calculating a Weighted Point Spread
§ For each point spread method, compare the projected
point spread to the actual margin of victory:
§ Calculate expected point spread for each game by
averaging the three methods, weighted by RMSE.
§ We can also calculate win probabilities from the
projected point spreads using logistic regression.
October 9, 2015 Statistical Analysis in Sports 22
Method RMSE Weight
GLS 10.774 0.331
Average Efﬁciencies 10.664 0.334
Net Efﬁciencies 10.651 0.335

23. October 9, 2015 Statistical Analysis in Sports 23

24. October 9, 2015 Statistical Analysis in Sports 24

25. In Game Win Probability
§ Adapted and expanded from Winston (2012) and Paine
(2012).
§ Model assumes the margin of victory of a given game
is ~N(PS, 10.612).
§ The mean and standard deviation of the distribution
over the course of a game are given by:
§ StDev = 10.612 / sqrt(40 / minutes remaining)
§ Mean = (PS * (minutes remaining / 40)) + (Margin * (40 /
minutes played))
§ The win probability is given by the proportion of the
distribution covering margins of victory that would
result in the team winning.
October 9, 2015 Statistical Analysis in Sports 25

26. In Game Win Probability
October 9, 2015 Statistical Analysis in Sports 26

27. In Game Win Probability
October 9, 2015 Statistical Analysis in Sports 27

28. Alternate Rating
Methods

29. Elo Ratings
§ Named after physics professor Arpad Elo.
§ Most widely used in chess and international soccer.
§ How it works:
§ Given a starting state of two teams, how is each team
expected to perform?
§ How did the teams actually perform?
§ Update the ratings with this new information.
October 9, 2015 Statistical Analysis in Sports 29

30. Calculating Elo
§ Long-run average rating of 1500.
§ All teams start out at a rating of 1300.
§ For a given game, Team A’s win probability is given by:
§ Team A’s rating is then updated using:
October 9, 2015 Statistical Analysis in Sports 30

31. Adding Margin of Victory to Elo
§ Big wins and losses are more impressive and usually
§ Where the MOV Factor is given by:
§ Complicated but corrects for autocorrelation problems
(favorites tend to win by more than they lose; Silver &
Fischer-Baum, 2015).
October 9, 2015 Statistical Analysis in Sports 31

32. Elo Ratings Pros/Cons
§ Pros:
§ Easy to calculate
§ Only need game scores (location can also be added in)
§ Track historical trends
§ Cons:
§ Ratings heavily dependent on performance in previous
seasons (can be less accurate early in season)
§ Ratings are not retroactively adjusted to account for team’s
being better/worst than expected.
October 9, 2015 Statistical Analysis in Sports 32

33. Correlated-Gaussian Ratings
§ Basically a standardized average margin of victory.
§ Developed by Oliver (2004) to estimate a team’s
expected winning percentage given their performance.
§ Not adjusted for strength of schedule, but can be.
§ The raw correlated-Gaussian rating can be used to
estimate a team’s “luck”:
§ Luck = Win% - CorGaus%
October 9, 2015 Statistical Analysis in Sports 33

34. References
James, B. (1981). Baseball Abstracts. Lawrence, KS: Privately Printed.
James, B. (1983). Baseball Abstracts. New York: Ballantine Books.
Kubatko, J. (2013). Pythagoras of the hardwood [Web log post]. Retrieved from http://statitudes.com/blog/
2013/09/09/pythagoras-of-the-hardwood/
Long, C. (2013). Baseball, chess, psychology, and psychometrics: Everyone uses the same damn rating system
psychology-and.html
Oliver, D. (2004). Basketball on paper: Rules and tools for performance analysis. Dulles, Virginia: Potomac Books,
Inc.
Paine, N. (2012). Are NFL playoff outcomes getting more random? [Web log post]. Retrieved from http://
www.footballperspective.com/are-nﬂ-playoff-outcomes-getting-more-random/
Pomeroy, K. (2012, June 8). Ratings glossary [Web log post]. Retrieved from http://kenpom.com/blog/index.php/
weblog/entry/ratings_glossary
Silver, N. & Fischer-Baum, R. (2015). How we calculate NBA Elo ratings [Web log post]. Retrieved from http://
ﬁvethirtyeight.com/features/how-we-calculate-nba-elo-ratings/
Smyth, D. & Heipp, B. (2009). Runs Per Win From Pythagenpat [Web log post]. Retrieved from http://
walksaber.blogspot.com/2009/01/runs-per-win-from-pythagenpat.html
Winston, W. (2012). Mathletics: How gamblers, managers, and sports enthusiasts use mathematics in baseball,
basketball, and football. Princeton, NJ: Princeton University Press.
October 9, 2015 Statistical Analysis in Sports 34