## Fractional Factorial Design

The number of experiments required in a full 2fc factorial design increases geometrically with k. For a process with 7 factors, for example 27 = 128 experiments are needed to estimate the 7 main effects, and 120 interactions. There are 21 two-factor, 35 three-factor, 35 four-factor, 21 five-factor, 7 six-factor and 1 seven-factor interactions . Fortunately, not all of these interactions are important. Furthermore, the main effects tend to be larger in absolute magnitude than the two-factor interactions, which in turn are greater in absolute magnitude than the three-factor interactions, and so on. This then permits neglecting higher order terms in the Taylor series expansion in Eq. 3.2. Consequently, the information collected by a full factorial design will have redundancies, and fewer experiments than the required number (2P) may be enough to extract all the relevant information. A popular experimental design approach that plans only part of the full factorial design is the fractional factorial design. The fractional factorial designs are named according to the fraction of the full design used, half-fraction indicating that only half of the experiments are conducted.

Consider a process where the effects of five factors are investigated. A full factorial design would necessitate 25 = 32 runs as listed in Table 3.5. One possible half-fraction design (16 runs) includes all runs indicated by an asterisk in the half-fraction column of Table 3.5. The half-fraction design for an experiment with five factors is designated as 25-1 to underline that the design has five variables, each at two levels, and only 24 = 16 runs are used

The selection of the specific 16 runs is important. One way to make the selection is to start with a full 24 design for the first four variables 1, 2, 3, and 4. Then, derive the column of signs for the 1234 interaction and use it to define the levels of variable 5. Thus, 5=1234 as displayed in Table 3.5. Because only 16 runs are carried out, 16 quantities can be estimated: the mean, 5 main factors and 10 two-factor interactions. But there are 10 three-factor interactions, 5 four-factor interactions and 1 five-factor interaction as well. Consider the three-factor interaction 123 written for the 16 runs in Table 3.5. They are identical to the two-factor interaction 45, hence 123 = 45. The remaining 16 runs not included in the half-fraction

Table 3.5. Full and half-fraction 25 factorial design

 Run