Chevy versus Ford NASCAR Race Effect Size – A Meta-Analysis.
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Transcript of Chevy versus Ford NASCAR Race Effect Size – A Meta-Analysis.
Chevy versus Ford NASCAR Race Effect Size – A Meta-Analysis
Data Description
• All 256 NASCAR Races for 1993-2000 Season• 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)
betterChevy means Positive better, Fordmean valuesNegative :Note
1 :SizeEffect
:Effects Brand gfor testin Statistic-Z
12
1 :Deviation Standard
2
1 :effects Brand No Under Fordfor SumRank Expected
2
)1( :i racein Ford/Chevyfor SumsRank
:i racein CarsChevy and Ford ofNumber
,,
,
,,
,
,,,,
,,
ii
i
i
iChevyiFord
ii
Ford,i
Ford,iiFordi
iiChevyiFordFord,i
iiFordFord,i
iiiChevyiFordiChevyiFord
iChevyiFordi
NdV
N
Z
NN
Zd
TZ
NNN
NN
NNTTTT
NNN
Chevy Effect by Race
-0.8000
-0.6000
-0.4000
-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
Ch
ev
y E
ffe
ct
Evidence that Chevrolet tends to do better than Ford
Histogram of Effect Sizes
0.0000
5.0000
10.0000
15.0000
20.0000
25.0000
30.0000
35.0000
40.0000
-0.2000 -0.1500 -0.1000 -0.0500 0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 0.4500 0.5000
Effect Size
Fre
qu
ency
Effect Sizes Appear to be approximately Normal
Combining Effect Sizes Across Races
• Weighted Average of Race-Specific Effect Sizes
• Weight Factor 1/V(di) = 1/Ni = 1/(NFord,i+NChevy,i)
)1001,.0585(.0208.0793.)0106(.96.10793. :Mean Populationfor CI %95
0106.8850
1
8850/11
0793.8850
49.701
1
1
1256
1
256
1
2
2256
1
256
1
2
2256
1
256
1
1256
1
256
1
256
1
256
1
w
ii
iii
iii
ii
iiw
iii
ii
iiiw
jj
i
jj
ii
dSE
NN
NNdVNNdV
NdNdwd
N
N
dV
dVw
Test for Homogeneity of Effect Sizes
sizesEffect ofity Heterogene of evidence No
7037.5.242 :value-P
5.242 :statisticTest
equal are allNot :...:
racefor Chevy for SizeEffect True
2255
256
12^
2
25610
P
d
ddQ
HH
i
ii
wi
iA
i
Average Effect by Year (Mean +/- 1SD)
-0.2000
-0.1000
0.0000
0.1000
0.2000
0.3000
0.4000
1992 1993 1994 1995 1996 1997 1998 1999 2000
Year
Av
era
ge
Eff
ec
t +
/-1
SD
Testing for Year Effects
races allfor Average Weighted
year for Average Weighted
year in race for effect Observed
races allfor Average Weighted
year for Average Weighted
year in race for effect True
year in Races ofNumber
(classes) years 8 :Notation
1 1
1
1 1
1
1
1
th
th
p
i
m
jijij
p
i
m
jij
m
jijij
m
jiji
ij
i
ij
i
ii
ii
dNNd
dNNid
ijd
i
ij
im
p
Testing for Year Effects
WBTWBT
p
iiW
p
iWW
m
jiijij
m
jij
iijW
B
p
iii
p
i
m
jiij
p
i
m
jij
iB
p
iiT
p
i
m
jijij
p
i
m
jij
ijT
ijiijij
ij
dfdfdfQQQ
mdfQQddN
d
ddQ
pdfddNddN
d
ddQ
mdfddN
d
ddQ
HjiHjiH
i
ii
i
ii
ii
111
2
12^
2
1
2
1 1
2
1 12^
2
11 1
2
1 12^
2
2*
1*
0
1
1
1
edunrestrict :,:,:
: regarding Models Three
Source df Q-Statistic P-value Null AlternativeBetween Years 7 15.27 0.0327 H0 H1Within Years 248 227.19 0.8243 H1 H2Total 255 242.46 0.7037 H0 H2
Chevy Effect Size vs Track Length
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 0.5 1 1.5 2 2.5 3
Track Length
Eff
ec
t S
ize
Chevy Effect Size vs Laps
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 100 200 300 400 500 600
Laps
Eff
ec
t S
ize
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 beta se(beta) t-stat P-val1993 0.0281 0.1239 0.2270 0.82061994 -0.0548 0.1235 -0.4440 0.65741995 -0.0368 0.1221 -0.3014 0.76341996 0.0090 0.1207 0.0745 0.94071997 -0.0916 0.1214 -0.7542 0.45141998 -0.0992 0.1202 -0.8256 0.40981999 -0.0389 0.1196 -0.3254 0.74522000 -0.0250 0.1159 -0.2157 0.8294
TrkLen 0.0004 0.0003 1.3402 0.1814Laps 0.0397 0.0520 0.7644 0.4454
RaceLen -0.0001 0.0002 -0.8560 0.3929
Controlling for all other predictors, none appear significant
2 – Tests for Sub-Models and Overall
0637.877.14' :StatisticTest
0... effectsYear No :
2023.361.4' :StatisticTest
0 effects sLength/LapTrack No :
0000.115.75' :StatisticTest
0... Effects No have Factors All :
^1^^
3
8130
^1^^
2
1110920
^1^^
1
1111
0
PdfQ
H
PdfQ
H
PdfQ
H
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