NBA球員得分之迴歸分析：以Steve Nash為例

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NBA

Steve Nash

The Analysis of Regression on NBA Players’ Scores─

Take Steve Nash for Example

D9963395 D0087675 D9963216 D9963161

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2007-2009 ( ) ( ) (Forward

method), (Backward method) (Stepwise

method)

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Abstract

In a basketball game, what people focus on is usually the result. However, what can affect players’ performance includes steals, errors, rebounds, assists and field goals. We take Steve Nash, one of the Lakers’ guards, as an analytical basis to understand the effects of steals, errors, rebounds, assists and field goals upon scores. The data range from 2007 to 2009.

This report focuses on all response variables and tries to find which one has more expansion in explanatory variables (scores). At first, we conducted fitness tests to recognize if a response variable has any relationship with explanatory variables. And then, after we collected and calculated the data, we were sure that they have relationships of linear regression. Then, we used three methods (forward method, backward method and stepwise method) to test the model. In addition to the three methods, we used other methods to make a comparison. Finally, we found that the greatest explanatory variables are field goal attempts and percentage. The field goal attempts and percentage contribute to the best model of regression.

After getting the model, we used the residual analysis to test if it has the relationship of normal distribution, if the variance is contact, and if the model of regression is the best model linear regression.

We came to the conclusion that when the other conditions are fixed, more field goal attempts and higher field goal percentage lead to more scores.

Keywor

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... 6 ... 8 2-1. ... 8 2-2. Correlation ... 13 ... 15 3-1. ... 15 3-2. ... 16 3-3. ... 19 3-4. ... 20 ... 21 4-1. (Forward)slentry=0.05 ... 21 4-2. (Backward)slentry=0.05 ... 22 4-3. slentry=0.05 slstay=0.05 ... 23 4-4. ... 25 4-5. ... 28 ... 29 5-1. ... 29 5-2. ... 33 5-3. ... 34 ... 38 ... 39 7-1. ... 39 ... 48

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1. ··· 9 2. ··· 9 3. ··· 10 4. ··· 10 5. ··· 11 6. ··· 11 7. ··· 12 8. ··· 30 9. ··· 31 10. (Nomal q-q plot) ··· 31 11. (X1&X2)(Nomal q-q plot) ··· 32 12. ··· 33 13. ··· 34 14. ··· 35 15. ··· 36

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1. ··· 8 2. ··· 13 3. ··· 15 4. ··· 19 5. ··· 20

6. Summary of Forward Selection ··· 21

7. Summary of Backward Elimination ··· 22

8. STEP1-1 ··· 23

9. STEP1-2 ··· 23

10. STEP2-1 ··· 24

11. STEP2-2 ··· 24

12. Summary of Stepwise Selection ··· 25

13. ··· 25

14. ··· 28

15. ··· 29

16. ··· 33

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NBA SBL

NBA

NBA 1949

50

Steve Nash Nash

Nash Nash

NBA

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( X )

X1 X2

X

( AIC SBC R2

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2-1.

Y：得分 x1：出手次數 x2：命中率 x3：籃板數

x4：失誤次數 x5：抄截次數 x6：助攻次數 N：場次 1.

Variable N Mean Std Dev Sum Minimum Maximum

Y (得分) 155 16.32903 7.27664 2531 0 37.00000 x1 (出手數) 155 11.69677 4.54469 1813 0 27.00000 x2 (命中率) 155 50.18774 16.82697 7779 0 100.00000 x3 (籃板) 155 3.25806 1.87200 505.00000 0 9.00000 x4 (失誤) 155 3.50323 1.98492 543.00000 0 10.00000 x5 (抄截) 155 0.69677 0.90722 108.00000 0 4.00000 x6 (助攻) 155 10.41290 3.75169 1614 2.00000 21.00000

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Y

1. 11-15 40 31-35 36-40 5 2 6-25

(1)

X

1 2. 6-10 63 26-30 1 6-20

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(2)

X

2 3. 40%-50% 41 0~10% 1 71%

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X

3 4. 3 36 9 1 2-5

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X

4 5. 3 32 8 3

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X

5 6. 0 80 4 3

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X

6

7.

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2-2. Correlation

2.

Pearson Correlation Coefficients, N = 155 Prob > |r| under H0: Rho=0

y x1 x2 x3 x4 x5 x6 y 得分 1.00000 0.77491 <.0001 0.49758 <.0001 0.09193 0.2553 0.15615 0.0523 0.10177 0.2076 0.06064 0.4535 x1 出手數 0.77491 <.0001 1.00000 0.02631 0.7452 0.04055 0.6164 0.13724 0.0886 0.15395 0.0558 0.09232 0.2532 x2 命中率 0.49758 <.0001 0.02631 0.7452 1.00000 0.04172 0.6062 0.08668 0.2835 0.00716 0.9296 -0.06564 0.4171 x3 籃板數 0.09193 0.2553 0.04055 0.6164 0.04172 0.6062 1.00000 0.09938 0.2186 0.05020 0.5351 0.20663 0.0099 x4 失誤 0.15615 0.0523 0.13724 0.0886 0.08668 0.2835 0.09938 0.2186 1.00000 0.08889 0.2714 0.08876 0.2721 x5 抄截 0.10177 0.2076 0.15395 0.0558 0.00716 0.9296 0.05020 0.5351 0.08889 0.2714 1.00000 0.07709 0.3404 x6 助攻 0.06064 0.4535 0.09232 0.2532 -0.06564 0.4171 0.20663 0.0099 0.08876 0.2721 0.07709 0.3404 1.00000 (1) r 1 -1 (2) r : (a)0.7< r ≤1 (b)0.3≤ r ≤0.7 (c) 0≤ r <0.3 (Y) (

)

0.77491 (Y) (

)

0.49758 (Y) (

)

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0.09193 (Y) (

)

0.15615 (Y) (

)

0.10177， (Y) (

)

0.06064 r

(Y) (

)

(Y) (

)

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3-1.

(Y) (X1) (X2) (X3) (X4) (X5) (X6) Yi= 0+ 1X1i+ 2X2i+ 3X3i+ 4X4i+ 5X5 i+ 6X6i+ i i=1,2,3, ,155 E(Yi)= 0+ 1X1i+ 2X2i+ 3X3i+ 4X4i+ 5X5 i+ 6X6i + i i=1,2,3, ,155 X1= X2= X3= X4= X5= X6= SAS 3. Variable Label DF Parameter Estimate Standard Error t Value Pr > |t| Intercept Intercept 1 -9.03176 1.24125 -7.28 <0.0001 x1 出手數 1 1.22000 0.05539 22.03 <0.0001 x2 命中率 1 0.20603 0.01474 13.98 <0.0001 x3 籃板數 1 0.14978 0.13503 1.11 0.2691 x4 失誤 1 0.02626 0.12664 0.21 0.8360 x5 抄截 1 -0.18163 0.27574 -0.66 0.5111 x6 助攻 1 0.02854 0.06773 0.42 0.6741

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0=-9.03176 1=1.22000 2=0.20603 3=0.14978 4=0.02626 5=-0.18163 6=0.02854 Model =-9.03176+1.22X1+0.20603X2+0.14978X3+0.02626X4-0.18163X5+0.02854X6

3-2.

(X1) (X2) (X3) (X4) (X5) (X6) (1) (X1) (Y) 0 0 p-value<0.0001< =0.05 0 (X1) (Y) (2) (X2) (Y) 0 0 p-value<0.0001< =0.05 =0 (X2) (Y)

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(3) (X3) (Y) 0 0 p-value=0.2691> =0.05 0 (X3) (Y) (4) (X4) (Y) 0 0 p-value=0.8360> =0.05 0 (X4) (Y) (5) (X5) (Y) 0 0 p-value=0.5111> =0.05 0 (X5) (Y) (6) (X6) (Y) 0 0 p-value=0.6741> =0.05

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3-3.

SAS 4. Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model 6 6774.22955 1129.03826 121.09 <0.0001 Error 148 1379.98980 9.32426 Corrected Total 154 8154.21935 H0 1= 2= 3= 4= 5= 6=0 H1 i 0 i= 1 2 3 4 5 6 p-value<0.0001< =0.05 H0 i 0 ( ) ( )

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3-4.

5.

Root MSE 3.05356 R-Square 0.8308

Dependent Mean 16.32903 Adj R-Sq 0.8239

CoeffVar 18.70022 R-Square=SSR/SSTO=0.8308

Adj R-Square= 1 MSE/MSTO=0.8239 R2

0.8308 83.08%

R2

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4-1.

(Forward)

slentry=0.05

SAS PROC REG (FORWARD)

F

( p-value )

F

X1= X2= X3=

X4= X5= X6=

6. Summary of Forward Selection

Summary of Forward Selection Step Variable Entered Label Number Vars In Partial R-Square Model R-Square C(p) F Value Pr > F 1 x1 出手數 1 0.6005 0.6005 198.380 229.97 <.0001 2 x2 命中率 2 0.2279 0.8284 1.1026 201.80 <.0001 Step X1 F 229.97( p-value<α=0.05) Step 5 (X2 X3 X4 X5 X6) X2 F 201.80(p-value<α=0.05) X3 X4 X5 X6 0.05 (X1 X2) :

=-8.31205+1.22062X

1

+0.20650X

2

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4-2.

(Backward)

slentry=0.05

SAS PROC REG

F p-value

X1= X2= X3=

X4= X5= X6=

7. Summary of Backward Elimination

Step F X4 F 0.04 (p-value=0.8360>0.05) Step (X1 X2 X3 X5 X6) F X6 F 0.19(P-Value=0.6630>0.05) Step (X1 X2 X3 X5) F X5 F 0.39(P-Value=0.5314>0.05) Step (X1 X2 X5) F X5 F 1.51(P-Value=0.0.2218>0.05)

Summary of Backward Elimination Step Variable Removed Label Number Vars In Partial R-Square Model R-Square C(p) F Value Pr > F 1 x4 失誤 5 0.0000 0.8307 5.0430 0.04 0.8360 2 x6 助攻 4 0.0002 0.8305 3.2324 0.19 0.6630 3 x5 抄截 3 0.0004 0.8301 1.6213 0.39 0.5314 4 x3 籃板數 2 0.0017 0.8284 1.1026 1.51 0.2218

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Step (X1 X2) F F 0.05 (X2 X4 X5)

=-8.31205+1.22062X

1

+0.20650X

2

4-3.

slentry=0.05 slstay=0.05

F ( p-value ) F (1) F (2) Step

Variable x1 Entered: R-Square = 0.6005 and C(p) = 198.3796

8. STEP1-1 Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model 1 4896.51500 4896.51500 229.97 <.0001 Error 153 3257.70436 21.29219 Corrected Total 154 8154.21935 9. STEP1-2

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Variable

Parameter Estimate

Standard

Error Type II SS F Value Pr > F Intercept 1.81644 1.02626 66.70291 3.13 0.0787

x1 1.24073 0.08182 4896.51500 229.97 <.0001

Step

Variable x2 Entered: R-Square = 0.8284 and C(p) = 1.1026

10. STEP2-1 11. STEP2-2 Variable Parameter Estimate Standard

Error Type II SS F Value Pr > F Intercept -8.31205 0.98175 660.04459 71.68 <.0001

x1 1.22062 0.05382 4735.76114 514.32 <.0001

x2 0.20650 0.01454 1858.10939 201.80 <.0001

All variables left in the model are significant at the 0.0500 level.

No other variable met the 0.0500 significance level for entry into the model.

Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model 2 6754.62439 3377.31219 366.79 <.0001 Error 152 1399.59497 9.20786 Corrected Total 154 8154.21935

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12. Summary of Stepwise Selection

Summary of Stepwise Selection Step Variable Entered Variable Removed Label Number Vars In Partial R-Square Model R-Square C(p) F Value Pr > F 1 x1 出手數 1 0.6005 0.6005 198.380 229.97 <.0001 2 x2 命中率 2 0.2279 0.8284 1.1026 201.80 <.0001 Step X1 F=229.97 (p-value<.0001<0.05) Step (X2 X3 X4 X5 X6) F X2 F=201.80 (p-value<.0001<0.05) X2 (X3 X4 X5 X6) (X1 X2)

=-8.31205+1.22062X

1

+0.20650X

2

4-4.

13.

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Number in

Model R-Square

Adjusted

R-Square C(p) AIC MSE SBC Variables in Model 1 0.6005 0.5979 198.3796 476.0297 21.29219 482.11655 x1 1 0.2476 0.2427 506.9996 574.1517 40.10037 580.23858 x2 1 0.0244 0.0180 702.1926 614.4178 51.99599 620.50464 x4 2 0.8284 0.8261 1.1026 347.0795 9.20786 356.2098 0 x1 x2 2 0.6042 0.5989 197.1731 476.6006 21.23557 485.73089 x1 x3 2 0.6030 0.5978 198.1684 477.0456 21.29663 486.17590 x1 x4 3 0.8301 0.8267 1.6213 347.5423 9.17737 359.7159 8 x1 x2 x3 3 0.8288 0.8254 2.6890 348.6518 9.24330 360.8255 3 x1 x2 x6 3 0.8287 0.8253 2.7790 348.7450 9.24886 360.9186 9 x1 x2 x5 4 0.8305 0.8260 3.2324 349.1362 9.21438 364.35335 x1 x2 x3 x5 4 0.8302 0.8257 3.4617 349.3758 9.22863 364.59292 x1 x2 x3 x6 4 0.8301 0.8256 3.5841 349.5035 9.23624 364.72059 x1 x2 x3 x4 5 0.8307 0.8250 5.0430 350.9380 9.26437 369.19856 x1 x2 x3 x5 x6 5 0.8306 0.8249 5.1776 351.0788 9.27279 369.33940 x1 x2 x3 x4 x5 5 0.8303 0.8246 5.4339 351.3467 9.28883 369.60727 x1 x2 x3 x4 x6 6 0.8308 0.8239 7.0000 352.8930 9.32426 374.19695 x1 x2 x3 x4 x5 x6

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(

)

Coefficient of Multiple Determination

=SSR/SSTO=1-SSE/SSTO Y X

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R-square

X1 X2 =0.8284 X1 X2 X3

X4 X5 X6 =0.8308 X3 X4 X5 X6

X1 X2

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(

)

Adjusted Coefficient of Multiple Determination =1-(SSE/n-p)/(SSTO/n-1)=1-(n-1/n-p)*(SSE/SSTO) X X SSE SSTO MSE SSE n-p X1 X2 X3 =0.8267 MSE=9.17737 X1 X2 X3

(3)Cp

n (total

mean squared error) Cp Cp

(p)

X1 X2 X4 X5 X6 Cp=6.2305 p(=6)

X1 X2 X4 X5 X6

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SBC AIC SBC AIC X1 X2 AIC=347.0795 SBC=356.20980 X1 X2

4-5.

95% 14. (RORWARD) X1 X2 (BACKWARD) X1 X2 (STEPWISE) X1 X2 (R2 ) X1 X2 (R2 a) X1 X2 X3 Cp X1 X2 X4 X5 X6 SBC AIC X1 X2 (X1) (X2)

=-8.31205+1.22062X

1

+0.20650X

2

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=-8.31205+1.22062X

1

+0.20650X

2

5-1.

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1

15.

Tests for Normality

Test Statistic p Value

Shapiro-Wilk W 0.987274 Pr < W 0.2015

Kolmogorov-Smirnov D 0.043675 Pr > D >0.1500

Cramer-von Mises W-Sq 0.067844 Pr > W-Sq >0.2500

Anderson-Darling A-Sq 0.462275 Pr > A-Sq >0.2500

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2

1

8.

2 1

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9.

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11. (X1&X2)(Nomal q-q plot) 3

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12.

5-2.

16. Dependent Variable: y Durbin-Watson D 1.956 Pr < DW 0.3920 Pr > DW 0.6080 Number of Observations 146

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Durbin-Watson D 1.956 1st Order Autocorrelation 0.020 D>DU D<DL DU>D>DL D=1.956>DU

5-3.

13.

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(1)

:

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(2)

:

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90% 1.4526392 -1.1799720 17. Quantile Estimate 100% Max 2.9169027 99% 2.6508793 95% 1.8220605 90% 1.4526392 75% Q3 0.5758795 50% Median -0.0703397 25% Q1 -0.6839298 10% -1.1799720 5% -1.6367895 1% -2.1100747 0% Min -2.1758434

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(R2 R2 a Cp SBC AIC )

=-8.31205+1.22062X

1

+0.20650X

2 =-8.31205+1.22062X1+0.20650X2 X1 X2 X1 X2=0 Y -8.31205 X2 X1 Y 1.22062 1.22062 X1 X2 Y 0.20650 0.20650 R-square=0.8239

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7-1.

Obs 出手數命中率籃板數助攻抄截失誤得分 n F9 1 1 100.0 2 5 0 1 3 1 2 11 36.4 9 14 2 5 13 2 3 9 22.2 6 9 0 1 6 3 4 10 40.0 5 10 0 0 12 4 5 11 45.5 2 8 0 2 16 5 6 17 23.5 5 10 0 5 14 6 7 6 83.3 3 11 1 7 14 7 8 14 42.9 4 18 3 4 17 8 9 18 61.1 3 8 1 0 36 9 10 9 22.2 3 11 1 3 10 10 11 8 25.0 2 10 1 4 5 11 12 10 50.0 2 9 0 1 12 12 13 14 64.3 3 9 0 1 23 13 14 10 50.0 4 5 0 6 15 14 15 13 69.2 2 11 1 6 23 15 16 16 56.3 0 8 0 1 22 16 17 9 77.8 2 8 1 2 18 17 18 13 53.8 4 13 0 3 21 18 19 7 57.1 3 8 0 2 8 19 20 12 58.3 5 14 0 4 19 20 21 14 50.0 6 15 1 3 17 21

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Obs 出手數命中率籃板數助攻抄截失誤得分 n F9 22 11 36.4 4 13 0 5 12 22 23 12 50.0 3 4 0 1 19 23 24 13 46.2 3 11 0 6 21 24 25 6 16.7 0 13 0 4 8 25 26 11 72.7 2 13 0 4 25 26 27 8 50.0 2 6 0 4 11 27 28 12 58.3 7 5 2 5 18 28 29 17 52.9 2 8 1 1 26 29 30 17 52.9 6 13 0 5 24 30 31 13 53.8 4 12 2 7 20 31 32 6 50.0 3 12 0 4 10 32 33 13 30.8 4 8 2 2 13 33 34 23 56.5 4 12 1 10 32 34 35 10 30.0 3 11 0 5 8 35 36 18 33.3 2 8 1 3 17 36 37 8 50.0 4 10 3 4 10 37 38 10 50.0 2 7 0 6 14 38 39 11 81.8 4 9 0 7 26 39 40 16 43.8 3 16 0 4 21 40 41 23 56.5 4 10 2 2 37 41 42 11 45.5 3 9 0 3 13 42 43 6 100.0 4 13 0 1 17 43 44 15 33.3 0 20 2 5 13 44 45 14 35.7 3 13 0 6 14 45

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Obs 出手數命中率籃板數助攻抄截失誤得分 n F9 142 14 57.1 8 15 0 7 20 142 143 7 85.7 3 7 0 8 16 143 144 9 33.3 1 10 0 2 8 144 145 10 60.0 3 8 0 2 14 145 146 10 60.0 3 7 0 4 17 146 147 10 40.0 4 3 0 2 10 147 148 11 27.3 2 6 0 6 12 148 149 9 33.3 5 7 1 1 16 149 150 7 28.6 0 5 0 3 6 150 151 5 40.0 3 6 1 3 7 151 152 7 71.4 2 11 1 4 12 152 153 10 60.0 3 7 1 1 20 153 154 16 56.3 4 9 0 7 24 154 155 11 45.5 2 13 0 3 13 155

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7777----2.SAS

2.SAS

odsrtffile="h:\picture.rtf";

PROCIMPORTOUT= WORK.master DATAFILE= "H:\第6組報告\nash.xls" DBMS=EXCEL REPLACE; RANGE="Sheet1$"; GETNAMES=YES; MIXED=NO; SCANTEXT=YES; USEDATE=YES; SCANTIME=YES; RUN; *迴規模式確立; data a; setwork.master ; label y=得分 x1='出手數' x2='命中率' x3='籃板數' x4='失誤' x5='抄截' x6='助攻' ;

procprintdata=a label; title'原始資料'; run; procglmdata=a ; model y=x1-x6; title'原始迴歸model'; run; proccorr; var y x1-x6; title'檢定是否有相關性'; run; procregdata=a;

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procregdata=a;

model y=x1-x6/selection=backward sls=0.05; title'backward';

procregdata=a;

model y=x1-x6/selection=stepwise sls=0.05sle=0.05; title'stepwise';

procrsquare data=a adjrsqcpaicsbcmse; model y=x1-x6 best=3;

title'其他選取變數方法'; run; data b; set a; proccorrdata=b; var y x1 x2; title'檢定是否有相關性'; data c; set b1; n=_n_; procunivariatenormalplot; var r; title'殘差檢定'; run; data d; set b; yhatt= -8.31205+1.22062*X1+0.20650*X2; residual=(y-yhatt); title'yhatt和殘差值'; procprintdata=d; run;

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1. (2004) SAS

2. (2009) SAS

3. NBA.COM