Understanding the NBA Study

1
Understanding the NBA Study


2
Roadmap

• Background What is a regression?
• Three key points
• Most of the analysis is not relevant
• NBA analysis Are blacks and whites treated
  differently?
• Our question is different Are blacks and whites
  treated differently by black and white referees?
• What does the NBA study show?
• The NBA analysis AGREES with our analysis
• Our analysis over the same period
• Also shows evidence of own-race bias over the
  past three years
• For another day Problems with the NBA study
• While we are analyzing the NBA statistical
  output, we do not endorse their methodology, and
  have not been allowed a chance to check their
  data.

3
Background What Is A Regression?

• Suppose we are interested in whether blacks or
  whites commit more fouls
• (Note This is not the Price-Wolfers question)
• Simplest approach
• Average fouls by white players 2.21
• Average fouls by black players 2.26
• Difference0.05
• Simple regression is equivalent
• Fouls 2.21 0.05Black player OR
• Fouls 2.26 0.05White player
• (Note that asking if blacks commit more fouls is
  equivalent to asking if whites commit fewer
  fouls)
• The advantage of more complex regressions is that
  we can take account of other factors. An example
• Fouls 0.05Minutes played 0.10 Black player

4
The NBA Study

• Analyze 2½ seasons of data Nov. 04 Jan. 07
• Price-Wolfers Initial study 1991/92-2003/04
• Price-Wolfers Update 2004/05-2006/07
• Analyze Fouls earned by a player in a game
• Price-Wolfers Analyze Fouls per 48 minutes
• Break data into four separate sub-samples
• Players who usually play 0-10, 10-20, 20-30 and
  30-40 minutes per game
• Price-Wolfers Analyze all the data jointly

5
Understanding the Question

• Price-Wolfers Do players earn fewer fouls under
  own-race referees?
• Not asking Do black players get different fouls
  than white players? (Or visa-versa)
• Not asking Do black referees blow more fouls
  than white referees? (Or visa-versa)
• Most of the NBA Analysis does not address our
  question
• Model 1 Do black players earn more fouls than
  whites?
• Model 2 Do black referees award more fouls than
  whites?
• Model 3 Do white players earn more fouls than
  blacks?
• Same as Model 1, but in reverse
• Model 4 Do white referees award more fouls than
  blacks?
• Same as Model 2, but in reverse
• Models 5-8 are all the same Asks two questions
• Do black players earn more fouls than whites?
• Do black referees give more fouls than whites?
• The models are all the same, simply rephrasing
  these questions instead in reverse (do white
  players in models 7, 8 do white referees in
  models 6,7)
• They are literally the same, and give literally
  the same answer (by construction)
• Model 10 asks two questions
• Do white players earn more fouls than blacks?
• Do white referees award more fouls than blacks?
• These regressions also take account of average
  differences in fouls by player, team, referee,
  and home-away status, by season

6
Models 11 and 13

• The key is the variable same race combo
• Price-Wolfers findSame racegtFewer fouls (i.e.
  negative effect)
• NBA analysis
• Group 1 Negative effect, but statistically
  insignificant
• Group 2 Statistically significant negative
  effects
• Group 3 Insignificant and small positive effect
• Group 4 No effect
• The only statistically significant impact agrees
  with our analysis!!!

7
Model 15 Taking Account of Position

• 0-10 minute players
• Own-race referee yield fewer fouls
• Statistically insignificant

• 0-10 minutes

• 10-20 minute players
• Own-race referee yield fewer fouls
• Statistically significant!!!

• 20-30 minute players
• Own-race referee yield slightly more fouls
• Statistically insignificant

• 30 minute players
• Own-race referee yield slightly fewer fouls
• Statistically insignificant

8
Model 9

• Simply asks
• What if you are of the same race of the referee?
• Finds mainly positive effects (contrary to
  Price-Wolfers)
• But
• White players are more likely to be of the same
  race as the referee
• White players earn more fouls
• This needs to be accounted for (which requires
  some work)

9
Inferences from the NBA Data

• Reconstructing the data
• Model 1 tells us the average fouls by white
  players (constant)
• Model 1 tells us the average difference between
  black and white players (coefficient on Player
  black)
• Model 2 tells us the average fouls call by white
  refs (constant)
• Model 2 tells us the average difference between
  black and white refs (coefficient on Official
  black)
• Model 9 tells us the average number of fouls in
  opposite-race interactions (constant)
• Model 9 tells us the average difference between
  own-race and opposite-race calls (coefficient on
  same race combo)
• Model 5 tells us the average level of fouls,
  taking account of both player and referee race.
• From these 7 facts, we can construct
• The proportion of the sample involving each type
  of interaction (bb, bw, ww, wb)
• The average number of fouls from each type of
  interaction.
• This involves no assumptions, simply mathematical
  inferences
• Simultaneous equations
• We have 7 facts, which give us 7 equations
• Plus an 8th equation the sum of the
  probabilities of each type of interaction must
  equal one.
• We have 8 unknowns
• We simply solve this system mathematically
  (computer code over the page)
• Because the NBA data are reported to three
  decimal places, this slightly limits our accuracy

10
Mathematica Code to Reconstruct the NBA Data

• (0-10 MINUTES)
• Solve
• (Get the data)
• M1_CONSTŠ2.728,
• M1_BLACKPLAYERŠ-0.167,
• M2_CONSTŠ2.593,
• M2_BLACKOFFICIAL0.051,
• M9_CONSTŠ2.619,
• M9_COMBOŠ-.006,
• M5_CONSTŠ2.706,
• M5_BLACKPLAYERŠ-.167,
• M5_BLACKOFFICIAL0.051,
• (Model 1)
• (p_bw mu_bwp_ww mu_ww)/(p_bwp_ww)ŠM1_CONST,
• (p_bb mu_bb p_wb mu_wb)/(p_bbp_wb)ŠM1_CONST
  M1_BLACKPLAYER,
• (Model 2)
• (p_ww mu_ww p_wb mu_wb)/(p_wwp_wb)ŠM2_CONST,
  
• (p_bw mu_bwp_bb mu_bb)/(p_bwp_bb)ŠM2_CONSTM
  2_BLACKOFFICIAL,

This is the mathematica code used to reconstruct
the NBA data for the 0-10 minute players. Similar
code was used for the other players
11
NBA Data Players who typically play 0-10
minutesFouls per player

• n6,235 observations
• Data calculated from NBA study (Note rounding
  may induce small errors.)

12
NBA Data Players who typically play 10-20
minutesFouls per player

• n35,266 observations
• Data calculated from NBA study (Note rounding
  may induce small errors.)

13
NBA Data Players who typically play 20-30
minutesFouls per player

• n51,440 observations
• Data calculated from NBA study (Note rounding
  may induce small errors.)

14
NBA Data Players who typically play 30
minutesFouls per player

• n55,264 observations
• Data calculated from NBA study (Note rounding
  may induce small errors.)

15
Understanding Model 9
An example using players who typically play 0-10
minutes

• The NBA analysis (model 9) compares
• The weighted average of the own-race cells
  (282.656392.582)/(2839) 2.613
• The weighted average of the opposite-race
  cells(292.53343.187)/(294) 2.612
• This is why the NBA find no effect
• But because white players are much more likely to
  face own-race referees (see above) , this
  confounds two facts
• White players earn fewer fouls under own-race
  referees
• Which should lead to a negative own-race effect
• White players earn more fouls than black players
• And because they are likely to have own-race
  referees, this leads to an offsetting positive
  bias to the own-race effect
• Our difference-in-difference analysis takes
  account of underlying differences between black
  and white players.

16
Summarizing the NBA Analysis

• Most of the models in the NBA analysis do not
  speak to our research question
• We ask about how players of different races have
  fouls called differentially by referees of
  different races
• Most of the NBA analysis (Models 1, 2, 3, 4, 5,
  6, 7, 8, 10, 12, 14) instead asks either about
  differences in foul calls by player race, or
  referee race
• Of those models which test own-race bias
• Model 11 Finds evidence of own-race bias
• Model 13 Finds evidence of own-race bias
• Not surprising it is the same as model 11
• Model 15 Finds evidence of own-race bias
• Model 9 As formulated, no evidence of own-race
  bias
• When corrected, evidence of own-race bias

17
Roadmap

• Background What is a regression?
• Three key points
• Most of the analysis is not relevant
• NBA analysis Are blacks and whites treated
  differently?
• Our question Are blacks and whites treated
  differently by black and white referees?
• What does the NBA study show?
• The NBA analysis AGREES with our analysis
• What is statistical proof?
• Our analysis over the same period
• For another day Problems with the NBA study
• While we are analyzing the NBA statistical
  output, we do not endorse their methodology, and
  have not been allowed a chance to check their
  data.

18
Price-Wolfers 1991/92 to 2003/04

• n266,984 player-game observations
• Player-game observations weighted by minutes
  played
• , , denote statistically significant at
  1, 5 and 10
• (Standard errors in parentheses)

19
Price-Wolfers 2004/05-2006/07 Update

• n71,759 player-game observations
• Player-game observations weighted by minutes
  played
• , , denote statistically significant at
  1, 5 and 10
• (Standard errors in parentheses)