5.
Process Improvement
5.3. Choosing an experimental design 5.3.3. How do you select an experimental design?


Center point, or 'Control' Runs  
Centerpoint runs provide a check for both process stability and possible curvature 
As mentioned earlier in this section, we add centerpoint runs
interspersed among the experimental setting runs for two purposes:


Centerpoint runs are not randomized  Centerpoint runs should begin and end the experiment, and should be dispersed as evenly as possible throughout the design matrix. The centerpoint runs are not randomized! There would be no reason to randomize them as they are there as guardians against process instability and the best way to find instability is to sample the process on a regular basis.  
Rough rule of thumb is to add 3 to 5 center point runs to your design  With this in mind, we have to decide on how many centerpoint runs to do. This is a tradeoff between the resources we have, the need for enough runs to see if there is process instability, and the desire to get the experiment over with as quickly as possible. As a rough guide, you should generally add approximately 3 to 5 centerpoint runs to a full or fractional factorial design.  
Table of randomized, replicated 2^{3} full factorial design with centerpoints 
In the following Table we have added three centerpoint runs to the
otherwise randomized design matrix, making a total of nineteen runs.


Preparing a worksheet for operator of experiment  To prepare a worksheet for an operator to use when running the experiment, delete the columns 'RandOrd' and 'Standard Order.' Add an additional column for the output (Yield) on the right, and change all '1', '0', and '1' to original factor levels as follows.  
Operator worksheet 
Note that the control (centerpoint) runs appear at rows 1, 10, and 19. This worksheet can be given to the person who is going to do the runs/measurements and asked to proceed through it from first row to last in that order, filling in the Yield values as they are obtained. 

Pseudo Center points  
Center points for discrete factors 
One often runs experiments in which some factors are nominal. For
example, Catalyst "A" might be the (1) setting, catalyst "B" might be
coded (+1). The choice of which is "high" and which is "low" is
arbitrary, but one must have some way of deciding which catalyst setting
is the "standard" one.
These standard settings for the discrete input factors together with center points for the continuous input factors, will be regarded as the "center points" for purposes of design. 

Center Points in Response Surface Designs  
Uniform precision  In an unblocked response surface design, the number of center points controls other properties of the design matrix. The number of center points can make the design orthogonal or have "uniform precision." We will only focus on uniform precision here as classical quadratic designs were set up to have this property.  
Variance of prediction  Uniform precision ensures that the variance of prediction is the same at the center of the experimental space as it is at a unit distance away from the center.  
Protection against bias  In a response surface context, to contrast the virtue of uniform precision designs over replicated centerpoint orthogonal designs one should also consider the following guidance from Montgomery ("Design and Analysis of Experiments," Wiley, 1991, page 547), "A uniform precision design offers more protection against bias in the regression coefficients than does an orthogonal design because of the presence of thirdorder and higher terms in the true surface.  
Controlling \( \alpha \) and the number of center points  Myers, Vining, et al, ["Variance Dispersion of Response Surface Designs," Journal of Quality Technology, 24, pp. 111 (1992)] have explored the options regarding the number of center points and the value of \( \alpha \)somewhat further: An investigator may control two parameters, \( \alpha \) and the number of center points (n_{c}), given k factors. Either set \( \alpha \) = 2^{(k/4)} (for rotatability) or \( \sqrt{k} \)  an axial point on perimeter of design region. Designs are similar in performance with \( \sqrt{k} \) preferable as k increases. Findings indicate that the best overall design performance occurs with \( \alpha \approx \sqrt{k} \) and 2 ≤ n_{c} ≤ 5. 