1 understanding the nba analysis understanding the nba study

1Understanding the NBA Analysis

Understanding the NBA Study


Roadmap

Background: What is a regression? Three key points:

1. 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?

2. What does the NBA study show?– The NBA analysis AGREES with our analysis

3. 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.


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– Difference=0.05

Simple regression is equivalent:– #Fouls = 2.21 + 0.05*Black player; OR– #Fouls = 2.26 – 0.05*White 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 = 0.05*Minutes played – 0.10 * Black player


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


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– Model 12 is literally the same as model 10 – it is just reprinted and relabeled model 12.– Model 14 asks:

» Do white players earn more fouls than black players?» Do white referees award more fouls than black refs?» Taking account of average differences in fouls by referee, team, home-away, and position (but not player)

Thus, the only potentially relevant models are: 9, 11, 13 and 15– Models 11 and 13 are – literally – the same – just reprinted and relabeled


Models 11 and 13

The key is the variable “same race combo”

Price-Wolfers find:Same race=>Fewer 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!!!


Model 15: Taking Account of Position

0-10 minutes0-10 minute players•Own-race referee yield fewer fouls

—Statistically insignificant

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

—Statistically significant!!!

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


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



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…)


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


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_BLACKOFFICIAL==0.051, M9_CONSTŠ2.619, M9_COMBOŠ-.006, M5_CONSTŠ2.706, M5_BLACKPLAYERŠ-.167, M5_BLACKOFFICIAL==0.051, (*Model 1*) (p_bw mu_bw+p_ww mu_ww)/(p_bw+p_ww)ŠM1_CONST, (p_bb mu_bb + p_wb mu_wb)/(p_bb+p_wb)ŠM1_CONST+M1_BLACKPLAYER, (*Model 2*) (p_ww mu_ww+ p_wb mu_wb)/(p_ww+p_wb)ŠM2_CONST, (p_bw mu_bw+p_bb mu_bb)/(p_bw+p_bb)ŠM2_CONST+M2_BLACKOFFICIAL, (*Model 9*) (p_wb mu_wb + p_bw mu_bw)/(p_wb+p_bw)ŠM9_CONST,(p_ww mu_ww + p_bb mu_bb)/(p_ww+p_bb)ŠM9_CONST+M9_COMBO, (*Model 5*) p_ww mu_ww + p_wb mu_wb + p_bw mu_bw +p_bb mu_bb Š M5_CONST+M5_BLACKPLAYER*(p_wb+p_bb)

+M5_BLACKOFFICIAL*(p_bw+p_bb), (*Identity*) p_ww+p_wb+p_bw+p_bbŠ1, (*Calculations*) diff_whiterefŠmu_wb-mu_ww, diff_blackref==mu_bb-mu_bw, diff_blackplayer==mu_wb-mu_bb, diff_whiteplayer==mu_ww-mu_bw, ddŠdiff_whiteref-diff_blackref } ,{p_ww, p_wb, p_bb, p_bw, mu_ww, mu_wb, mu_bw, mu_bb, diff_whiteref, diff_blackref, diff_blackplayer, diff_whiteplayer, dd}]

{{diff_blackplayer®-0.0497269,diff_whiteplayer®-0.531512,dd®0.481785,diff_whiteref®-0.123092,diff_blackref®-0.604878,mu_ww®2.65563,mu_wb®2.53253,mu_bw®3.18714,mu_bb®2.58226,p_ww®0.279324,p_bw®0.0440296,p_wb®0.289304,p_bb®0.387343}}

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


NBA Data: Players who typically play 0-10 minutesFouls per player

n=6,235 observations Data calculated from NBA study (Note: rounding may induce small errors.)

Black Players

White Players

Difference: Black – White

Foul Rate White referee

2.533 2.656 -0.123

Black referee

2.582 3.187 -0.605

Difference: White referee - Black referee

-0.050

-0.532

Diff-in-diff +0.482



Black Players

White Players



3.089 2.994 0.095

Black referee

3.094 3.081 0.013


-0.005

-0.087

Diff-in-diff +0.083





Black Players

White Players



3.353 3.479 -0.126

Black referee

3.372 3.488 -0.116


-0.019

-0.009

Diff-in-diff -0.010


NBA Data: Players who typically play 30+ minutesFouls per player


Black Players

White Players



3.433 3.479 -0.046

Black referee

3.466 3.499 -0.033


-0.033

-0.021

Diff-in-diff -0.012


Understanding Model 9

Black Players

White Players



2.533 (29%)

2.656 (28%)

-0.123

Black referee

2.582 (39%)

3.187 (4%)

-0.605


-0.050

-0.532

Diff-in-diff +0.482

(Numbers in parentheses are sample proportions)

The NBA analysis (model 9) compares:– The weighted average of the own-race cells

=(28%*2.656+39%*2.582)/(28%+39%) = 2.613– The weighted average of the opposite-race cells

=(29%*2.533+4%*3.187)/(29%+4%) = 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.

An example using players who typically play 0-10 minutes


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


Roadmap

Background: What is a regression? Three key points:

1. 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?

2. 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.


Price-Wolfers: 1991/92 to 2003/04 Black

Players White

Players Difference:

Black – White Foul Rate

Slope: Δ(Black-White) /

Δ%White Refs

0% White Refs 4.418 (.043)

5.245 (.094)

-0.827 (.106)

33% White Ref 4.317 (.016)

4.992 (.035)

-0.675 (.038)

0.455 (.331)

67% White Refs 4.335 (.010)

4.989 (.023)

-0.654 (.025)

0.064 (.137)

100% White Refs 4.322 (.013)

4.897 (.029)

-0.574 (.032)

0.240** (.121)

Average slope: ΔFouls / Δ%White Refs

-0.022 (.027)

-0.204***

(.066)

Diff-in-diff 0.182*** (.066)

[p=.006] n=266,984 player-game observations

Player-game observations weighted by minutes played ***, **, * denote statistically significant at 1%, 5% and 10% (Standard errors in parentheses)


Price-Wolfers: 2004/05-2006/07 Update Black

Players White

Players Difference:

Black – White Foul Rate

Slope: Δ(Black-White) /

Δ%White Refs

0% White Refs 4.437 (0.063)

5.126 (0.120)

-0.690 (0.139)

33% White Ref 4.401 (0.026)

4.938 (0.049)

-0.537 (0.056)

0.458 (0.443)

67% White Refs 4.366 (0.021)

4.797 (0.040)

-0.430 (0.045)

0.321 (0.215)

100% White Refs 4.276 (0.034)

4.693 (0.065)

-0.417 (0.071)

0.038 (0.256)

Average slope: ΔFouls / Δ%White Refs

-0.162*** (0.053)

-0.405*** (0.107)

Diff-in-diff 0.243** (0.116) [p=.035]

n=71,759 player-game observations Player-game observations weighted by minutes played ***, **, * denote statistically significant at 1%, 5% and 10% (Standard errors in parentheses)

1 understanding the nba analysis understanding the nba study

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