271 PROJECT - Linear Regression Report

Linear Regression with No Selection (All Variables)

Dependent Variable = Pwins (Percentage of wins) For linear regression, our dependent variable is the percentage of wins and because in some

seasons the number of games played was not the same for all teams.o During the 2012-2013 season, a game between the Boston Celtics and Indiana Pacers

was cancelled due to the Boston Marathon bombings. As a result, both teams played 81 games instead of the standard 82 games.

o A lockout occurred at the beginning of the 2011-2012 season. As a result, the season was shortened to 66 games.

The Model: Pwins = β0 + β1*FG% + β2*3P% + β3*2P% + β4*FT% + β5*ORG/G + β6*DRB/G + β7*TRB/G + β8*AST/G + β9*STL/G + β10*BLK/G + β11*TOV/G + β12*PF/G + β13*PTS/G + β14*Age + β15*SE_Ind + β16*AT_Ind + β17*CE_Ind + β18*NW_Ind + β19*PA_Ind + β20*Coach Change

Fitted Model: *Note TRB/G = ORB/G + DRB/GPwins = -4.68452 + 0.71935*FG% + 2.34154*3P% + 5.83236*2P% + 1.02186*FT% + 0.05388*ORG/G + 0.04154*DRB/G + 0*TRB/G + -0.01051*AST/G + 0.04506*STL/G + 0.00796*BLK/G + -0.04208*TOV/G + -0.00130*PF/G + -0.01500*PTS/G + 0.01897*Age + 0.01776*SE_Ind + -0.00619*AT_Ind + 0.03977*CE_Ind + 0.02268*NW_Ind + -0.02821*PA_Ind + -0.01975*Coach Change

Overall Significance Test of Linear Regression with No Selection:The hypotheses are

HO: β1= β2=…= β20=0vs.

Ha: βj ≠ 0 for at least one jWith F-value = 30.25 and p-value < 0.0001, we reject the null hypothesis and accept the alternative hypothesis. That is, regression is overall significant.

Linear Regression Results

The REG ProcedureModel: Linear_Regression_Model

Dependent Variable: Pwins

Number of Observations Read120Number of Observations Used120

Analysis of Variance

Source DF

Sum ofSquare

sMean

SquareF ValuePr > FModel 19 2.489460.13102 30.25<.0001Error 100 0.433120.00433 Corrected Total119 2.92259

Root MSE 0.06581R-Square0.8518

Dependent Mean 0.50001Adj R-Sq 0.8236

Coeff Var 13.16214

TRB/G =ORB/G + DRB/GParameter Estimates

VariableDF

ParameterEstimate

StandardErrort ValuePr > |t|

Intercept 1 -4.68452 0.49244 -9.51<.0001FG% 1 0.71935 1.34471 0.53 0.59393P% 1 2.34154 0.46297 5.06<.00012P% 1 5.83236 1.30060 4.48<.0001FT% 1 1.02186 0.31148 3.28 0.0014ORB/G B 0.05388 0.00848 6.36<.0001DRB/G B 0.04154 0.00582 7.14<.0001TRB/G 0 0 . . .AST/G 1 -0.01051 0.00499 -2.11 0.0376STL/G 1 0.04506 0.00821 5.49<.0001BLK/G 1 0.00796 0.01022 0.78 0.4379TOV/G 1 -0.04208 0.00755 -5.57<.0001PF/G 1 -0.00130 0.00570 -0.23 0.8205PTS/G 1 -0.01500 0.00338 -4.44<.0001Age 1 0.01897 0.00484 3.92 0.0002SE_Ind 1 0.01776 0.02551 0.70 0.4879AT_Ind 1 -0.00619 0.02310 -0.27 0.7893CE_Ind 1 0.03977 0.02442 1.63 0.1066NW_Ind 1 0.02268 0.02271 1.00 0.3204PA_Ind 1 -0.02821 0.02340 -1.21 0.2309Coach Change 1 -0.01975 0.01758 -1.12 0.2640

With a comparison of the kernel and normal density, we conclude that normal distribution is a good candidate for residuals.

As predicted Pwins increases the variability of residuals seems to remain constant. Therefore, constancy of variance is maintained.

Observing the Q-Q plot of residuals for Pwins, normality of residuals is overall supported.

The scatterplots of residuals by all regressors for Pwins confirms that constancy of variance is not violated.

Correlation Analysis of 14 variables

FG%, 3P%, 2P%, FT%, ORG/G, DRB/G, TRB/G, AST/G, STL/G, BLK/G, TOV/G, PF/G, PTS/G, Age

Correlation Analysis

The CORR Procedure

14 Variables:

FG% 3P% 2P% FT% ORB/G DRB/G TRB/G AST/G STL/G BLK/G TOV/G PF/G PTS/G Age

Simple StatisticsVariable N Mean Std

DevSumMinimumMaximum

FG% 120

0.455300.01591 54.63600 0.41400 0.49600

3P% 120

0.353460.02036 42.41500 0.29500 0.41200

2P% 120

0.485280.01956 58.23300 0.43900 0.53600

FT% 120

0.757140.02847 90.85700 0.66000 0.82800

ORB/G 120

11.102181.23061 1332 7.71212 13.86364

DRB/G 120

30.755021.43451 3691 27.19512 34.15854

TRB/G 120

41.857191.69235 5023 38.41463 46.66667

AST/G 120

21.463891.58296 2576 18.54545 26.67073

STL/G 120

7.506120.85929900.73468 5.58537 9.56098

BLK/G 120

4.987490.75810598.49915 3.58537 8.16667

TOV/G 120

14.400420.95643 1728 11.18182 17.04878

PF/G 120

20.245401.39452 2429 16.80303 23.21212

PTS/G 120

98.600004.29665 11832 87.00000 110.20000

Age 120

26.610001.81564 3193 23.20000 31.30000

Pearson Correlation Coefficients, N = 120 Prob > |r| under H0: Rho=0

FG% 3P% 2P% FT%ORB/

GDRB/

GTRB/

GAST/

GSTL/

GBLK/

GTOV/

G PF/GPTS/

G Age

FG%

1.00000

0.52002

<.0001

0.93448

<.0001

0.08249

0.3704

-0.458

67<.000

1

0.33466

0.0002

-0.049

860.588

7

0.52813

<.0001

0.15094

0.0999

0.12250

0.1826

-0.093

520.309

6

0.05577

0.5452

0.69499

<.0001

0.38689

<.0001

3P% 0.52002

<.000

1.00000

0.41832

<.000

0.12665

0.168

-0.377

18

0.33645

0.000

0.01092

0.905

0.36544

<.000

0.00186

0.983

-0.050

70

-0.199

32

-0.045

67

0.51702

<.000

0.33172

0.000

1 1 1 <.0001

2 8 1 9 0.5823

0.0291

0.6204

1 2

2P%

0.93448

<.0001

0.41832

<.0001

1.00000

0.01919

0.8352

-0.475

55<.000

1

0.36139

<.0001

-0.039

470.668

7

0.50014

<.0001

0.14177

0.1225

0.08240

0.3709

-0.037

980.680

4

-0.017

790.847

0

0.71801

<.0001

0.43475

<.0001

FT%

0.08249

0.3704

0.12665

0.1681

0.01919

0.8352

1.00000

-0.258

950.004

3

0.00825

0.9287

-0.181

300.047

5

0.00690

0.9404

0.02094

0.8205

0.12382

0.1779

-0.171

550.061

0

0.18035

0.0487

0.21850

0.0165

-0.018

480.841

2

ORB/G

-0.458

67<.000

1

-0.377

18<.000

1

-0.475

55<.000

1

-0.258

950.004

3

1.00000

-0.200

570.028

1

0.55714

<.0001

-0.338

280.000

2

0.03940

0.6692

0.13213

0.1503

0.16217

0.0768

0.04809

0.6020

-0.129

740.157

8

-0.412

22<.000

1

DRB/G

0.33466

0.0002

0.33645

0.0002

0.36139

<.0001

0.00825

0.9287

-0.200

570.028

1

1.00000

0.70179

<.0001

0.25803

0.0044

-0.154

020.093

0

0.30405

0.0007

0.13732

0.1348

-0.260

010.004

1

0.34918

<.0001

0.26934

0.0029

TRB/G

-0.049

860.588

7

0.01092

0.9058

-0.039

470.668

7

-0.181

300.047

5

0.55714

<.0001

0.70179

<.0001

1.00000

-0.027

270.767

5

-0.101

910.268

1

0.35381

<.0001

0.23432

0.0100

-0.185

430.042

6

0.20164

0.0272

-0.071

450.438

1

AST/G

0.52813

<.0001

0.36544

<.0001

0.50014

<.0001

0.00690

0.9404

-0.338

280.000

2

0.25803

0.0044

-0.027

270.767

5

1.00000

0.14165

0.1228

-0.023

710.797

1

-0.091

220.321

7

-0.154

630.091

7

0.36671

<.0001

0.33373

0.0002

STL/G

0.15094

0.0999

0.00186

0.9839

0.14177

0.1225

0.02094

0.8205

0.03940

0.6692

-0.154

020.093

0

-0.101

910.268

1

0.14165

0.1228

1.00000

0.16842

0.0659

0.05306

0.5649

0.09926

0.2807

0.19097

0.0367

0.01131

0.9024

BLK/G

0.12250

0.1826

-0.050

700.582

3

0.08240

0.3709

0.12382

0.1779

0.13213

0.1503

0.30405

0.0007

0.35381

<.0001

-0.023

710.797

1

0.16842

0.0659

1.00000

0.18934

0.0383

0.03436

0.7095

0.10221

0.2667

-0.040

610.659

7TOV/G

-0.093

-0.199

-0.037

-0.171

0.16217

0.13732

0.23432

-0.091

0.05306

0.18934

1.00000

0.17368

-0.019

-0.274

520.309

6

320.029

1

980.680

4

550.061

0

0.0768

0.1348

0.0100

220.321

7

0.5649

0.0383

0.0578

290.834

3

500.002

4

PF/G

0.05577

0.5452

-0.045

670.620

4

-0.017

790.847

0

0.18035

0.0487

0.04809

0.6020

-0.260

010.004

1

-0.185

430.042

6

-0.154

630.091

7

0.09926

0.2807

0.03436

0.7095

0.17368

0.0578

1.00000

0.19334

0.0344

-0.358

51<.000

1

PTS/G

0.69499

<.0001

0.51702

<.0001

0.71801

<.0001

0.21850

0.0165

-0.129

740.157

8

0.34918

<.0001

0.20164

0.0272

0.36671

<.0001

0.19097

0.0367

0.10221

0.2667

-0.019

290.834

3

0.19334

0.0344

1.00000

0.14008

0.1270

Age

0.38689

<.0001

0.33172

0.0002

0.43475

<.0001

-0.018

480.841

2

-0.412

22<.000

1

0.26934

0.0029

-0.071

450.438

1

0.33373

0.0002

0.01131

0.9024

-0.040

610.659

7

-0.274

500.002

4

-0.358

51<.000

1

0.14008

0.1270

1.00000

Principal Component Analysis of 14 Variables

After our initial analysis of the eigenvalues of the correlation matrix table and scree plot, we decided to include the first 9 principal components into the regression model.

Including the first 9 principal components would result in 92.52% of total variability.

Principal Components Analysis

The PRINCOMP Procedure

Observations 120Variables 14Simple Statistics

FG% 3P% 2P% FT%ORB/

GDRB/

GTRB/

GAST/

GSTL/

GBLK/

GTOV/

G PF/GPTS/

G AgeMean

0.455300000

0

0.353458333

3

0.485275000

0

0.757141666

7

11.1021753

2

30.7550170

9

41.8571924

1

21.4638890

0

7.50612231

2

4.98749292

8

14.4004158

6

20.2453983

1

98.6000000

0

26.6100000

0

StD

0.015908667

9

0.020358920

5

0.019556409

5

0.028474460

31.23060569

1.43450737

1.69235097

1.58295848

0.85928592

3

0.75810211

00.95643164

1.39452040

4.29664712

1.81563703

Correlation Matrix

FG% 3P% 2P% FT%ORB/

GDRB/

GTRB/

GAST/

GSTL/

GBLK/

GTOV/

GPF/GPTS/

G AgeFG% 1.000 0.520 0.934 0.082 -.45870.3347 -.0499 0.528 0.1500.1225 -.0935 0.055 0.695 0.386

0 0 5 5 1 9 8 0 9

3P%0.520

01.000

00.418

30.126

7 -.37720.33640.010

90.365

40.001

9 -.0507 -.1993-.045

70.517

00.331

7

2P%0.934

50.418

31.000

00.019

2 -.47550.3614 -.03950.500

10.141

80.0824 -.0380-.017

80.718

00.434

7

FT%0.082

50.126

70.019

21.000

0 -.25890.0083 -.18130.006

90.020

90.1238 -.17150.180

40.218

5-.018

5ORB/G

-.4587

-.3772

-.4755

-.25891.0000 -.2006

0.5571-.3383

0.03940.13210.1622

0.0481

-.1297

-.4122

DRB/G

0.3347

0.3364

0.3614

0.0083 -.20061.0000

0.7018

0.2580

-.15400.30410.1373

-.2600

0.3492

0.2693

TRB/G

-.0499

0.0109

-.0395

-.18130.55710.7018

1.0000-.0273

-.10190.35380.2343

-.1854

0.2016

-.0714

AST/G

0.5281

0.3654

0.5001

0.0069 -.33830.2580 -.0273

1.0000

0.1417 -.0237 -.0912

-.1546

0.3667

0.3337

STL/G

0.1509

0.0019

0.1418

0.02090.0394 -.1540 -.1019

0.1417

1.00000.16840.0531

0.0993

0.1910

0.0113

BLK/G

0.1225

-.0507

0.0824

0.12380.13210.3041

0.3538-.0237

0.16841.00000.1893

0.0344

0.1022

-.0406

TOV/G

-.0935

-.1993

-.0380

-.17150.16220.1373

0.2343-.0912

0.05310.18931.0000

0.1737

-.0193

-.2745

PF/G0.055

8-.045

7-.017

80.180

40.0481 -.2600 -.1854-.15460.099

30.03440.17371.000

00.193

3-.358

5PTS/G

0.6950

0.5170

0.7180

0.2185 -.12970.3492

0.2016

0.3667

0.19100.1022 -.0193

0.1933

1.0000

0.1401

Age0.386

90.331

70.434

7-.018

5 -.41220.2693 -.07140.333

70.011

3 -.0406 -.2745-.358

50.140

11.000

0Eigenvalues of the Correlation Matrix

Eigenvalue DifferenceProportionCumulativ

e14.065470951.81316095 0.2904 0.290422.252309990.54489566 0.1609 0.451331.707414330.53337314 0.1220 0.573241.174041200.14391996 0.0839 0.657151.030121230.08729108 0.0736 0.730760.942830150.27659578 0.0673 0.798070.666234370.07859120 0.0476 0.845680.587643170.06072172 0.0420 0.887690.526921460.07830904 0.0376 0.9252100.448612420.04973914 0.0320 0.9573110.398873270.23317606 0.0285 0.9857120.165697210.13186697 0.0118 0.997610.033830240.03383024 0.0024 1.0000

3140.00000000 0.0000 1.0000

Eigenvectors

PRIN

1PRIN

2PRIN

3PRIN

4PRIN

5PRIN

6PRIN

7PRIN

8PRIN

9PRIN

10PRIN

11PRIN

12PRIN

13PRIN

14

FG%0.446

639-.0047

450.149

684-.1131

14-.0780

21-.0220

70-.2575

76-.1327

260.016

236-.1550

610.024

4870.559

188-.5806

520.000

000

3P%0.339

533-.0285

16-.0458

160.186

816-.1674

860.264

2690.258

6270.572

728-.4447

81-.1642

050.283

2690.147

4240.165

3530.000

000

2P%0.441

9400.010

5920.106

649-.1770

62-.0965

99-.0870

58-.3013

22-.1514

680.226

967-.1589

58-.0103

130.106

0720.736

9290.000

000

FT%0.083

150-.1551

240.249

9630.673

2250.326

902-.0160

470.291

282-.1531

740.369

6610.075

9500.192

8200.228

1560.078

9090.000

000ORB/G

-.292193

0.345676

0.092142

-.156102

-.001237

0.507098

-.085929

-.143283

0.064428

0.128719

0.393546

0.239970

0.093467

0.485068

DRB/G

0.252942

0.445245

-.211513

0.237322

-.039561

-.171710

0.132348

0.108022

0.057769

0.083208

-.495800

-.003152

-.025940

0.565440

TRB/G

0.001934

0.628769

-.112286

0.087653

-.034433

0.223192

0.049700

-.012625

0.095817

0.164130

-.134091

0.171825

0.045978

-.667074

AST/G

0.325174

-.027921

-.072677

-.244070

0.028266

0.011355

0.536633

-.580255

-.345197

0.273160

0.069691

-.055262

0.036642

0.000000

STL/G

0.064729

-.025044

0.343099

-.472033

0.561504

0.166420

0.272201

0.346045

0.147673

0.002046

-.299830

0.076979

0.013846

0.000000

BLK/G

0.043764

0.363206

0.202524

0.136621

0.545665

-.272094

-.321951

-.079053

-.489537

-.125743

0.197204

-.173856

0.014977

0.000000

TOV/G

-.086003

0.300286

0.247978

-.229852

-.259514

-.612793

0.314375

0.199052

0.208234

0.024090

0.400331

0.032779

-.038861

0.000000

PF/G-.0494

67-.0901

560.604

1640.122

861-.2936

18-.0117

71-.1777

670.076

894-.2702

020.586

055-.2457

050.055

3080.074

5200.000

000PTS/G

0.363966

0.142035

0.312140

0.051180

-.153772

0.319789

-.025667

-.019186

0.260889

-.085605

0.128445

-.683100

-.248701

0.000000

Age0.281

936-.1005

18-.3845

14-.0880

500.236

372-.0749

23-.2587

350.258

8600.178

9560.652

4760.304

004-.0720

06-.0633

530.000

000

Linear Regression with the First 9 Principal Components

With an adjusted R-square of 0.7548, we were unsatisfied with this particular model.






Source DF

Sum ofSquare

sMean




Coeff Var 15.52064 Parameter Estimates

VariableDF

ParameterEstimate


Intercept 1 0.51011 0.01895 26.91<.0001PRIN1 1 0.05697 0.00392 14.53<.0001PRIN2 1 0.02483 0.00527 4.71<.0001PRIN3 1 -0.01735 0.00618 -2.81 0.0059PRIN4 1 -0.00243 0.00677 -0.36 0.7210PRIN5 1 0.04030 0.00741 5.44<.0001

PRIN6 1 0.03171 0.00789 4.02 0.0001PRIN7 1 -0.03789 0.00921 -4.11<.0001PRIN8 1 0.02935 0.00981 2.99 0.0035PRIN9 1 0.00910 0.01069 0.85 0.3966SE_Ind 1 -0.01325 0.02883 -0.46 0.6469AT_Ind 1 -0.00224 0.02635 -0.08 0.9325CE_Ind 1 0.01277 0.02801 0.46 0.6494NW_Ind 1 0.01072 0.02647 0.40 0.6864PA_Ind 1 -0.06513 0.02615 -2.49 0.0143Coach Change 1 -0.00317 0.02021 -0.16 0.8756

Linear Regression with the First 11 Principal Components

We were unsatisfied with the previous model that included the first 9 principal components. Therefore, we decided to include principal components 10 & 11.

Including the first 11 principal components would result in 98.57% of total variability However, this model had an adjusted R-square of 0.7534 which was very similar to the previous

model. In this case, there was no benefit in adding principal components 10 & 11 to the model.






Source DF

Sum ofSquare

sMean





VariableDF

ParameterEstimate


Intercept 1 0.50966 0.01903 26.78<.0001PRIN1 1 0.05692 0.00394 14.46<.0001PRIN2 1 0.02482 0.00533 4.65<.0001PRIN3 1 -0.01746 0.00621 -2.81 0.0059PRIN4 1 -0.00250 0.00680 -0.37 0.7136PRIN5 1 0.04031 0.00743 5.42<.0001PRIN6 1 0.03244 0.00800 4.05<.0001PRIN7 1 -0.03743 0.00926 -4.04 0.0001PRIN8 1 0.02953 0.00984 3.00 0.0034PRIN9 1 0.00879 0.01072 0.82 0.4142PRIN10 1 0.01193 0.01151 1.04 0.3026PRIN11 1 -0.00633 0.01185 -0.53 0.5942SE_Ind 1 -0.00879 0.02956 -0.30 0.7669AT_Ind 1 -0.00399 0.02685 -0.15 0.8820CE_Ind 1 0.00929 0.02828 0.33 0.7433NW_Ind 1 0.01345 0.02665 0.50 0.6148PA_Ind 1 -0.06669 0.02661 -2.51 0.0138Coach Change 1 -0.00104 0.02045 -0.05 0.9595

Linear Regression with Principal Components 1-9, 12, and 13

After including the first 9 principal components as well as principal components 12 & 13, we observed a much better adjusted R-square value of 0.8237.






Source DF

Sum ofSquare

sMean





Variable DF ParameterStandardt ValuePr > |t|

Estimate ErrorIntercept 1 0.49606 0.01634 30.35<.0001PRIN1 1 0.05710 0.00332 17.18<.0001PRIN2 1 0.02221 0.00448 4.95<.0001PRIN3 1 -0.01599 0.00524 -3.05 0.0029PRIN4 10.00003589 0.00576 0.01 0.9950PRIN5 1 0.04077 0.00629 6.49<.0001PRIN6 1 0.03119 0.00670 4.66<.0001PRIN7 1 -0.03448 0.00783 -4.40<.0001PRIN8 1 0.02536 0.00834 3.04 0.0030PRIN9 1 0.01368 0.00909 1.50 0.1356PRIN12 1 0.09077 0.01570 5.78<.0001PRIN13 1 0.10673 0.03360 3.18 0.0020SE_Ind 1 0.01564 0.02502 0.63 0.5331AT_Ind 1 -0.00259 0.02272 -0.11 0.9094CE_Ind 1 0.04234 0.02423 1.75 0.0836NW_Ind 1 0.01985 0.02262 0.88 0.3821PA_Ind 1 -0.02860 0.02302 -1.24 0.2169Coach Change 1 -0.02084 0.01740 -1.20 0.2337

The Model: Pwins = β0 + β1*PRIN1 + β2*PRIN2 + β3*PRIN3 + β4*PRIN4 + β5*PRIN5 + β6*PRIN6 + β7*PRIN7 + β8*PRIN8 + β9*PRIN9 + β10*PRIN12 + β11*PRIN13 + β12*SE_Ind + β13*AT_Ind + β14*CE_Ind + β15*NW_Ind + β16*PA_Ind + β17*Coach Change

Fitted model using linear regression:Pwins = 0.49606 + 0.05710*PRIN1 + 0.02221*PRIN2 – 0.01599*PRIN3 + 0.00003589*PRIN4 + 0.04077*PRIN5 + 0.03119*PRIN6 – 0.03448*PRIN7 + 0.02536*PRIN8 + 0.01368*PRIN9 + 0.09077*PRIN12 + 0.10673*PRIN13 + 0.01564*SE_Ind – 0.00259*AT_Ind + 0.04234*CE_Ind + 0.01985*NW_Ind – 0.02860*PA_Ind – 0.02084*Coach Change

Overall Significance Test of Final Model for Linear Regression:The hypotheses are

HO: β1= β2=…= β17=0vs.

Ha: βj ≠ 0 for at least one jWith F-value = 33.71 and p-value < 0.0001, we reject the null hypothesis and accept the alternative hypothesis. That is, the final model for linear regression is overall significant.

With a comparison of the kernel and normal density, we conclude that normal distribution is a good candidate for residuals.

As predicted Pwins increases the variability of residuals seems to remain constant. Therefore, constancy of variance is maintained.

Observing the Q-Q plot of residuals for Pwins, normality of residuals is overall supported.

The scatterplots of residuals by all regressors for Pwins confirms that constancy of variance is not violated.

271 PROJECT - Linear Regression Report

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