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