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EXAMPLE D9SAS Program data^ cement;input^ x1-x4 y;datalines;7 26^ 6 60^ 78.51 29 15 52^ 74.311 56^ 8 20 104.311 31^ 8 47^ 87.67 52^ 6 33^ 95.911 55^ 9 22 109.23 71 17^ 6 102.71 31 22 44^ 72.52 54 18 22^ 93.121 47^ 4 26 115.91 40 23 34^ 83.811 66^ 9 12 113.310 68^ 8 12 109.4;^ run ; title^ c=darkpurple h=^2 'Regression : Variable Subset Selection Techniques';symbol^ v=dot^ h=^1 c=red; proc^ reg^ corr^ ; model^ y = x1-x4/selection=f
sle= .05 ; model y = x1-x4/selection=b sls= .1 ; model y = x1-x4/selection=stepwise^ sle= .15^ sls= .15 ; model y = x1-x4/selection=rsquare^ sse^ cp;model y = x1-x4/selection=rsquare^ start=^1 stop=^3 best=^2^ sse^ mse^ aic^ cp;plot cp.np./cmallows=blue vaxis=^0 to^^8 by^^1^ ctext=blue^ caxes=darkred;title 'Regression : Variable Subset Selection Techniques';
run ;
SAS Log 2 data^ cement; 3 input^ x1-x4^ y; 4 datalines;NOTE:^ The^ data^ set^ WORK.CEMENT
has^13 observations^ and^5 variables.
NOTE:^ DATA^ statement^ used (Total process
time):
real^ time^ 0.01 secondscpu^ time^ 0.00 seconds 18 ; run; 1920 proc^ reg^ corr^ ; 21 model^ y^ =^ x1-x4/selection=f
sle=.05;
22 model^ y^ =^ x1-x4/selection=b
sls=.1;
23 model^ y^ =^ x1-x4/selection=stepwise sle=.15 sls=.15; 24 model^ y^ =^ x1-x4/selection=rsquare
sse cp;
25 model^ y^ =^ x1-x4/selection=rsquare
start=1^ stop=3^ best=
sse^ mse^ aic^ cp;
27 title^ 'Regression^ :^ Variable
Subset^ Selection Techniques';
28 run;
The^ REG^ ProcedureModel:^ MODEL1Dependent^ Variable:^ y Number of Observations Read^13 Number of Observations Used^13 Forward^ Selection: Step^1 Variable x4 Entered: R-Square^ = 0.6745^ and^ C(p)^ = 138.7308Analysis^ of^ VarianceSum^ of^ Mean Source^ DF^
Squares^ Square^ F Value
Pr^ >^ F Model^1
1831.89616^ 1831.^
22.80^ 0.
Error^11
883.86692^ 80.
Corrected^ Total^12
2715.76308Parameter StandardVariable Estimate Error^ Type^ II^ SS^ F^ Value
Pr^ >^ F Intercept^ 117.56793^ 5.
40108 499.16^ <.
x4^ -0.73816^ 0.
1831.89616^ 22.80^ 0.0006Bounds on condition^ number: 1, 1
Forward^ Selection: Step^2
Variable^ x1^ Entered:^ R-Square^
=^ 0.9725^ and C(p)^ =^ 5.4959Analysis of VarianceSum of^ Mean Source^ DF^
Squares^ Square^ F Value
Pr^ >^ F Model^2
2641.00096^ 1320.^
176.63^ <.
Error^10
74.76211^ 7.
Corrected^ Total^12
2715.76308Parameter StandardVariable Estimate Error^ Type^ II^ SS^ F^ Value
Pr^ >^ F Intercept^ 103.09738^ 2.
17615 2356.10^ <.
x1^ 1.43996^ 0.
809.10480^ 108.22^ <.
x4^ -0.61395^ 0.
1190.92464^ 159.30^ <.0001Bounds on condition number: 1.0641, 4. ----------------------------------------------------------------------------------------------------No^ other^ variable^ met
the^ 0.0500 significance level
for^ entry^ into^ the^ model. Summary of Forward Selection Variable^ Number^ Partial
Model Step^ Entered^ Vars^ In^
R-Square^ R-Square^ C(p)^
F^ Value^ Pr >^ F 1 x4^1
0.6745^ 0.6745^ 138.
22.80^ 0.
2 x1^2
0.2979^ 0.9725^ 5.
108.22^ <.
Backward^ Elimination:^ Step^1 Variable x3 Removed:^ R-Square^ =^ 0.9823^ and C(p)
=^ 3.0182Analysis of VarianceSum of Mean Source^ DF^
Squares^ Square^ F Value
Pr^ >^ F Model^3
2667.79035^ 889.^
166.83^ <.
Error^9
47.97273^ 5.
Corrected^ Total^12
2715.76308Parameter StandardVariable Estimate Error^ Type^ II^ SS^ F^ Value
Pr^ >^ F Intercept^ 71.64831^ 14.
136.81003^ 25.67^ 0.
x1^ 1.45194^ 0.
820.90740^ 154.01^ <.
x2^ 0.41611^ 0.
26.78938^ 5.03^ 0.
x4^ -0.23654^ 0.
9.93175^ 1.86^ 0.2054Bounds on condition number: 18.94,^ 116.
Backward Elimination: Step 2Variable x4 Removed:^ R-Square^ =^ 0.9787^ and C(p)
=^ 2.6782Analysis of VarianceSum of Mean Source^ DF^
Squares^ Square^ F Value
Pr^ >^ F Model^2
2657.85859^ 1328.^
229.50^ <.
Error^10
57.90448^ 5.
Corrected^ Total^12
2715.76308Parameter StandardVariable Estimate Error^ Type^ II^ SS^ F^ Value
Pr^ >^ F Intercept^ 52.57735^ 2.
3062.60416^ 528.91^ <.
x1^ 1.46831^ 0.
848.43186^ 146.52^ <.
x2^ 0.66225^ 0.
1207.78227^ 208.58^ <.0001Bounds on condition number: 1.0551, 4. ----------------------------------------------------------------------------------------------------All^ variables^
left in the^ model are^ significant
at^ the 0.1000^ level. Summary of Backward Elimination Variable^ Number^ Partial
Model Step^ Removed^ Vars^ In^
R-Square^ R-Square^ C(p)^
F^ Value^ Pr >^ F 1 x3^3
0.0000^ 0.9823^ 3.
0.02^ 0.
2 x4^2
0.0037^ 0.9787^ 2.
1.86^ 0.
Stepwise^ Selection: Step 2Variable x1 Entered:^ R-Square^ =^ 0.9725^ and C(p)
=^ 5.4959Analysis of VarianceSum of Mean Source^ DF^
Squares^ Square^ F Value
Pr^ >^ F Model^2
2641.00096^ 1320.^
176.63^ <.
Error^10
74.76211^ 7.
Corrected^ Total^12
2715.76308Parameter StandardVariable Estimate Error^ Type^ II^ SS^ F^ Value
Pr^ >^ F Intercept^ 103.09738^ 2.
17615 2356.10^ <.
x1^ 1.43996^ 0.
809.10480^ 108.22^ <.
x4^ -0.61395^ 0.
1190.92464^ 159.30^ <.0001Bounds on condition number: 1.0641, 4.
Stepwise^ Selection: Step 3Variable x2 Entered:^ R-Square^ =^ 0.9823^ and C(p)
=^ 3.0182Analysis of VarianceSum of Mean Source^ DF^
Squares^ Square^ F Value
Pr^ >^ F Model^3
2667.79035^ 889.^
166.83^ <.
Error^9
47.97273^ 5.
Corrected^ Total^12
Parameter^ StandardVariable Estimate^ Error^ Type^
II^ SS^ F^ Value^ Pr^ >^ F Intercept^ 71.64831^ 14.
136.81003^ 25.67^ 0.
x1^ 1.45194^ 0.
820.90740^ 154.01^ <.
x2^ 0.41611^ 0.
26.78938^ 5.03^ 0.
x4^ -0.23654^ 0.
9.93175^ 1.86^ 0.2054Bounds on condition number: 18.94,^ 116.
Stepwise^ Selection: Step 4Variable x4 Removed:^ R-Square^ =^ 0.9787^ and C(p)
=^ 2.6782Analysis of VarianceSum of Mean Source^ DF^
Squares^ Square^ F Value
Pr^ >^ F Model^2
2657.85859^ 1328.^
229.50^ <.
Error^10
57.90448^ 5.
Corrected^ Total^12
2715.76308Parameter StandardVariable Estimate Error^ Type^ II^ SS^ F^ Value
Pr^ >^ F Intercept^ 52.57735^ 2.
3062.60416^ 528.91^ <.
x1^ 1.46831^ 0.
848.43186^ 146.52^ <.
x2^ 0.66225^ 0.
1207.78227^ 208.58^ <.0001Bounds on condition number: 1.0551, 4. ----------------------------------------------------------------------------------------------------All^ variables^
left in the^ model are^ significant
at^ the 0.1500^ level. No^ other^ variable^ met^ the 0.
significance^ level^ for^ entry into
the model.
Regression^ : Variable Subset^ Selection
Techniques^
The^ REG^ ProcedureModel:^ MODEL4Dependent^ Variable:^ y R-Square^ Selection MethodNumber of Observations Read^13 Number of Observations Used^13 Number inModel R-Square C(p)^ SSE^ Variables
in Model 1 0.6745^ 138.^
883.86692^ x 1 0.6663^ 142.^
906.33634^ x 1 0.5339^ 202.^
1265.68675^ x 1 0.2859^ 315.^
1939.40047^ x ----------------------------------------------------------------------^2 0.9787^ 2.
57.90448^ x1^ x2 2 0.9725 5.4959 74.76211^ x1^ x4 2 0.9353 22.3731 175.73800^ x3^ x4 2 0.8470 62.4377 415.44273^ x2^ x3 2 0.6801 138.2259 868.88013^ x2^ x4 2 0.5482 198.0947 1227.07206^ x1^ x ----------------------------------------------------------------------^3 0.9823^ 3.
47.97273^ x1^ x2^ x4 3 0.9823 3.0413 48.11061^ x1^ x2^ x3 3 0.9813 3.4968 50.83612^ x1^ x3^ x4 3 0.9728 7.3375 73.81455^ x2^ x3^ x ----------------------------------------------------------------------^4 0.9824^ 5.
47.86364^ x1^ x2^ x3^ x
Regression^ : Variable Subset^ Selection
Techniques^ Number^ inModel^ R-Square^ C(p)
AIC^ MSE^
- The REG ProcedureModel: MODEL5Dependent Variable: y R-Square Selection MethodNumber of Observations Read^13 Number of Observations Used
- 1 0.6745 138. SSE Variables in Model
- 58.8516 80.35154 883.
- x
- 1 0.6663 142.
- 59.1780 82.39421 906.
- x
- ---------------------------------------------------------------------------------------------------^2 0.9787 2.
- 25.4200 5.
- 57.90448 x1 x
- 2 0.9725 5.
- 28.7417 7.47621 74.
- x1 x
- ---------------------------------------------------------------------------------------------------^3 0.9823 3.
- 24.9739 5.
- 47.97273 x1 x2 x
- 3 0.9823 3.
- 25.0112 5.34562 48.
- x1 x2 x
