FACTORIAL DESIGNS AND THEIR FLEXIBILITY

Dave Doehlert

When factorial designs are introduced to a newcomer to Design of Experiments, there is an implication of rigidity that's unnecessarily restrictive.

The example that's easy to see on paper is the 2-level, 2-factor factorial design:

             Hi o----------o
                |          |      Trials     x1     x2
                |          |         1       Lo     Lo
             x2 |          |         2       Hi     Lo
                |          |         3       Lo     Hi
                |          |         4       Hi     Hi
             Lo o----------o
                Lo   x1    Hi
            

The implication in early teaching of DOE was that the trials at those four corners had to be run exactly at those four corners. The good news is that they can be run at positions somewhat away from the corners. These placements are not as good, but they are ok:

                  o      OR       o
          __________             __________
         |          |           |          |
         | o        |           |        o |
         |          |           |          |
         |          |           |          |
         |          | o         |          | o
       o |__________|           |__________|
                                o              

There is some benefit to doing the runs exactly at the corners or very close:

• The effects of x1 and x2 and their interaction are more clearly separated from one another;
• The confidence limits on predictions will be better.

But you have some room to maneuver. When might you need to move a point away from a corner?

Consider the upper right-hand point: Hi/Hi. When the factors are both set at their high limits, there is some risk that the process will not run at all or, if x1 is time and x2 is temperature, then overcooking might result in a sample whose performance can't be measured in units proportional to quality of product; it is spoiled.

If you can't run that corner, you can mend your design by pulling that corner in toward the center, perhaps 30% of the way toward the center, and then it might well run ok. Analysis using a general purpose regression program will probably handle the data quite well. This problem, that a corner might not run, is even more likely when you have more factors: 4, 7, 10 or more. But it can happen at any location in any problem.

Another situation in which you might not be able to run exactly at the corners is when you do not have complete control of your factors in your lab or pilot plant. You expect to have excellent control, say of temperatures, ultimately in the plant process. But in your lab equipment you have less control.

So you plan the factorial with Lo and Hi factor temperatures. And then you run as nearly at the corners as you can. Perhaps during the run you can measure the factor level, temperature, to see what you are actually getting for temperature. Then enter the temperature that you actually achieve into your regression analysis. These levels probably will not be exactly at the corners. If you are close to corners, say no more than 15 or 20% of the edge length away from the corner, then the analysis will proceed ok.

We were given the impression as students that the runs had to be exactly at the corners because those who were teaching us wanted to show us simple sums and differences for analysis. Now with electronic computing available, we don't have to stay with those simple analyses from the 1920s.

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