5.
Process Improvement
5.3. Choosing an experimental design 5.3.3. How do you select an experimental design? 5.3.3.4. Fractional factorial designs


We can run a fraction of a full factorial experiment and still be able to estimate main effects 
Consider the twolevel, full factorial design for three factors, namely
the 2^{3} design. This implies eight runs (not counting
replications or center points). Graphically, as shown
earlier, we can represent the 2^{3}
design by the following cube:
FIGURE 3.4: A 2^{3}
Full Factorial Design;


Tabular representation of the design 
In tabular form, this design (also showing eight observations
'y_{j}'
(j = 1,...,8) is given by


Responses in standard order  The rightmost column of the table lists 'y_{1}' through 'y_{8}' to indicate the responses measured for the experimental runs when listed in standard order. For example, `y_{1}' is the response (i.e., output) observed when the three factors were all run at their 'low' setting. The numbers entered in the 'y' column will be used to illustrate calculations of effects.  
Computing X_{1} main effect 
From the entries in the table we are able to compute all 'effects' such
as main effects, firstorder 'interaction' effects, etc. For example,
to compute the main effect estimate
'c_{1}' of factor X_{1}, we compute the
average response at all runs with X_{1} at the 'high'
setting, namely (1/4)(y_{2} + y_{4}
+ y_{6} + y_{8}), minus the average
response of all runs with X_{1} set at 'low,' namely
(1/4)(y_{1} + y_{3} + y_{5}
+ y_{7}). That is,


Can we estimate X1 main effect with four runs?  Suppose, however, that we only have enough resources to do four runs. Is it still possible to estimate the main effect for X_{1}? Or any other main effect? The answer is yes, and there are even different choices of the four runs that will accomplish this.  
Example of computing the main effects using only four runs 
For example, suppose we select only the four light (unshaded) corners
of the design cube. Using these four runs (1, 4, 6 and 7), we can still
compute c_{1} as follows:


Alternative runs for computing main effects  We could also have used the four dark (shaded) corners of the design cube for our runs and obtained similiar, but slightly different, estimates for the main effects. In either case, we would have used half the number of runs that the full factorial requires. The half fraction we used is a new design written as 2^{31}. Note that 2^{31} = 2^{3}/2 = 2^{2} = 4, which is the number of runs in this halffraction design. In the next section, a general method for choosing fractions that "work" will be discussed.  
Example of how fractional factorial experiments often arise in industry 
Example: An engineering experiment calls for running three
factors, namely Pressure, Table speed, and Down force, each at a
'high' and a 'low' setting, on a production tool to determine
which has the greatest effect on product uniformity. Interaction
effects are considered negligible, but uniformity measurement error
requires that at least two separate runs (replications) be made at
each process setting. In addition, several 'standard setting' runs
(centerpoint runs) need to be made at regular intervals during the
experiment to monitor for process drift. As experimental time and
material are limited, no more than 15 runs can be planned.
A full factorial 2^{3} design, replicated twice, calls for 8x2 = 16 runs, even without centerpoint runs, so this is not an option. However a 2^{31} design replicated twice requires only 4x2 = 8 runs, and then we would have 158 = 7 spare runs: 3 to 5 of these spare runs can be used for centerpoint runs and the rest saved for backup in case something goes wrong with any run. As long as we are confident that the interactions are negligbly small (compared to the main effects), and as long as complete replication is required, then the above replicated 2^{31} fractional factorial design (with center points) is a very reasonable choice. On the other hand, if interactions are potentially large (and if the replication required could be set aside), then the usual 2^{3} full factorial design (with center points) would serve as a good design. 