1. An Intelligence Approach to Evaluation of Sports Teamsby
Edward Kambour, Ph.D. 1

2. Agenda College Football Linear Model Generalized Linear Model
Intelligence (Bayesian) Approach
Results
Other Sports
Future Work

3. General Background Goals Forecast winners of future games
Beat the Bookie!
Estimate the outcome of unscheduled games
What’s the probability that Iowa would have beaten Ohio St?
Generate reasonable rankings

4. Major College Football
No playoff system
“Computer rankings” are an element of the BCS
114 teams
12 games for each in a season

5. Linear Model
Rothman (1970’s), Harville (1977), Stefani (1977), …, Kambour (1991), …, Sagarin???
Response, Y, is the net result (point-spread)
Parameter, , is the vector of ratings
For a game involving teams i and j,
E[Y] = i - j

6. Linear Model (cont.)
Let X be a row vector with
E[Y]=X

7. Regression Model Notes
Least Squares  Normality, Homogeneity
College Football
Estimate 100 parameters
Sample size for a full season is about 600
Design Matrix is sparse and not full rank

8. Home-field Advantage Generic Advantage (Stefani, 1980)
Force i to be home team and j the visiting team
Add an intercept term to X
Adds one more parameter to estimate
UAB = Alabama
Rice = Texas A&M
Team Specific Advantage
Doubles the number of parameters to estimate

9. Linear Model Issues Normality Homogeneity
Lots of parameters, with relatively small sample size
Overfitting
The bookie takes you to the cleaners!

10. Linear Model Issues (cont.)
Should we model point differential
A and B play twice
A by 34 in first, B by 14 in the second
A by 10 each time
Running up the score (or lack thereof)
BCS: Thou shalt not use margin of victory in thy ratings!

11. Logistic Regression Rothman (1970s) Linear Model Use binary variable
Winning is all that matters
Avoid margin of victory
Coin Flips

12. Logistic Regression Issues
Still have sample size issues
Throw away a lot of information
Undefeated teams

13. Transformations Transform the differentials to normality
Power transformations
Rothman logistic transform
Transforms points to probabilities for logistic regression
“Diminishing returns” transforms
Downweights runaway scores

14. Power Transforms Transform the point-spread Y = sign(Z)|Z|a
a = 1  straight margin of victory
a = 0  just win baby
a = 0  Poisson or Gamma “ish”

15. Maximum Likelihood Transform
seasons
MLE = 0.98
Power
-2ln(likelihood)
0.1
52487
0.3
41213
0.5
35128
0.67
32597
0.8
31418 1
31193

16. Predicting the Score Model point differential
Y1 = Si – Sj
Additionally model the sum of the points scored
Y2 = Si + Sj
Fit a similar linear model (different parameter estimates)
Forecast home and visitors score
H = (Y1 + Y2 )/2, V = (Y2 - Y1)/2

17. Another Transformation Idea
Scores (touchdowns or field goals) are arrivals, maybe Poisson
Final score = 7 times a Poisson + 3 times a Poisson + …
Transform the scores to homogeneity and normality first
The differences (and sums) should follow suit

18. Square Root Transform
Since the score is “similar” to a linear combination of Poissons, square root should work
Transformation
Why k?
For small Poisson arrival rates, get better performance (Anscombe, 1948)

19. Likelihood Test LRT: No transformation vs. square root with fitted k
Used College Football results from
k = 21
Transformation was significantly better
p-value = , chi-square = 9.26

20. Predicting the Score with Transform
Model point differential
Additionally model the sum of the points scored
Forecast home and visitors score
H = ((Y1 + Y2 )/2)2 , V = ((Y2 - Y1)/2)2
Note the point differential is the product

21. Unresolved Linear Model Issues
Overfitting
History
Going into the season, we have a good idea as to how teams will do
The best teams tend to stay the best
The worst teams tend to stay the worst
Changes happen
Kansas State

22. Intelligence Model Concept Model
The ratings and home-ads for year t are similar to those of year t-1. There is some drift from one year to the next.
Model

23. Intelligence Model (Details)
Notation
L teams
M seasons of data
Ni games in the ith season
Xi : the Ni by 2L “X” matrix for season i
Yi : the Ni vector of results for season i
i : the Ni vector of results for season I

24. Details (cont.)
Data Distribution:
For all i = 1, 2, …, M

25. Details (cont.)
Prior Distribution

26. Details (finally, the end)
The Posterior Distribution of M and -2 is closed form and can be calculated by an iterative method
The Predictive Distribution for future results (transformed sum or difference) is straight-forward correlated normal (given the variance)

27. Forecasts For Scores For the point-spread Simply untransform
E[Z2] = Var[Z] + E[Z]2
For the point-spread
Product of two normals
Simulate results

28. Enhanced Model Fit the prior parameters Hierarchical models
Drifts and initial variances
No closed form for posterior and predictive distributions (at least as far as I know)
The complete conditionals are straight-forward, so Gibbs sampling will work (eventually)

29. Results (www.geocities.com/kambour/football.html)
2002 Final Rankings
Team
Rating
Home
Miami
72.23 (1.03)
0.21 (0.04)
Kansas St
72.04 (1.04)
0.44 (0.03)
USC
71.95 (1.03)
0.04 (0.03)
Oklahoma
71.85 (1.02)
0.18 (0.03)
Texas
71.57 (1.03)
0.36 (0.03)
Georgia
71.49 (1.03)
0.02 (0.03)
Alabama
71.45 (1.03)
-0.09 (0.03)
Iowa
71.30 (1.03)
Florida St
71.29 (1.02)
0.43 (0.03)
Virginia Tech
71.25 (1.03)
0.12 (0.03)
Ohio St
71.18 (1.03)
0.27 (0.03)

30. Results 2002 Final Rankings Team Rating Home Miami 72.23 0.21
Kansas St
72.04
0.44
USC
71.95
0.04
Oklahoma
71.85
0.18
Texas
71.57
0.36
Georgia
71.49
0.02
Alabama
71.45
-0.09
Iowa
71.30
Florida St
71.29
0.43
Virginia Tech
71.25
0.12
Ohio St
71.18
0.27

31. Results 2002 Final Rankings Team Rating Home Miami 72.23 0.21
Kansas St
72.04
0.44
USC
71.95
0.04
Oklahoma
71.85
0.18
Texas
71.57
0.36
Georgia
71.49
0.02
Alabama
71.45
-0.09
Iowa
71.30
Florida St
71.29
0.43
Virginia Tech
71.25
0.12
Ohio St
71.18
0.27

32. Bowl Predictions Ohio St 17 Miami Fl (-13) 31 0.8255 0.5228
Washington St
Oklahoma (-6.5)
Iowa
USC (-6)
NC State (E)
Notre Dame
Florida St (+4)
Georgia

33. 2002 Final Record Picking Winners Against the Vegas lines Best Bets
522 –
Against the Vegas lines
367 – 307 –
Best Bets
9 –
In 2001,

34. ESPN College Pick’em (http://games.espn.go.com/cpickem/leader)
1. Barry Schultz
2. Jim Dobbs
3. Michael Reeves
4. Fup Biz
5. Joe *
6. Rising Cream
7. Intelligence Ratings

35. Ratings System Comparison (http://tbeck. freeshell. org/fb/awards2002
Todd Beck
Ph.D. Statistician
Rush Institute
Intelligence Ratings – Best Predictors

36. College Football Conclusions
Can forecast the outcome of games
Capture the random nature
High variability
Sparse design
Scientists should avoid BCS
Statistical significance is impossible
Problem Complexity
Other issues

37. NFL Similar to College Football Square root transform is applicable
Drift is a little higher than College Football
Better design matrix
Small sample size
Playoff

38. NFL Results (www.geocities.com/kambour/NFL.html)
2002 Final Rankings (after the Super Bowl)
Team
Rating
Home
Tampa Bay
70.72
0.29
Oakland
70.57
0.28
Philadelphia
70.55
0.10
New England
70.16
0.12
Atlanta
70.13
0.20
NY Jets
70.10
-0.01
Pittsburgh
69.95
Green Bay
69.92
Kansas City
69.90
0.51
Denver
69.89
0.50
Miami
0.49

39. 2002 Final NFL Record Picking Winners Against the Vegas lines
162 – 104 –
Against the Vegas lines
135 – 128 –
Best Bets
9 –

40. NFL Europe Similar to College and NFL Square root transform
Dramatic drift
Teams change dramatically in mid-season
Few teams
Better design matrix

41. College Basketball Transform? A lot more games
Much more normal (Central Limit Theorem)
A lot more games
Intersectional games
Less emphasis on programs than in College Football
More drift
NCAA tournament

42. NCAA Basketball Pre-tournament Ratings
Team
Rating
Home
Arizona
100.06
3.97
Kentucky
99.33
4.32
Kansas
95.89
3.85
Texas
93.42
4.44
Duke
92.90
4.66
Oklahoma
90.19
4.31
Florida
90.65
3.99
Wake Forest
88.70
3.65
Syracuse
88.50
3.49
Xavier
87.89
3.37
Louisville
87.88
4.16

43. NBA Similar to College Basketball A lot more games – fewer teams
Normal – No transformation
A lot more games – fewer teams
Playoffs are completely different from regular season
Regular season – very balanced, strong home court
Post season – less balanced, home court lessened

44. Hockey Transform A lot more games History matters
Rare events = “Poissonish”
Square root with k around 1
A lot more games
History matters
Playoffs seem similar to regular season
Balance

45. Soccer Similar to hockey Transform Not a lot of games
Square root with low k
Not a lot of games
Friendlys versus cup play
Home pitch is pronounced
Varies widely

46. Soccer Results Correctly forecasted 2002 World Cup final
Brazil over Germany
Correctly forecasted US run to quarter-finals
Won the PROS World Cup Soccer Pool

47. Future Enhancements Hierarchical Approaches
Conferences
More complicated drift models
Correlations
Individual drifts
Drift during the season
Mean correcting drift
More informative priors