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