Rough rule of thumb is to add 3 to 5 center point runs to your design

As mentioned earlier in this section, we add center point runs interspersed among the experimental setting runs for two purposes: 

1. To provide a measure of process stability and inherent variability
2. To check for curvature.

With this in mind, we have to decide on how many center point 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 center point runs to a full or fractional factorial design. 

In the following Table we have added three center point runs to the otherwise randomized design matrix giving a total of nineteen runs. 

 
 

	Random Order	Standard Order	SPEED	FEED	DEPTH
1	?	?	0	0	0
2	1	5	-1	-1	1
3	2	15	-1	1	1
4	3	9	-1	-1	-1
5	4	7	-1	1	1
6	5	3	-1	1	-1
7	6	12	1	1	-1
8	7	6	1	-1	1
9	8	4	1	1	-1
10	?	?	0	0	0
11	9	2	1	-1	-1
12	10	13	-1	-1	1
13	11	8	1	1	1
14	12	16	1	1	1
15	13	1	-1	-1	-1
16	14	14	1	-1	1
17	15	11	-1	1	-1
18	16	10	1	-1	-1
19	?	?	0	0	0

To prepare a worksheet for an operator to use when running the experiment, delete the columns ’RandOrd’ and ‘StdOrd.’  Add an additional column for the output (Yield) on the right, and change all ‘-1’, ‘0’, and ‘1’ to original factor levels as follows. 
 
 

	Random Order	Standard Order	SPEED	FEED	DEPTH
1	?	?	0	0	0
2	1	5	-1	-1	1
3	2	15	-1	1	1
4	3	9	-1	-1	-1
5	4	7	-1	1	1
6	5	3	-1	1	-1
7	6	12	1	1	-1
8	7	6	1	-1	1
9	8	4	1	1	-1
10	?	?	0	0	0
11	9	2	1	-1	-1
12	10	13	-1	-1	1
13	11	8	1	1	1
14	12	16	1	1	1
15	13	1	-1	-1	-1
16	14	14	1	-1	1
17	15	11	-1	1	-1
18	16	10	1	-1	-1
19	?	?	0	0	0

Note that the control (center point) 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 

One often runs experiments where 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

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. 

Uniform precision ensures that the variance of prediction is the same at the center of the experimental space as it is at unit distance away from the center. 

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 third order and higher terms in the true surface.” 

Myers, Vining, et al, [“Variance Dispersion of Response Surface Designs,” Journal of Quality Technology, 24, pp. 1-11  (1992)] have explored the options regarding the number of center points and the value of a somewhat further: An investigator may control two parameters, a and the number of center points (n_c), given k factors.  Either set a = 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 best overall design performance occurs with .