Example 7.9 Mixed-Level Design by Collapsing Factors
[See FACTEX6C in the SAS/QC Sample Library]You can construct a mixed-level design by collapsing factors—that is, by replacing a factor with 
levels by a factor with 
levels, where 
. Orthogonality is retained in the sense that estimates of different effects are uncorrelated, although not all estimates have equal variance; refer to Section 6 of Chakravarti (1956). This method has been used by Addelman (1962) to derive main effects plans for factors with mixed numbers of levels and by Margolin (1967) to construct plans that consider two-factor interactions.
You can use the value specification in the OUTPUT statement as a convenient tool for collapsing factors. For example, the following statements create a 27-run design for two 2-level factors (x1 and x2) and two 3-level factors (x3 and x4) such that all main effects and two-factor interactions are uncorrelated:
proc factex;
factors x1-x4 / nlev = 3;
size design=27;
model r=4;
output out=MixedLevel x1 nvals=(-1 1 -1)
x2 nvals=(-1 1 -1);
proc print data=MixedLevel;
run;
The mixed-level design is a three-quarter fraction with resolution 5; refer to Margolin (1967). The design is displayed in Output 7.9.1.
Design of Resolution V in 27 Runs
| Obs | x1 | x2 | x3 | x4 |
|---|---|---|---|---|
| 1 | -1 | -1 | -1 | -1 |
| 2 | -1 | -1 | 0 | 1 |
| 3 | -1 | -1 | 1 | 0 |
| 4 | -1 | 1 | -1 | 1 |
| 5 | -1 | 1 | 0 | 0 |
| 6 | -1 | 1 | 1 | -1 |
| 7 | -1 | -1 | -1 | 0 |
| 8 | -1 | -1 | 0 | -1 |
| 9 | -1 | -1 | 1 | 1 |
| 10 | 1 | -1 | -1 | 1 |
| 11 | 1 | -1 | 0 | 0 |
| 12 | 1 | -1 | 1 | -1 |
| 13 | 1 | 1 | -1 | 0 |
| 14 | 1 | 1 | 0 | -1 |
| 15 | 1 | 1 | 1 | 1 |
| 16 | 1 | -1 | -1 | -1 |
| 17 | 1 | -1 | 0 | 1 |
| 18 | 1 | -1 | 1 | 0 |
| 19 | -1 | -1 | -1 | 0 |
| 20 | -1 | -1 | 0 | -1 |
| 21 | -1 | -1 | 1 | 1 |
| 22 | -1 | 1 | -1 | -1 |
| 23 | -1 | 1 | 0 | 1 |
| 24 | -1 | 1 | 1 | 0 |
| 25 | -1 | -1 | -1 | 1 |
| 26 | -1 | -1 | 0 | 0 |
| 27 | -1 | -1 | 1 | -1 |