[See FACTEX10 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 into a design of resolution . 1 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 fraction of a factorial design with factors A, B, C, D, E, and F. The following statements construct a 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 fraction of a complete factorial—that is, 8 (=). The design, which is saved in the data set Original, is displayed in Output 7.4.1.
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.
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.
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 |