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


Construction of a 2^{31} half fraction design by staring with a 2^{2} full factorial design 
First note that, mathematically, 2^{31} = 2^{2}.
This gives us the first step, which is to start with a regular
2^{2} full factorial design. That is, we start with the
following design table.


Assign the third factor to the interaction column of a 2^{2} design 
This design has four runs, the right number for a halffraction of a
2^{3}, but there is no column for factor X_{3}. We
need to add a third column to take care of this, and we do it by adding
the X_{1}*X_{2} interaction column. This
column is, as you will recall from full factorial designs, constructed by
multiplying the row entry for X_{1} with that of
X_{2} to obtain the row entry for
X_{1}*X_{2}.


Design table with X_{3} set to X_{1}*X_{2} 
We may now substitute 'X_{3}' in place of
'X_{1}*X_{2}' in this table.


Design table with X_{3} set to X_{1}*X_{2} 
Note that the rows of Table 3.14 give the darkshaded corners of the
design in Figure 3.4. If we had
set X_{3} = X_{1}*X_{2}
as the rule for generating the third column of our 2^{31}
design, we would have obtained:


Main effect estimates from fractional factorial not as good as full factorial  This design gives the lightshaded corners of the box of Figure 3.4. Both 2^{31} designs that we have generated are equally good, and both save half the number of runs over the original 2^{3} full factorial design. If c_{1}, c_{2}, and c_{3} are our estimates of the main effects for the factors X_{1}, X_{2}, X_{3} (i.e., the difference in the response due to going from "low" to "high" for an effect), then the precision of the estimates c_{1}, c_{2}, and c_{3} are not quite as good as for the full 8run factorial because we only have four observations to construct the averages instead of eight; this is one price we have to pay for using fewer runs.  
Example  Example: For the 'Pressure (P), Table speed (T), and Down force (D)' design situation of the previous example, here's a replicated 2^{31} in randomized run order, with five centerpoint runs ('000') interspersed among the runs. This design table was constructed using the technique discussed above, with D = P*T.  
Design table for the example 
