Chevy versus Ford NASCAR Race
Effect Size – A Meta-Analysis
Data Description
• All 256 NASCAR Races for 1993-2000 Seasons
• Race Finishes Among all Ford and Chevy Drivers (Ranks)
– Ford: 5208 Drivers (20.3 per race)
– Chevrolet: 3642 Drivers (14.2 per race)
• For each race, Compute Wilcoxon Rank-Sum Statistic
(Large-sample Normal Approximation)
• Effect Size = Z/SQRT(NFord + NChevy)
Wilcoxon Rank-Sum Test (Large-Sample)
Number of Ford and Chevy Cars in race i : N i  N Ford,i  N Chevy ,i
N i ( N i  1) 

T

T

 Ford,i Chevy ,i

2


N Ford,i  N i  1
Expected Rank Sum for Ford Under No Brand effects :  Ford,i 
2
N Ford,i N Chevy ,i  N i  1
Standard Deviation :  Ford,i 
12
TFord,i   Ford,i
Z - Statistic for testin g Brand Effects : Z i 
Rank Sums for Ford/Chevy in race i : TFord,i
TChevy ,i
 Ford,i
Effect Size : d i 
Zi
Z
 i
N Ford,i  N Chevy ,i
Ni
V d i  
1
Ni
Note : Negative values mean Ford better, Positive means Chevy better
Chevy Effect by Race
0.8000
0.6000
0.4000
Chevy Effect
0.2000
0.0000
-0.2000
-0.4000
-0.6000
-0.8000
1
9
17
25
33
41
49
57
65
73
81
89
97 105 113 121 129 137 145 153 161 169 177 185 193 201 209 217 225 233 241 249
Race Number
Evidence that Chevrolet tends to do better than Ford
Histogram of Effect Sizes
40.0000
35.0000
30.0000
Frequency
25.0000
20.0000
15.0000
10.0000
5.0000
0.0000
-0.2000
-0.1500
-0.1000
-0.0500
0.0000
0.0500
0.1000
0.1500
0.2000
0.2500
0.3000
0.3500
Effect Size
Effect Sizes Appear to be approximately Normal
0.4000
0.4500
0.5000
Combining Effect Sizes Across Races
• Weighted Average of Race-Specific Effect Sizes
• Weight Factor  1/V(di) = Ni = (NFord,i+NChevy,i)
wi 
1 V d i 

Ni
 1 V d   N
256
j
j 1
256
j 1


d w   wi d i    N i 
i 1
 i 1 
256
 
256
 2 256
j
1 256
 di Ni 
i 1
701.49
 .0793
8850
 2 256
1




 256 
2
2 1 
V d w    N i   N i V d i     N i   N i      N i   1 / 8850
 i 1  i 1
 i 1  i 1
 N i   i 1 
1
SE d w 
 .0106
8850
95% CI for Population Mean : .0793  1.96(.0106)  .0793  .0208  (.0585,.1001)
 
256
256
Test for Homogeneity of Effect Sizes
 i  True Effect Size for Chevy for race i
H 0 : 1  ...   256 H A : Not all  i are equal
256
Test statistic : Q  
i 1

P - value : P 
2
255
d  d 
2
w
i
^ 2
 d i 
 242.5

 242.5  .7037
No evidence of Heterogene ity of Effect sizes
Average Effect by Year (Mean +/- 1SD)
0.4000
0.3000
Average Effect +/-1SD
0.2000
0.1000
0.0000
-0.1000
-0.2000
1992
1993
1994
1995
1996
Year
1997
1998
1999
2000
Testing for Year Effects
Notation : p  8 years (classes)
mi  Number of Races in year i
 ij  True effect for j th race in year i
 i   Weighted Average for year i
    Weighted Average for all races
d ij  Observed effect for j th race in year i
 mi

d i   Weighted Average for year i   N ij 
 j 1 
mi
N
j 1


 Weighted Average for all races   N ij 
 i 1 j 1 
p
d 
1
mi
ij
1
d ij
p
mi
 N
i 1 j 1
ij
d ij
Testing for Year Effects
Three Models regarding  ij :
H 0 :  ij   * i, j
p
mi
QT  
d
mi
QB  
2
d 

mi
  N ij  dij  d  
p
^ 2

j 1
dfT   mi  1
i 1
2
p
mi
p
  N ij  di   d     N i   d i   d  
2
i 1 j 1
i 1
2
df B  p  1
ij
2
ij
p
2
ij
d 
d d 


  N d
 d 
mi
H 2 :  ij unrestricted
i 1 j 1
 di   d  
i 1 j 1
QWi
 d  
^ 2
i 1 j 1
p
ij
H1 :  ij   i* i, j
i
^ 2
mi
j 1
ij
ij
 di  
2
p
QW   QWi
i 1
p
dfW    mi  1
i 1
ij
QT  QB  QW
Source
Between Years
Within Years
Total
dfT  df B  dfW
df
7
248
255
Q-Statistic
15.27
227.19
242.46
P-value
0.0327
0.8243
0.7037
Null
H0
H1
H0
Alternative
H1
H2
H2
Chevy Effect Size vs Track Length
0.8
0.6
Effect Size
0.4
0.2
0
-0.2
-0.4
-0.6
0
0.5
1
1.5
Track Length
2
2.5
3
Chevy Effect Size vs Laps
0.8
0.6
Effect Size
0.4
0.2
0
-0.2
-0.4
-0.6
0
100
200
300
Laps
400
500
600
Testing for Year and Race/Track Effects
• Regression Model Relating Effect Size to:
–
–
–
–
Season (8 Dummy Variables (No Intercept))
Track Length
Number of Laps
Race Length (Track Length x # of Laps)
• Weighted Least Squares with weighti = Ni
Regression Coefficients/t-tests
Variable
1993
1994
1995
1996
1997
1998
1999
2000
TrkLen
Laps
RaceLen
beta
0.0281
-0.0548
-0.0368
0.0090
-0.0916
-0.0992
-0.0389
-0.0250
0.0004
0.0397
-0.0001
se(beta)
0.1239
0.1235
0.1221
0.1207
0.1214
0.1202
0.1196
0.1159
0.0003
0.0520
0.0002
t-stat
0.2270
-0.4440
-0.3014
0.0745
-0.7542
-0.8256
-0.3254
-0.2157
1.3402
0.7644
-0.8560
P-val
0.8206
0.6574
0.7634
0.9407
0.4514
0.4098
0.7452
0.8294
0.1814
0.4454
0.3929
Controlling for all other predictors, none appear significant
C2 – Tests for Sub-Models and Overall
H 10 : All Factors have No Effects 1  ...  11  0
^
^ 1 ^
Test Statistic : Q1   '     75.5 df  11 P  .0000
H 02 : No Track Length/Lap s effects  9  10  11  0
^
^ 1 ^
Test Statistic : Q2   '     4.61 df  3 P  .2023
H 03 : No Year effects 1  ...   8  0
^
^ 1 ^
Test Statistic : Q3   '     14.77 df  8 P  .0637
Sources
• Hedges, L.V. and I. Olkin (1985). Statistical Methods for
Meta-Analysis, Academic Press, Orlando, FL.
• Winner, L. (2006). “NASCAR Winston Cup Race Results for
1975-2003,” Journal of Statistical Education, Volume 14, #3
www.amstat.org/publications/jse/v14n3/datasets.winner.html