Progetto Statistics ENG

8/12/2019 Progetto Statistics ENG

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Statistical analysis of performances of LeBron James, Dwayne Wade andChris Bosh, the Miami Heat s Big Three , during NBA season 2012-13

Summary

Introduction to the statistical analysis done

Dataset presentation

Descriptive analysis of data

Verification of normality of variables

Computation of confidence intervals for the scored points average of the BigThree

Hypothesis testing on the difference between points scored by the Big Three inwins and losses

Linear regression model:

o Linear relation between LeBron James performance and the scored pointsby Dwayne Wade e Chris Bosh

o Linear relation between the sum of the points by the Big Three and Miamiwinning margin

o Linear relation between points scored individually by LeBron James,Dwayne Wade and Chris Bosh and Miami winning margin

o Linear relation between points scored by overall Miami team and the sumof the Big Three


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Introduction to the statistical analysis : how does the Big Three influenceMiamis wins?

In the 2010 Summer, LeBron James joined Chris Bosh e Dwayne Wade in Miami Heatteam, arranging a players core with athletic means and huge basketball capabilities.These three players are members of USA national basketball team and a lot of criticismarose when they decided to stay together in one team. Put three players of this leveltogether seemed to ruin the basketball game. Miami Heat goal was clear : win thechampionship and then the trophy. The first year they failed. Then they won in 2012and 2013 and while we are writing (mid May) they are in the playoffs. In these twoyears LeBron was honored with two MVP (most valuable player) awards. He was seen

as Miami leader and the main reason for Miami wins thanks to his exceptionalperformances whereas Wade and Bosh were put on a secondary level.

Our statistical analysis main aim is to determine if there is statistical evidence to statethat the King is essential for Miamis wins o if anyway a linear correlation existsbetween his performances and the final result of a game. We have also taken intoaccount Wade and Bosh and we have wanted to study if there are evidences to statethat LeBron performances influence their ones. In the end we have tried to understandif the Big Three is necessary for Miamis wins or if there are other factors that l eads theteam to win.


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DATASET

To perform proper statistical tests, we considered all the games played by all themembers of the Big Three during the regular season and playoffs of 2012-13, since atthis time we are unable to get full data about 2013-14 season still in progress.

ame Date Result Margin AssistsLJ TurnoverLJ ReboundsLJ PointsLJ PointsDW PointsCB PointsMIA1 30/10/2012 W 13 3 0 10 26 29 19 120

2 02/11/2012 L -20 5 5 7 23 15 12 104

3 03/11/2012 W 3 11 0 9 20 14 40 119

4 05/11/2012 W 25 1 3 11 23 22 18 124

5 07/11/2012 W 30 8 2 12 20 22 8 103

6 09/11/2012 W 6 9 2 11 21 9 24 95

7 11/11/2012 L -18 6 2 10 20 8 22 104

8 12/11/2012 W 3 6 0 10 38 19 24 113

9 14/11/2012 L -7 7 4 5 30 6 11 107

10 15/11/2012 L -5 12 4 7 27 16 14 98

11 17/11/2012 W 9 3 1 7 21 13 24 97

12 21/11/2012 W 7 8 1 10 28 28 24 113

13 24/11/2012 W 2 5 7 6 30 18 23 110

14 29/11/2012 W 5 7 4 9 23 19 18 105

15 01/12/2012 W 13 6 2 9 21 34 8 102

16 04/12/2012 L -4 11 3 13 26 24 20 105

17 06/12/2012 L -20 9 2 10 31 13 12 112

18 08/12/2012 W 16 7 4 5 24 26 13 106

19 10/12/2012 W 9 6 3 7 27 26 14 101

20 12/12/2012 L -2 5 4 3 31 14 21 97

21 15/12/2012 W 30 5 2 10 23 13 12 102

22 18/12/2012 W 11 11 0 6 22 24 15 103

23 20/12/2012 W 15 5 5 9 24 19 17 110

24 22/12/2012 W 16 7 4 9 30 21 9 105

25 25/12/2012 W 6 9 3 8 29 21 16 103

26 26/12/2012 W 13 8 3 12 27 29 14 105

27 28/12/2012 L -10 5 3 6 35 3 28 109

28 29/12/2012 L -19 7 6 6 26 24 12 104

29 31/12/2012 W 2 11 3 8 36 21 22 112

30 02/01/2013 W 10 9 4 12 32 27 17 119

31 04/01/2013 L -7 2 2 6 30 18 14 96

32 06/01/2013 W 28 7 1 2 24 14 17 99

33 08/01/2013 L -10 4 7 10 22 30 14 87

34 10/01/2013 L -2 9 4 10 15 18 29 92

35 12/01/2013 W 29 7 1 5 20 11 16 128

36 14/01/2013 L -7 6 3 4 32 11 16 104

37 16/01/2013 W 17 10 1 7 25 15 11 92

38 17/01/2013 W 9 8 2 7 39 27 7 99

39 23/01/2013 W 7 11 3 10 31 35 12 123

40 25/01/2013 W 22 7 2 7 23 29 14 110

41 27/01/2013 L -2 7 3 16 34 20 16 100

42 30/01/2013 W 20 7 4 9 24 21 16 105

43 01/02/2013 L -13 3 3 6 28 17 13 102

44 03/02/2013 W 15 7 3 8 30 23 28 100

45 04/02/2013 W 5 8 5 8 31 20 23 99

46 06/02/2013 W 6 5 4 6 32 31 12 114


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47 08/02/2013 W 22 6 5 5 30 20 9 111

48 10/02/2013 W 10 4 4 7 32 30 12 107

49 12/02/2013 W 13 9 1 6 30 24 32 117

50 14/02/2013 W 10 7 4 12 39 13 20 110

51 20/02/2013 W 13 11 2 6 24 20 6 103

52 21/02/2013 W 19 7 4 12 26 17 12 86

53 23/02/2013 W 24 11 2 10 16 33 13 114

54 24/02/2013 W 4 8 2 3 28 24 7 109

55 26/02/2013 W 12 16 2 8 40 39 15 141

56 01/03/2013 W 7 10 1 8 18 22 13 98

57 03/03/2013 W 6 7 2 11 29 20 16 99

58 04/03/2013 W 16 4 7 10 20 32 11 97

59 06/03/2013 L 1 2 2 3 26 24 17 97

60 08/03/2013 W 9 5 3 10 25 22 16 102

61 10/03/2013 W 14 7 4 6 13 23 24 105

62 12/03/2013 W 17 7 4 7 15 23 14 98

63 13/03/2013 W 4 8 4 7 27 21 10 98

64 15/03/2013 W 13 7 2 10 28 20 28 107

65 17/03/2013 W 17 8 3 12 22 24 18 108

66 18/03/2013 W 2 12 5 7 37 16 13 105

67 20/03/2013 W 3 10 4 12 25 11 11 98

68 22/03/2013 W 14 8 3 8 29 19 5 103

69 24/03/2013 W 32 10 4 8 32 22 15 109

70 25/03/2013 W 14 11 3 9 24 13 12 108

71 27/03/2013 L -4 3 4 7 32 18 21 101

72 29/03/2013 W 19 6 3 4 36 17 10 108

73 02/04/2013 L 19 5 1 6 23 20 23 102

74 05/04/2013 W 10 3 4 8 18 20 18 89

75 06/04/2013 W 19 7 2 4 27 11 16 106

76 09/04/2013 W 11 7 3 7 28 22 12 94

77 12/04/2013 W 8 9 1 6 20 21 17 109

78 14/04/2013 W 12 6 3 7 24 15 12 105

79 21/04/2013 W 23 8 5 10 27 16 15 110

80 23/04/2013 L 12 6 4 8 19 21 10 98

81 25/04/2013 W 13 6 5 5 22 4 16 104

82 28/04/2013 W 11 7 5 8 30 16 10 88

83 06/05/2013 L -7 7 2 8 24 14 9 93

84 08/05/2013 W 37 9 3 5 19 15 13 115

85 10/05/2013 W 10 7 1 8 25 10 20 104

86 13/05/2013 W 23 8 5 7 27 6 14 88

87 15/05/2013 L -3 8 2 7 23 18 12 94

88 22/05/2013 W 1 10 4 10 30 19 17 103

89 24/05/2013 L -4 3 5 8 36 14 17 97

90 26/05/2013 W 18 3 0 4 22 18 15 114

91 28/05/2013 L -7 5 2 6 24 16 7 99

92 30/05/2013 W 11 6 3 8 30 10 7 90

93 01/06/2013 L -14 6 4 7 29 10 5 91

94 03/06/2013 W 23 4 2 8 32 21 9 99

95 06/06/2013 L -4 10 2 18 18 17 13 92

96 09/06/2013 W 19 7 3 8 17 10 12 103

97 11/06/2013 L -37 6 2 11 15 16 12 113

98 13/06/2013 W 16 4 2 11 33 32 20 109

99 16/06/2013 L -10 8 3 6 25 25 16 114

100 18/06/2013 W 3 11 6 10 32 14 10 103

101 20/06/2013 W 7 4 2 12 37 23 10 95


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Analysis of ReboundsLJ variable

Sample mean 8.089109

Variance 7.34198

Standar deviation 2.709609

Median 8

Min 2

Max 18

Range 16

Quantiles

0% 25% 50% 75% 100%

1 5 7 9 16


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Analysis of TurnoversLJ variable

Sample mean 3.009901

Variance 2.389901

Standard deviation 1.54593

Median 3

Min 0

Max 7

Range 7

Quantiles

0% 25% 50% 75% 100%

0 2 3 4 7


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Analysis of PointsLJ, PointsDW and PointsCB variables

POINTS SCORED BY LEBRON JAMESSample mean 26.46535

Variance 34.81129


Median 26

Min 13

Max 40

Range 27

POINTS SCORED BY DWAYNE WADESample mean 19.38614

Variance 48.51941


Median 20

Min 3Max 39

Range 36

POINTS SCORED BY CHRIS BOSH

Sample mean 15.40594Variance 37.12356


Median 14

Min 5

Max 40

Range 35


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Normality check of considered variables


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Computation of confidence intervals for scored points mean of the Big Three

The histrogram and the qqplot are passing, it is possible to assume normality of data.From the qqplot the variable seems to be discrete, thanks to the fact that the sampledimension (large) we can use asymptotic confidence intervals without doing theassumption of normality of data.

Confidence intervals for all variable : 0.9

CI LeBron James scored points meanmeanLJ meanLJ[1] 26.46535> sdLJ sdLJ[1] 5.900109> CI.alphaLJ CI.alphaLJ[1] 25.49968 27.43101

CI Dwayne Wade scored points mean> meanDW meanDW[1] 19.38614> sdDW sdDW[1] 6.965587> CI.alphaDW CI.alphaDW[1] 18.24609 20.52619

CI Chris Bosh scored points mean> meanCB meanCB

[1] 15.40594> sdCB sdCB[1] 6.092911> CI.alphaCB CI.alphaCB[1] 14.40872 16.40316

Hypothesis testing on the difference of scored points by the Big Three in

Miamis wins and losses


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I take into account a new random variable PointsSum, sum of the scored points duringone game. The mean of this variable is bigger in wins than in losses?

Now we compute an evaluation of PointsSum mean in wins, in losses, hence thedifference between the means.

mean.v mean.v

[1] 62.39189> mean.p mean.p[1] 58.14815> mean.diff mean.diff[1] 4.243744

At this point we evaluate the normality of data separately for each group, before we goon.


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Data seems to have a good normality,

hence we proceed computing confidence intervals for means difference of 0.95.> CI.alpha CI.alpha[1] -0.5159391 9.0034266

The confidence interval includes 0, it could not exist any significant difference betweenthe two means, we will verify with a test.


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H0: MeanPointsSumWins = MeanPointsSumLossesH1: MeanPointsSumWins > MeanPointsSumLosses

We compute statistic test and 0.95 quantile of a t-Student a (n.losses + n.wins -2) dof,

in this way I am able to compute the critical region of the unilateral test for the twosamples of 0.05.

> t.alpha t.alpha[1] 1.660391> T.0 T.0[1] 1.769132

The statistic test is in the critical region, at 0.05 we have evidence to reject H 0 and statethat a difference exists between the means of the two sub-samples. To be moreprecise, we compute p-value of the unilateral test with unknown variance.

p p[1] 0.03997566

The p-value is quite low, we have evidence to reject H 0, hence to state that the mean ofthe sum of points in wins is bigger than in losses.


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Linear regression models:

1) Linear relation between LeBron James and Dwayne Wade - Chris Bosh pointsscored

Predictors: points, assists, rebounds, turnovers by LeBron JamesResponse : points sum of Dwayne Wade - Chris Bosh

Call:lm(formula = PointsW.B ~ PointsLJ + ReboundsLJ + AssistsLJ + TurnoversLJ)

Residuals:Min 1Q Median 3Q Max

-18.324 -6.340 -1.758 6.128 20.203

Coefficients:Estimate Std. Error t value Pr(>|t|)

(Intercept) 30.66267 5.36985 5.710 1.26e-07 ***PointsLJ 0.07160 0.15047 0.476 0.6353ReboundsLJ 0.60188 0.33054 1.821 0.0717 .AssistsLJ 0.03768 0.34481 0.109 0.9132TurnoversLJ -0.96367 0.57594 -1.673 0.0975 .---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Residual standard error: 8.797 on 96 degrees of freedomMultiple R-squared: 0.06077, Adjusted R-squared: 0.02163F-statistic: 1.553 on 4 and 96 DF, p-value: 0.1932

R-squared is really low, p-value of the test is high, no predictor is significant, we canstate that there is no linear correlation between LeBron James performance and Bosh-Wade points.


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2) Linear relation between points sum of Big Three and Miami wins margin

Predictors: points sum by the Big ThreeResponse: Miami win margin

Call:lm(formula = Margin ~ SumBigThree)


-44.173 -6.953 1.708 8.149 29.692


(Intercept) 5.71383 7.38270 0.774 0.441SumBigThree 0.03393 0.11871 0.286 0.776

Residual standard error: 12.8 on 99 degrees of freedomMultiple R-squared: 0.0008243, Adjusted R-squared: -0.009268F-statistic: 0.08167 on 1 and 99 DF, p-value: 0.7756

R-squared is really low, p-value is high, we can state that there is no linear correlationbetween points sum by the Big Three and Miami win margin. This result is not strange,

the fact that these players score a lot of points in a single game doesnt give us anyinformation about win margin, since really often in a game with a lot of points themargin is little even if the performances of the Big Three are exceptional.


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3) Linear relation between points scored by James, Wade, Bosh separately and Miamiwin margin.

Predictors: points scored by James, Wade, Bosh separatelyResponse: Miami win margin

Call:lm(formula = Margin ~ PointsLJ + PointsDW + PointsCB)


-46.869 -5.634 0.513 7.411 28.562


(Intercept) 9.38538 7.55921 1.242 0.2174PointsLJ -0.25410 0.21462 -1.184 0.2393PointsDW 0.33083 0.18221 1.816 0.0725 .PointsCB -0.08321 0.20807 -0.400 0.6901---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Residual standard error: 12.63 on 97 degrees of freedom

Multiple R-squared: 0.04633, Adjusted R-squared: 0.01684F-statistic: 1.571 on 3 and 97 DF, p-value: 0.2014

The only predictor that seems to be significant is PointsDW, so we decide to take otherpredictors out of the regression model

Call:lm(formula = Margin ~ PointsDW)


-43.697 -6.667 0.363 7.686 30.626


(Intercept) 1.5226 3.7251 0.409 0.6836PointsDW 0.3234 0.1809 1.787 0.0769 .---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1


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Residual standard error: 12.6 on 99 degrees of freedomMultiple R-squared: 0.03126, Adjusted R-squared: 0.02148F-statistic: 3.195 on 1 and 99 DF, p-value: 0.07694

Also this result is not suprising, since James is quite continuous player, hisperformances are always really good. Wade is quite irregular due to physical problems,but Miami takes advantage when he scores a lot of points, so the win margin seems tobe connected to Wade scored points.

At this point thanks to dispersion chart we analyze how good is the model and weverify the normality and the homoscedasticity of residuals.


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Among selected models this one is the best working one, but it has some problems.The residuals seems normal and homoscedastic but R-squared index is low, also Wade

scored points dont seem to be connected to Miami win margin.

4) Linear relation between Miami points and Big Three points

Predictors: Big Three points in a single gameResponse: Miami points in a single game

Call:lm(formula = PointsMIA ~ SumBigThree)


-18.7926 -6.1266 -0.1266 4.2074 28.8734


(Intercept) 82.63700 4.80426 17.201 < 2e-16 ***SumBigThree 0.35084 0.07725 4.542 1.58e-05 ***---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Residual standard error: 8.329 on 99 degrees of freedomMultiple R-squared: 0.1724, Adjusted R-squared: 0.1641F-statistic: 20.63 on 1 and 99 DF, p-value: 1.576e-05


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Also in this case, even if the predictor is significant (low p-value) and residuals seemshomoscedastic and normal, R-squared is really low, the variability of the phenomenoncan be explained better with internal variability of the group instead of variabilitybetween groups, as the scatterplot shows. This fact means that it is not possible tocreate a linear correlation between Big Three points summed up and Miami points. Theexplanation is logic : although these three champions of Miami Heat are a bigadvantage, in basketball team game is a key aspect. All this negative tests lead us tostate that there is no statistical evidence to consider these three player as fundamentalfor Miami Heat. Wins cant be explained only by LeBron James , Wade and Boshperformances, also a battleship like Miami Heat has to rely on its overall team.

Luca BazzucchiFilippo Campolmi

Progetto Statistics ENG

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