The FACTEX Procedure

Example 7.8 Mixed-Level Design Using Pseudo-Factors

Note: See Mixed-Level Designs Using Pseudo-Factors in the SAS/QC Sample Library.

If the numbers of levels for the factors of the mixed-level design are all powers of the same prime power q, you can construct the design by using pseudo-factors, where the levels of k q-level pseudo-factors are associated with the levels of a single derived factor with $q^{k}$ levels. Refer to Section 5 of Chakravarti (1956) and see the section Types of Factors for details.

For example, the following statements create a design for one 4-level factor (A) and three 2-level factors (B, C, and D) in 16 runs (a half replicate):

proc factex;
   factors A1 A2 B C D;
   model estimate     =(B C D   A1|A2                  )
         nonnegligible=(B|C|D@2 A1|A2|B A1|A2|C A1|A2|D);
   size design=16;
   output out=DesignA [A1 A2]=A cvals = ('A' 'B' 'C' 'D');
run;
proc print;
   var A B C D;
run;

The levels of two 2-level pseudo-factors (A1 and A2) are used to represent the four levels of A. Hence the three degrees of freedom associated with A are given by the main effects of A1 and A2 and their interaction A1*A2, and you can thus refer to (A1|A2) as the main effect of A.

The MODEL statement specifies that the main effects of all factors are to be estimable, and that all of the two-factor interactions between B, C, and D, in addition to the interactions between each of these and (A1|A2), are to be nonnegligible. As a result, the mixed-level design has resolution 4. The design is saved in the data set DesignA, combining the levels of the two pseudo-factors, A1 and A2, to obtain the levels of the 4-level factor A. The data set DesignA is listed in Output 7.8.1.

Output 7.8.1: $4\times 2^3$ Design of Resolution 4 in 16 Runs

Obs	A	B	C	D
1	A	-1	-1	1
2	A	-1	1	-1
3	A	1	-1	-1
4	A	1	1	1
5	C	-1	-1	-1
6	C	-1	1	1
7	C	1	-1	1
8	C	1	1	-1
9	B	-1	-1	-1
10	B	-1	1	1
11	B	1	-1	1
12	B	1	1	-1
13	D	-1	-1	1
14	D	-1	1	-1
15	D	1	-1	-1
16	D	1	1	1

Obs	A	B	C	D
1	A	-1	-1	1
2	A	-1	1	-1
3	A	1	-1	-1
4	A	1	1	1
5	C	-1	-1	-1
6	C	-1	1	1
7	C	1	-1	1
8	C	1	1	-1
9	B	-1	-1	-1
10	B	-1	1	1
11	B	1	-1	1
12	B	1	1	-1
13	D	-1	-1	1
14	D	-1	1	-1
15	D	1	-1	-1
16	D	1	1	1

Obs	A	B	C	D
1	A	-1	-1	1
2	A	-1	1	-1
3	A	1	-1	-1
4	A	1	1	1
5	C	-1	-1	-1
6	C	-1	1	1
7	C	1	-1	1
8	C	1	1	-1
9	B	-1	-1	-1
10	B	-1	1	1
11	B	1	-1	1
12	B	1	1	-1
13	D	-1	-1	1
14	D	-1	1	-1
15	D	1	-1	-1
16	D	1	1	1

Obs	A	B	C	D
1	A	-1	-1	1
2	A	-1	1	-1
3	A	1	-1	-1
4	A	1	1	1
5	C	-1	-1	-1
6	C	-1	1	1
7	C	1	-1	1
8	C	1	1	-1
9	B	-1	-1	-1
10	B	-1	1	1
11	B	1	-1	1
12	B	1	1	-1
13	D	-1	-1	1
14	D	-1	1	-1
15	D	1	-1	-1
16	D	1	1	1