Slide 1

Chaughule, Thorat - Statistical Analysis/Design of Experiments

requires eight runs per replicate. The eight treatment combinations corresponding to
these runs are - (abc), a, b, ab, c, ac, bc and abc. It should be observed here that the
treatment combinations are designed in such order that factors are introduced one by
one with each new factor being combined with the preceding terms. This order of writ-
ing the treatments is called the standard order or Yates' order. The design matrix for the
23 design is shown in Figure 8. 3. The design matrix can be constructed by following the
standard order for the treatment combinations to obtain the columns for the main ef-
fects and then multiplying the main effects columns to obtain the interaction columns.

The full factorial design is the most popular first-order design, in which every factor
is experimentally studied at only two levels. Due to their simplicity and relatively low
cost, full factorial designs are very useful for preliminary studies or in the initial steps of
optimization, while fractional designs are almost mandatory when the problem involves
a large number of factors. However, since only two levels are used, the models that can
be fit to these designs are somewhat restricted. Consequently, if a more sophisticated
model is required, as for the location of an optimum set of experimental conditions, then
one must resort to designs for second-order models, which employ more than two factor
levels to allow fitting of a full quadratic polynomial (Ferreiraa et al, 2007a).

Figure 8.3. 3N Full Factorial

8.6.2. Fractional Factorial Designs

As the number of factors in a factorial design increases, the number of runs for even
a single replicate of the 2k design becomes very large. For example, a single replicate of
an eight factor two level experiment would require 256 runs. Fractional factorial designs
can be used in these cases to draw out valuable conclusions from fewer runs. The basis
of fractional factorial designs is the scarcity of effects principle. The principle states that,
most of the time, responses are affected by a small number of main effects and lower
order interactions, while higher order interactions are relatively unimportant. Fraction-
al factorial designs are used for screening experiments during the initial stages of expe-
rimentation. At these stages, a large number of factors have to be investigated and the
focus is on the main effects and two factor interactions. These designs obtain informa-
tion about main effects and lower order interactions with fewer experiment runs by
confounding these effects with unimportant higher order interactions. As an example,
consider a 28 design that requires 256 runs. This design allows investigation of 8 main
effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to