Example of a Half-Fraction Factorial Design

[See FACTEX3 in the SAS/QC Sample Library]Often you do not have the resources for a full factorial design. In this case, a fractional factorial design is a reasonable alternative, provided that the effects of interest can be estimated.

Box, Hunter, and Hunter (1978) describe a fractional factorial design for studying a chemical reaction to determine what percentage of the chemicals responded in a reactor. The researchers identified the following five treatment factors that were thought to influence the percentage of reactant:

• the feed rate of the chemicals (FeedRate), ranging from 10 to 15 liters per minute

• the percentage of the catalyst (Catalyst), ranging from 1% to 2%

• the agitation rate of the reactor (AgitRate), ranging from 100 to 120 revolutions per minute

• the temperature (Temperature), ranging from 140 to 180 degrees Celsius

• the concentration (Concentration), ranging from 3% to 6%

The complete factorial design requires 32 runs, but the experimenter can afford only 16 runs.

Suppose that all main effects and two-factor interactions are to be estimated. An appropriate design for this situation is a design of resolution 5 (denoted as ), in which no main effect or two-factor interaction is aliased with any other main effect or two-factor interaction but in which two-factor interactions are aliased with three-factor interactions. This design loses the ability to estimate interactions between three or more factors, but this is usually not a serious loss. For more on resolution, see the section Resolution.

You can use the following statements to construct a 16-run factorial design that has five factors and resolution 5:

proc factex;
   factors FeedRate Catalyst AgitRate Temperature Concentration;
   size design=16;
   model resolution=5;
   output out=Reaction FeedRate      nvals=(10 15)
                       Catalyst      nvals=(1 2)
                       AgitRate      nvals=(100 120)
                       Temperature   nvals=(140 180)
                       Concentration nvals=(3 6);
proc print;
run;

The design saved in the Reaction data set is listed in Figure 7.5.

Figure 7.5 Half-Fraction of a Design for Reactors

Obs	FeedRate	Catalyst	AgitRate	Temperature	Concentration
1	10	1	100	140	6
2	10	1	100	180	3
3	10	1	120	140	3
4	10	1	120	180	6
5	10	2	100	140	3
6	10	2	100	180	6
7	10	2	120	140	6
8	10	2	120	180	3
9	15	1	100	140	3
10	15	1	100	180	6
11	15	1	120	140	6
12	15	1	120	180	3
13	15	2	100	140	6
14	15	2	100	180	3
15	15	2	120	140	3
16	15	2	120	180	6

The use of a half-fraction causes some interaction terms to be confounded with each other. You can use the EXAMINE statement with the ALIASING option to determine which interaction terms are aliased, as follows:

proc factex;
   factors FeedRate Catalyst AgitRate Temperature Concentration;
   size design=16;
   model resolution=5;
   examine aliasing;
run;

The alias structure summarizes the estimability of all main effects and two- and three-factor interactions. Figure 7.6 indicates that each of the three-factor interactions is confounded with a two-factor interaction. Thus, if a particular three-factor interaction is believed to be significant, the aliased two-factor interaction cannot be estimated with this half-fraction design.