7.2.3. Stepwise Regression
Stepwise Regression provides an answer to the question of which independent variables to include in the regression equation.
The simplest way to isolate the effects of various independent variables on the variation of dependent variable would be to start with one independent variable and run a series of regressions adding one independent variable at a time. An alternative would be to start with all independent variables and omit one at a time. Indeed, these are the two basic procedures most commonly used in Stepwise Regression, but with a difference. Rather than adding or omitting variables randomly it is possible to introduce a statistically meaningful criterion to rank the sequence. The enter/omit criteria used here are the Ftoenter, Ftoremove and Tolerance parameters.
As in Linear Regression, it is possible to create interaction terms, dummy variables, lag/lead terms, select multiple dependent variables and run regressions on subsamples defined by several factor columns (see 7.2.1.1. Linear Regression Variable Selection). However, a weights option is not included. The set of independent variables selected or created are the candidates for inclusion in the regression equation. The stepwise procedure will not consider columns that are not in the Variables Selected list.
7.2.3.1. Stepwise Selection Criteria
The next dialogue is for selecting the Tolerance, Ftoenter and Ftoremove thresholds. One of Forward Selection or Backward Selection methods is also specified on this dialogue.
The values suggested by the program are the most commonly used limits. Of course, it is possible to enter any value of choice by editing the number in the field. UNISTAT allows entry of Fvalues only as enter / remove thresholds. If you wish to enter tail probability values instead, the corresponding Fvalues can be calculated easily using the Statistics 1 → Distribution Functions → Critical Value procedure. The complement of the desired tail probability value (1 – α) should be entered in the Probability dialogue, and numerator and denominator degrees of freedom should be entered as 1 and 100,000 (representing infinity) respectively. The critical value obtained in this way can then be used in the Stepwise Regression procedure.
FtoEnter: The Ftoenter statistic of an independent variable is the Fstatistic for testing the significance of the regression coefficient it would have if it were in the regression equation. If this calculated value is above the one specified by the user, then the variable can enter the equation. The default value is 3.8416, corresponding to a tail probability value of 0.05 (with 1 and 100,000 degrees of freedom) and it must always be greater than the Ftoremove value. If you wish to change this default value permanently, enter and edit the following line in the [Options] section of Documents\Unistat10\Unistat10.ini file:
StepwiseFtoEnter=3.8416
FtoRemove: The Ftoremove statistic of an independent variable which is already in the regression equation is the Fstatistic for testing the significance of its regression coefficient. If this calculated value is below the one specified by the user then the variable is removed from the equation. The default value is 2.7056, corresponding to a tail probability value of 0.10 (with 1 and 100,000 degrees of freedom) and it must always be less than the Ftoenter value. If you wish to change this default value permanently, enter and edit the following line in the [Options] section of Documents\Unistat10\Unistat10.ini file:
StepwiseFtoRemove=2.7056
Tolerance: In order to avoid highly correlated variables and also to prevent accumulation of rounding errors, a Tolerance value is specified. The Tolerance of a variable which is not in the equation is defined as 1 – Rsquared where R is the multiple correlation between the variable and all variables which are in the regression equation. If you wish to change this default value permanently, enter and edit the following line in the [Options] section of Unistat10.ini:
StepwiseTolerance=0.001
Forward/Backward Selection: If the Forward Selection method is employed, then the program will first run a regression with the most likely candidate, and then successively introduce other variables or omit existing ones. If the Backward Selection method is selected, then the program will first run a regression with all independent variables included and then proceed with the omission process. In this case, the output will also include a full regression output in the beginning.
It is important to emphasise that neither Ftoenter or Ftoremove, nor the Tolerance of a variable (either in the equation or not) remains the same when a variable is added to or removed from the regression equation. Therefore, whenever an addition or omission takes place, all variables, regardless of being in the equation or not, are made subject to the above checks. When the last of the independent variables is tried for entry or removal and no variables can be entered or removed, then the selection process is terminated.
7.2.3.2. Stepwise Regression Output Options
The full output can be substantial, as a large amount of statistics are reported for each step. These include the standard error, multiple correlation, Rsquared, adjusted Rsquared, change in Rsquared, Analysis of Variance. The regression coefficient, its standard error, tstatistic, its tail probability and the calculated Ftoremove value are displayed for each independent variable. Partial correlation, Tolerance and Ftoenter values of variables which are not in the equation are also displayed.
At the end of the selection process, a summary table gives the multiple correlation, Rsquared and Fstatistic for each step.
7.2.3.3. Stepwise Regression Example
Example 20.1e on p. 436 from Zar, J. H. (2010).
Open REGRESS, select Statistics 1 → Regression Analysis → Stepwise Regression and select temperature, cm, mm and min (C1 to C4) as [Variable]s and ml (C5) as [Dependent]. Select Backward Selection and accept the Tolerance levels given in the next dialogue to obtain the following output:
Stepwise Regression
Dependent Variable: ml
Valid Number of Cases: 33, 0 Omitted
Backward Selection
Tolerance: 0.001
FtoEnter: 3.8416 (5.0%)
FtoRemove: 2.7056 (10.0%)
All uncorrelated variables entered
Standard Error 
Multiple Correlation 
Rsquared 
Adjusted Rsquared 
Change in Rsquared 
0.4238 
0.8117 
0.6589 
0.6102 
0.6589 
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Regression 
9.717 
4 
2.429 
13.524 
0.0000 
Error 
5.030 
28 
0.180 


Variables in Equation 
Coefficient 
Std Error 
tStatistic 
Prob 
FtoRemove 
Constant 
2.9583 




Temperature 
0.1293 
0.0213 
6.0751 
0.0000 
36.9063 
cm 
0.0188 
0.0563 
0.3338 
0.7410 
0.1114 
mm 
0.0462 
0.2073 
0.2230 
0.8252 
0.0497 
min 
0.2088 
0.0670 
3.1141 
0.0042 
9.6979 
Step 1: Variable Removed: mm
Standard Error 
Multiple Correlation 
Rsquared 
Adjusted Rsquared 
Change in Rsquared 
0.4168 
0.8114 
0.6583 
0.6230 
0.0006 
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Regression 
9.708 
3 
3.236 
18.625 
0.0000 
Error 
5.039 
29 
0.174 


Variables in Equation 
Coefficient 
Std Error 
tStatistic 
Prob 
FtoRemove 
Constant 
2.6725 




Temperature 
0.1305 
0.0203 
6.4232 
0.0000 
41.2572 
cm 
0.0154 
0.0533 
0.2892 
0.7745 
0.0837 
min 
0.2045 
0.0632 
3.2356 
0.0030 
10.4694 
Variables not in Equation 
Partial Corr 
Tolerance 
FtoEnter 
mm 
0.0421 
0.8518 
0.0497 
Step 2: Variable Removed: cm
Standard Error 
Multiple Correlation 
Rsquared 
Adjusted Rsquared 
Change in Rsquared 
0.4104 
0.8108 
0.6573 
0.6345 
0.0010 
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Regression 
9.694 
2 
4.847 
28.775 
0.0000 
Error 
5.053 
30 
0.168 


Variables in Equation 
Coefficient 
Std Error 
tStatistic 
Prob 
FtoRemove 
Constant 
2.5520 




Temperature 
0.1324 
0.0189 
6.9993 
0.0000 
48.9907 
min 
0.2013 
0.0613 
3.2850 
0.0026 
10.7910 
Variables not in Equation 
Partial Corr 
Tolerance 
FtoEnter 
mm 
0.0261 
0.9176 
0.0198 
cm 
0.0536 
0.8652 
0.0837 
Summary Table
Dependent Variable: ml
Step 
In/Out 
Variable 
Multiple Corr 
Rsquared 
FStat 
Prob 
1 
Out 
mm 
0.8114 
0.6583 
18.6251 
0.0000 
2 
Out 
cm 
0.8108 
0.6573 
28.7748 
0.0000 