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