Original Program from program editor.
****************************************************; *** Multiple Regression Example ***; *** ***; ****************************************************; options ps=256 ls=99 nocenter nodate nonumber; title1 'Generic multiple regression introduction'; data one; infile cards missover; input Y X1 X2 X3; cards; run; 1 2 9 2 3 4 6 5 5 7 7 9 3 3 5 5 6 5 8 9 4 3 4 2 2 2 3 6 8 6 2 1 9 7 5 3 3 8 2 4 5 7 3 7 6 9 1 4 ; proc print data=one; title2 'Raw data listing'; run; title2 'All possible models with PROC REG'; proc reg data=one; model y = x1/ ss1 ss2; run; proc reg data=one; model y = x2 / ss1 ss2; run; proc reg data=one; model y = x3 / ss1 ss2; run; proc reg data=one; model y = x1 x2 / ss1 ss2; run; proc reg data=one; model y = x1 x3 / ss1 ss2; run; proc reg data=one; model y = x2 x3 / ss1 ss2; run; proc reg data=one; model y = x1 x2 x3 / ss1 ss2; run; proc mixed data=one; model y = x1 x2 x3 / solution HType=1 2 3; title2 'Multiple Regression with PROC MIXED'; run; proc glm data=one; model y = x1 x2 x3 / solution ss1 ss2 ss3 ss4; title2 'Multiple Regression with PROC GLM'; run;
Below is output from the SAS log (bold) and output from the SAS Output window.
1 ****************************************************;
2 *** Multiple Regression Example ***;
3 *** ***;
4 ****************************************************;
5 options ps=256 ls=99 nocenter nodate nonumber;
6 title1 'Generic multiple regression introduction';
7
8 data one; infile cards missover;
9 input Y X1 X2 X3;
10 cards;
NOTE: The data set WORK.ONE has 12 observations and 4 variables.
NOTE: DATA statement used (Total process time):
real time 0.03 seconds
cpu time 0.03 seconds
10 ! run;
23 ;
24 proc print data=one; title2 'Raw data listing'; run;
NOTE: There were 12 observations read from the data set WORK.ONE.
NOTE: The PROCEDURE PRINT printed page 1.
NOTE: PROCEDURE PRINT used (Total process time):
real time 0.01 seconds
cpu time 0.01 seconds
Generic multiple regression introduction
Raw data listing
Obs Y X1 X2 X3
1 1 2 9 2
2 3 4 6 5
3 5 7 7 9
4 3 3 5 5
5 6 5 8 9
6 4 3 4 2
7 2 2 3 6
8 8 6 2 1
9 9 7 5 3
10 3 8 2 4
11 5 7 3 7
12 6 9 1 4
26 title2 'All possible models with PROC REG';
27 proc reg data=one; model y = x1/ ss1 ss2; run;
NOTE: 12 observations read.
NOTE: 12 observations used in computations.
NOTE: The PROCEDURE REG printed page 2.
NOTE: PROCEDURE REG used (Total process time):
real time 0.04 seconds
cpu time 0.04 seconds
Generic multiple regression introduction
All possible models with PROC REG
The REG Procedure
Model: MODEL1
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 23.97763 23.97763 6.16 0.0325
Error 10 38.93904 3.89390
Corrected Total 11 62.91667
Root MSE 1.97330 R-Square 0.3811
Dependent Mean 4.58333 Adj R-Sq 0.3192
Coeff Var 43.05377
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| Type I SS Type II SS
Intercept 1 1.37613 1.41242 0.97 0.3529 252.08333 3.69640
X1 1 0.61089 0.24618 2.48 0.0325 23.97763 23.97763
28 proc reg data=one; model y = x2 / ss1 ss2; run;
NOTE: 12 observations read.
NOTE: 12 observations used in computations.
NOTE: The PROCEDURE REG printed page 3.
NOTE: PROCEDURE REG used (Total process time):
real time 0.03 seconds
cpu time 0.03 seconds
Generic multiple regression introduction
All possible models with PROC REG
The REG Procedure
Model: MODEL1
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 4.11526 4.11526 0.70 0.4224
Error 10 58.80141 5.88014
Corrected Total 11 62.91667
Root MSE 2.42490 R-Square 0.0654
Dependent Mean 4.58333 Adj R-Sq -0.0281
Coeff Var 52.90691
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| Type I SS Type II SS
Intercept 1 5.68743 1.49393 3.81 0.0034 252.08333 85.22336
X2 1 -0.24089 0.28795 -0.84 0.4224 4.11526 4.11526
29 proc reg data=one; model y = x3 / ss1 ss2; run;
NOTE: 12 observations read.
NOTE: 12 observations used in computations.
NOTE: The PROCEDURE REG printed page 4.
NOTE: PROCEDURE REG used (Total process time):
real time 0.01 seconds
cpu time 0.01 seconds
Generic multiple regression introduction
All possible models with PROC REG
The REG Procedure
Model: MODEL1
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 0.23689 0.23689 0.04 0.8498
Error 10 62.67978 6.26798
Corrected Total 11 62.91667
Root MSE 2.50359 R-Square 0.0038
Dependent Mean 4.58333 Adj R-Sq -0.0959
Coeff Var 54.62385
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| Type I SS Type II SS
Intercept 1 4.84809 1.54176 3.14 0.0104 252.08333 61.97728
X3 1 -0.05574 0.28671 -0.19 0.8498 0.23689 0.23689
30 proc reg data=one; model y = x1 x2 / ss1 ss2; run;
NOTE: 12 observations read.
NOTE: 12 observations used in computations.
NOTE: The PROCEDURE REG printed page 5.
NOTE: PROCEDURE REG used (Total process time):
real time 0.03 seconds
cpu time 0.03 seconds
Generic multiple regression introduction
All possible models with PROC REG
The REG Procedure
Model: MODEL1
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 24.07446 12.03723 2.79 0.1141
Error 9 38.84220 4.31580
Corrected Total 11 62.91667
Root MSE 2.07745 R-Square 0.3826
Dependent Mean 4.58333 Adj R-Sq 0.2454
Coeff Var 45.32619
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| Type I SS Type II SS
Intercept 1 1.07558 2.49743 0.43 0.6768 252.08333 0.80049
X1 1 0.63159 0.29369 2.15 0.0600 23.97763 19.95921
X2 1 0.04187 0.27955 0.15 0.8842 0.09684 0.09684
31 proc reg data=one; model y = x1 x3 / ss1 ss2; run;
NOTE: 12 observations read.
NOTE: 12 observations used in computations.
NOTE: The PROCEDURE REG printed page 6.
NOTE: PROCEDURE REG used (Total process time):
real time 0.03 seconds
cpu time 0.03 seconds
Generic multiple regression introduction
All possible models with PROC REG
The REG Procedure
Model: MODEL1
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 25.37078 12.68539 3.04 0.0980
Error 9 37.54588 4.17176
Corrected Total 11 62.91667
Root MSE 2.04249 R-Square 0.4032
Dependent Mean 4.58333 Adj R-Sq 0.2706
Coeff Var 44.56341
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| Type I SS Type II SS
Intercept 1 1.91576 1.73473 1.10 0.2981 252.08333 5.08794
X1 1 0.63161 0.25732 2.45 0.0365 23.97763 25.13390
X3 1 -0.13650 0.23621 -0.58 0.5775 1.39316 1.39316
32 proc reg data=one; model y = x2 x3 / ss1 ss2; run;
NOTE: 12 observations read.
NOTE: 12 observations used in computations.
NOTE: The PROCEDURE REG printed page 7.
NOTE: PROCEDURE REG used (Total process time):
real time 0.03 seconds
cpu time 0.03 seconds
Generic multiple regression introduction
All possible models with PROC REG
The REG Procedure
Model: MODEL1
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 4.13721 2.06860 0.32 0.7363
Error 9 58.77946 6.53105
Corrected Total 11 62.91667
Root MSE 2.55559 R-Square 0.0658
Dependent Mean 4.58333 Adj R-Sq -0.1419
Coeff Var 55.75837
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| Type I SS Type II SS
Intercept 1 5.62891 1.87021 3.01 0.0147 252.08333 59.16272
X2 1 -0.24662 0.31913 -0.77 0.4595 4.11526 3.90032
X3 1 0.01784 0.30776 0.06 0.9550 0.02195 0.02195
33 proc reg data=one; model y = x1 x2 x3 / ss1 ss2; run;
NOTE: 12 observations read.
NOTE: 12 observations used in computations.
34
NOTE: The PROCEDURE REG printed page 8.
NOTE: PROCEDURE REG used (Total process time):
real time 0.03 seconds
cpu time 0.01 seconds
Generic multiple regression introduction
All possible models with PROC REG
The REG Procedure
Model: MODEL1
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 26.18995 8.72998 1.90 0.2078
Error 8 36.72672 4.59084
Corrected Total 11 62.91667
Root MSE 2.14262 R-Square 0.4163
Dependent Mean 4.58333 Adj R-Sq 0.1974
Coeff Var 46.74817
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t| Type I SS Type II SS
Intercept 1 1.14454 2.57778 0.44 0.6688 252.08333 0.90503
X1 1 0.70578 0.32202 2.19 0.0598 23.97763 22.05274
X2 1 0.13483 0.31918 0.42 0.6838 0.09684 0.81916
X3 1 -0.18621 0.27431 -0.68 0.5164 2.11548 2.11548
35 proc mixed data=one; model y = x1 x2 x3 / solution HType=1 2 3;
36 title2 'Multiple Regression with PROC MIXED';
37 run;
NOTE: The PROCEDURE MIXED printed page 9.
NOTE: PROCEDURE MIXED used (Total process time):
real time 0.01 seconds
cpu time 0.01 seconds
Generic multiple regression introduction
Multiple Regression with PROC MIXED
The Mixed Procedure
Model Information
Data Set WORK.ONE
Dependent Variable Y
Covariance Structure Diagonal
Estimation Method REML
Residual Variance Method Profile
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Residual
Dimensions
Covariance Parameters 1
Columns in X 4
Columns in Z 0
Subjects 1
Max Obs Per Subject 12
Observations Used 12
Observations Not Used 0
Total Observations 12
Covariance Parameter Estimates
Cov Parm Estimate
Residual 4.5908
Fit Statistics
-2 Res Log Likelihood 49.7
AIC (smaller is better) 51.7
AICC (smaller is better) 52.3
BIC (smaller is better) 51.7
Solution for Fixed Effects
Standard
Effect Estimate Error DF t Value Pr > |t|
Intercept 1.1445 2.5778 8 0.44 0.6688
X1 0.7058 0.3220 8 2.19 0.0598
X2 0.1348 0.3192 8 0.42 0.6838
X3 -0.1862 0.2743 8 -0.68 0.5164
Type 1 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
X1 1 8 5.22 0.0516
X2 1 8 0.02 0.8881
X3 1 8 0.46 0.5164
Type 2 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
X1 1 8 4.80 0.0598
X2 1 8 0.18 0.6838
X3 1 8 0.46 0.5164
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
X1 1 8 4.80 0.0598
X2 1 8 0.18 0.6838
X3 1 8 0.46 0.5164
38 proc glm data=one; model y = x1 x2 x3 / solution ss1 ss2 ss3 ss4;
39 title2 'Multiple Regression with PROC GLM';
40 run;
41
NOTE: The PROCEDURE GLM printed pages 10-11.
NOTE: PROCEDURE GLM used (Total process time):
real time 0.01 seconds
cpu time 0.01 seconds
NOTE: SAS Institute Inc., SAS Campus Drive, Cary, NC USA 27513-2414
NOTE: The SAS System used:
real time 1.42 seconds
cpu time 0.43 seconds
Generic multiple regression introduction
Multiple Regression with PROC GLM
The GLM Procedure
Number of observations 12
Dependent Variable: Y
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 3 26.18994658 8.72998219 1.90 0.2078
Error 8 36.72672008 4.59084001
Corrected Total 11 62.91666667
R-Square Coeff Var Root MSE Y Mean
0.416264 46.74817 2.142625 4.583333
Source DF Type I SS Mean Square F Value Pr > F
X1 1 23.97762646 23.97762646 5.22 0.0516
X2 1 0.09683730 0.09683730 0.02 0.8881
X3 1 2.11548283 2.11548283 0.46 0.5164
Source DF Type II SS Mean Square F Value Pr > F
X1 1 22.05273729 22.05273729 4.80 0.0598
X2 1 0.81916182 0.81916182 0.18 0.6838
X3 1 2.11548283 2.11548283 0.46 0.5164
Source DF Type III SS Mean Square F Value Pr > F
X1 1 22.05273729 22.05273729 4.80 0.0598
X2 1 0.81916182 0.81916182 0.18 0.6838
X3 1 2.11548283 2.11548283 0.46 0.5164
Source DF Type IV SS Mean Square F Value Pr > F
X1 1 22.05273729 22.05273729 4.80 0.0598
X2 1 0.81916182 0.81916182 0.18 0.6838
X3 1 2.11548283 2.11548283 0.46 0.5164
Standard
Parameter Estimate Error t Value Pr > |t|
Intercept 1.144541173 2.57777758 0.44 0.6688
X1 0.705779391 0.32202071 2.19 0.0598
X2 0.134827053 0.31918191 0.42 0.6838
X3 -0.186211994 0.27431463 -0.68 0.5164