The FACTEX Procedure

Example 7.4 Fold-Over Design

Note: See A Fold-Over Design in the SAS/QC Sample Library.

Folding over a fractional factorial design is a method for breaking the links between aliased effects in a design. Folding over a design means adding a new fraction identical to the original fraction except that the signs of all the factors are reversed. The new fraction is called a fold-over design. Combining a fold-over design with the original fraction converts a design of odd resolution r into a design of resolution r + 1. [16] For example, folding over a resolution 3 design yields a resolution 4 design. You can use the FACTEX procedure to construct the original design fraction and a DATA step to generate the fold-over design.

Consider a $\frac{1}{8}$ fraction of a $2^6$ factorial design with factors A, B, C, D, E, and F. The following statements construct a $2^{6-3}_\mr {III}$ design: 

proc factex;
   factors A B C D E F;
   size fraction=8;
   model resolution=3;
   examine aliasing;
   output out=Original;
run;

title 'Original Design';
proc print data=Original;
run;

The option FRACTION=8 of the SIZE statement specifies a $\frac{1}{8}$ fraction of a complete factorial—that is, 8 (=$\frac{1}{8}2^6$). The design, which is saved in the data set Original, is displayed in Output 7.4.1.

Output 7.4.1: A $2^{6-3}_\mr {III}$ Design

Original Design

Obs A B C D E F
1 -1 -1 -1 -1 1 1
2 -1 -1 1 1 -1 -1
3 -1 1 -1 1 -1 1
4 -1 1 1 -1 1 -1
5 1 -1 -1 1 1 -1
6 1 -1 1 -1 -1 1
7 1 1 -1 -1 -1 -1
8 1 1 1 1 1 1 




            

            


            
Since the design is of resolution 3, the alias structure in Output 7.4.2 indicates that all the main effects are confounded with the two-factor interactions. 
            

            
  

            

               
Output 7.4.2: Alias Structure for a $2^{6-3}_\mr {III}$ Design
               

               

                  

                     
                     

                        
                        
                        
The FACTEX Procedure

                        
                        

                        
                        

                           
                           

                              
                              




Aliasing Structure
                                       
                                    




A = C*F = D*E
                                       
                                    



B = C*E = D*F
                                       
                                    



C = A*F = B*E
                                       
                                    



D = A*E = B*F
                                       
                                    



E = A*D = B*C
                                       
                                    



F = A*C = B*D
                                       
                                    



A*B = C*D = E*F
                                       
                                    






                           
                        

                        



                     
                     
                  





            

            


            
To separate the main effects and the two-factor interactions, augment the original design with a 1/8 fraction in which the
               signs of all the factors are reversed. The combined design (original design and fold-over design) of resolution 4 breaks the
               alias links between the main effects and the two-factor interactions. The fold-over design can be created by using the following
               DATA step: 
            


data FoldOver;
   set Original;
   A=-A; B=-B; C=-C;
   D=-D; E=-E; F=-F;
run;
title 'Fold-Over Design';
proc print data=FoldOver;
run;



Here, the DATA step creates the fold-over fraction by reversing the signs of the values of the factors in the original fraction.
               The fold-over design is displayed in Output 7.4.3. 
            

            
  

            

               
Output 7.4.3: A $2^{6-3}_\mr {III}$ Design with Signs Reversed
               

               

                  

                     
                     

                        
                        
                        

Fold-Over Design
                              
                           



                           
                           

                              
                              












Obs
                                       
                                       A
                                       
                                       B
                                       
                                       C
                                       
                                       D
                                       
                                       E
                                       
                                       F
                                       
                                    




1
                                       
                                       1
                                       
                                       1
                                       
                                       1
                                       
                                       1
                                       
                                       -1
                                       
                                       -1
                                       
                                    



2
                                       
                                       1
                                       
                                       1
                                       
                                       -1
                                       
                                       -1
                                       
                                       1
                                       
                                       1
                                       
                                    



3
                                       
                                       1
                                       
                                       -1
                                       
                                       1
                                       
                                       -1
                                       
                                       1
                                       
                                       -1
                                       
                                    



4
                                       
                                       1
                                       
                                       -1
                                       
                                       -1
                                       
                                       1
                                       
                                       -1
                                       
                                       1
                                       
                                    



5
                                       
                                       -1
                                       
                                       1
                                       
                                       1
                                       
                                       -1
                                       
                                       -1
                                       
                                       1
                                       
                                    



6
                                       
                                       -1
                                       
                                       1
                                       
                                       -1
                                       
                                       1
                                       
                                       1
                                       
                                       -1
                                       
                                    



7
                                       
                                       -1
                                       
                                       -1
                                       
                                       1
                                       
                                       1
                                       
                                       1
                                       
                                       1
                                       
                                    



8
                                       
                                       -1
                                       
                                       -1
                                       
                                       -1
                                       
                                       -1
                                       
                                       -1
                                       
                                       -1
                                       
                                    






                           
                        

                        



                     
                     
                  





            

            


            

               
            

            





                  
[16] This is not true if the original design has even resolution.