data curve;
input (y x) (2.0 2.0);
x2=x**2;
x3=x**3;
cards;
3 2
4 2
6 2
5 2
7 4
10 4
10 4
13 6
14 6
15 6
16 8
17 8
21 8
1810
1910
2010
1912
2012
2112
2014
2214
2314
2116
2216
2316
2218
2318
2418
2220
2320
2420
2222
2322
2422
2024
2124
2224
1726
1826
1926
1528
1628
1728
;
PROC REG SIMPLE CORR;
MODEL Y=X / ADJRSQ STB;
PLOT Y*X RESIDUAL.*X RESIDUAL.*PREDICTED. / HPLOTS=2 VPLOTS =2;
MODEL Y=X X2 X3 / ADJRSQ STB SELECTION=FORWARD;
PLOT RESIDUAL.*X RESIDUAL.*PREDICTED. / HPLOTS=2 VPLOTS =2;
MODEL Y=X X2 / ADJRSQ STB SELECTION=FORWARD;
PLOT RESIDUAL.*X RESIDUAL.*PREDICTED. / HPLOTS=2 VPLOTS =2;
RUN;
The SAS System 18:38 Tuesday, December 21, 1999 1
Descriptive Statistics
Variables Sum Mean Uncorrected SS
INTERCEP 43 1 43
X 632 14.697674419 12184
Y 761 17.697674419 14901
X2 12184 283.34883721 6128992
X3 264608 6153.6744186 3665720704
Variables Variance Std Deviation
INTERCEP 0 0
X 68.930232558 8.3024232943
Y 34.120708749 5.841293414
X2 63730.232558 252.44847506
X3 48509743.415 6964.8936399
Correlation
CORR X Y X2 X3
X 1.0000 0.6378 0.9716 0.9223
Y 0.6378 1.0000 0.4456 0.3089
X2 0.9716 0.4456 1.0000 0.9864
X3 0.9223 0.3089 0.9864 1.0000
The SAS System 18:38 Tuesday, December 21, 1999 2
Model: MODEL1
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 1 582.91592 582.91592 28.112 0.0001
Error 41 850.15385 20.73546
C Total 42 1433.06977
Root MSE 4.55362 R-square 0.4068
Dep Mean 17.69767 Adj R-sq 0.3923
C.V. 25.73005
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 11.102564 1.42458391 7.794 0.0001
X 1 0.448718 0.08463058 5.302 0.0001
Standardized
Variable DF Estimate
INTERCEP 1 0.00000000
X 1 0.63777764
The SAS System 18:38 Tuesday, December 21, 1999 3
-+----+----+----+----+----+----+-- -+----+----+----+----+----+----+-
| | | |
25 + + 10 + +
| 1 1 1 | | |
| 2 2 2 2 2 1 | | |
| 1 1 1 1 | | |
20 + 1 1 1 1 + | 1 1 |
| 1 1 1 | 5 + 1 1 1 1 +
Y | 1 1 2 1 | R | 1 1 1 1 1 |
| 1 1 | E | 1 1 1 1 1 1 1 |
15 + 1 1 + S | 1 1 1 1 |
| 1 | I | 1 1 1 |
| 1 | D 0 + 1 1 +
| | U | 1 1 |
10 + 2 + A | 1 |
| | L | 2 |
| 1 | | 1 |
| 1 | -5 + 1 +
5 + 1 + | 1 1 1 |
| 1 | | 1 1 |
| 1 | | 1 1 |
| | | 1 1 |
0 + + -10 + +
| | | |
-+----+----+----+----+----+----+-- -+----+----+----+----+----+----+-
0 5 10 15 20 25 30 0 5 10 15 20 25 30
X X
-+----+----+----+----+----+----+-
| |
10 + +
| |
| |
| |
| 1 1 |
5 + 1 1 1 1 +
R | 1 1 1 1 1 |
E | 1 1 1 1 1 1 1 |
S | 1 1 1 1 |
I | 1 1 1 |
D 0 + 1 1 +
U | 1 1 |
A | 1 |
L | 2 |
| 1 |
-5 + 1 +
|1 1 1 |
|1 1 |
|1 1 |
|1 1 |
-10 + +
| |
-+----+----+----+----+----+----+-
12 14 16 18 20 22 24
PRED
The SAS System 18:38 Tuesday, December 21, 1999 4
Forward Selection Procedure for Dependent Variable Y
Step 1 Variable X Entered R-square = 0.40676032 C(p) =421.85048378
DF Sum of Squares Mean Square F Prob>F
Regression 1 582.91592129 582.91592129 28.11 0.0001
Error 41 850.15384615 20.73545966
Total 42 1433.06976744
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 11.10256410 1.42458391 1259.45941714 60.74 0.0001
X 0.44871795 0.08463058 582.91592129 28.11 0.0001
Bounds on condition number: 1, 1
--------------------------------------------------------------------------------
Step 2 Variable X2 Entered R-square = 0.94871141 C(p) = 2.84287117
DF Sum of Squares Mean Square F Prob>F
Regression 2 1359.56963809 679.78481905 369.95 0.0001
Error 40 73.50012935 1.83750323
Total 42 1433.06976744
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 0.23652806 0.67763418 0.22387358 0.12 0.7289
X 2.57643943 0.10651623 1075.06926573 585.07 0.0001
X2 -0.07201908 0.00350306 776.65371681 422.67 0.0001
Bounds on condition number: 17.87568, 71.5027
--------------------------------------------------------------------------------
Step 3 Variable X3 Entered R-square = 0.94979641 C(p) = 4.00000000
DF Sum of Squares Mean Square F Prob>F
Regression 3 1361.12452453 453.70817484 245.95 0.0001
Error 39 71.94524291 1.84474982
Total 42 1433.06976744
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP -0.46672105 1.02359880 0.38352416 0.21 0.6509
X 2.82999335 0.29608270 168.53194413 91.36 0.0001
X2 -0.09283382 0.02294212 30.20527830 16.37 0.0002
X3 0.00046711 0.00050879 1.55488644 0.84 0.3642
Bounds on condition number: 763.7028, 3561.54
--------------------------------------------------------------------------------
All variables have been entered into the model.
The SAS System 18:38 Tuesday, December 21, 1999 5
Summary of Forward Selection Procedure for Dependent Variable Y
Variable Number Partial Model
Step Entered In R**2 R**2 C(p) F Prob>F
1 X 1 0.4068 0.4068 421.8505 28.1120 0.0001
2 X2 2 0.5420 0.9487 2.8429 422.6679 0.0001
3 X3 3 0.0011 0.9498 4.0000 0.8429 0.3642
The SAS System 18:38 Tuesday, December 21, 1999 6
Model: MODEL2
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 3 1361.12452 453.70817 245.946 0.0001
Error 39 71.94524 1.84475
C Total 42 1433.06977
Root MSE 1.35822 R-square 0.9498
Dep Mean 17.69767 Adj R-sq 0.9459
C.V. 7.67454
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 -0.466721 1.02359880 -0.456 0.6509
X 1 2.829993 0.29608270 9.558 0.0001
X2 1 -0.092834 0.02294212 -4.046 0.0002
X3 1 0.000467 0.00050879 0.918 0.3642
Standardized
Variable DF Estimate
INTERCEP 1 0.00000000
X 1 4.02236303
X2 1 -4.01208329
X3 1 0.55695811
The SAS System 18:38 Tuesday, December 21, 1999 7
-+----+----+----+----+----+----+-- --+------+------+------+------+---
| | | |
6 + + 6 + +
| | | |
| | | |
| 1 | | 1 |
4 + + 4 + +
| | | |
R | | R | |
E | | E | |
S 2 + 1 + S 2 + 1 +
I | 1 1 1 | I | 1 1 1 |
D | 1 1 1 1 1 1 | D | 1 1 1 21 |
U | 2 1 1 1 1 1 | U | 2 1 1 1 1 1 |
A 0 + 1 1 1 1 1 1 1 1 + A 0 + 1 1 1 1 22 +
L | 1 1 1 1 1 | L | 1 1 1 1 1 |
| 1 1 1 1 1 1 | | 1 1 1 1 2 |
| 1 | | 1 |
-2 + 1 1 1 1 + -2 + 1 1 11 +
| 1 | | 1 |
| | | |
| | | |
-4 + + -4 + +
| | | |
-+----+----+----+----+----+----+-- --+------+------+------+------+---
0 5 10 15 20 25 30 5 10 15 20 25
X PRED
The SAS System 18:38 Tuesday, December 21, 1999 8
Forward Selection Procedure for Dependent Variable Y
Step 1 Variable X Entered R-square = 0.40676032 C(p) =423.66794559
DF Sum of Squares Mean Square F Prob>F
Regression 1 582.91592129 582.91592129 28.11 0.0001
Error 41 850.15384615 20.73545966
Total 42 1433.06976744
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 11.10256410 1.42458391 1259.45941714 60.74 0.0001
X 0.44871795 0.08463058 582.91592129 28.11 0.0001
Bounds on condition number: 1, 1
--------------------------------------------------------------------------------
Step 2 Variable X2 Entered R-square = 0.94871141 C(p) = 3.00000000
DF Sum of Squares Mean Square F Prob>F
Regression 2 1359.56963809 679.78481905 369.95 0.0001
Error 40 73.50012935 1.83750323
Total 42 1433.06976744
Parameter Standard Type II
Variable Estimate Error Sum of Squares F Prob>F
INTERCEP 0.23652806 0.67763418 0.22387358 0.12 0.7289
X 2.57643943 0.10651623 1075.06926573 585.07 0.0001
X2 -0.07201908 0.00350306 776.65371681 422.67 0.0001
Bounds on condition number: 17.87568, 71.5027
--------------------------------------------------------------------------------
All variables have been entered into the model.
Summary of Forward Selection Procedure for Dependent Variable Y
Variable Number Partial Model
Step Entered In R**2 R**2 C(p) F Prob>F
1 X 1 0.4068 0.4068 423.6679 28.1120 0.0001
2 X2 2 0.5420 0.9487 3.0000 422.6679 0.0001
The SAS System 18:38 Tuesday, December 21, 1999 9
Model: MODEL3
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 2 1359.56964 679.78482 369.950 0.0001
Error 40 73.50013 1.83750
C Total 42 1433.06977
Root MSE 1.35555 R-square 0.9487
Dep Mean 17.69767 Adj R-sq 0.9461
C.V. 7.65945
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 0.236528 0.67763418 0.349 0.7289
X 1 2.576439 0.10651623 24.188 0.0001
X2 1 -0.072019 0.00350306 -20.559 0.0001
Standardized
Variable DF Estimate
INTERCEP 1 0.00000000
X 1 3.66197847
X2 1 -3.11251373
The SAS System 18:38 Tuesday, December 21, 1999 10
-+----+----+----+----+----+----+-- --+------+------+------+------+---
| | | |
6 + + 6 + +
| | | |
| 1 | | 1 |
| | | |
4 + + 4 + +
| | | |
R | | R | |
E | | E | |
S 2 + 1 1 + S 2 + 1 1 +
I | 1 | I | 1 |
D | 1 1 1 1 1 1 1 1 | D | 1 1 11 1 21 |
U | 2 1 1 1 | U | 2 1 1 1 |
A 0 + 1 1 1 1 1 1 1 1 1 1 + A 0 + 1 1 11 1 1 22 +
L | 1 1 1 | L | 1 1 1 |
| 1 1 1 1 1 1 | | 1 1 1 1 2 |
| 1 1 | | 1 1 |
-2 + 1 1 1 1 + -2 + 1 1 11 +
| 1 | | 1 |
| | | |
| | | |
-4 + + -4 + +
| | | |
-+----+----+----+----+----+----+-- --+------+------+------+------+---
0 5 10 15 20 25 30 5 10 15 20 25
X PRED