How do orthogonal axis values change with the number of center points?

[#1]

In JMP response surface design, when the design type is set to 'CCD - Orthogonal Blocked,' the number of center points is assigned by default, and rotatability and orthogonal axis values are obtained. If I modify the 'number of center points' manually, the orthogonal axis values change. Could you explain the principle behind the change in orthogonal axis values and how to obtain these altered values?For example, for a 3-factor CCD orthogonal block design, the default number of center points is 6, at which the orthogonal axis value is 1.633; if you custom modify the number of center points to 3, then the orthogonal axis value will become 1.897.I would be very grateful if someone is willing to answer.

[#2]

Hi @DualArmadillo50,

Welcome in the Community !

You can find the answer about the orthogonal axial value (and its link to number of centre points) in the related JMP Help section which present the calculation of this value: Choose a Design

Hope this answer will help you, 

[#3]

Thank you for your reply. I would like to know if the formula you mentioned is this one:

In addition, the shaft value calculated according to this formula does not match the actual output in JMP. I would like to know if there are other processing steps applied to the data beyond the formula.

[#4]

Yes, I was mentionning this specific formula, but I have confused your situation of Orthogonal Blocks CCD with the Orthogonal CCD (without blocks). The formula I mentioned is applicable for Orthogonal CCD, but not appropriate for Orthogonally Blocked CCD.
As an example for a 3-factors Orthogonal CCD, I have :

• 8 factorial runs (denoted with +/- signs only)
• 6 axial runs (denoted with a/A signs)
• 9 centre points (by default, denoted with 000)

Using the formula:

I can find the same axial value than the one displayed by JMP (1,668) for orthogonal CCD.

For orthogonal blocks CCD, the axial value is increased compared to orthogonal CCD to ensure that there is no correlation between blocks and main effects (+ interactions):

If you force the axial value for an orthogonal blocks CCD back to the orthogonal one (or any other value) for a design with blocks (according to the number of centre points), then you'll degrade the performance of this modified design (here with a calculated value of 1,525 (orthogonal CCD) for a 3-factors CCD with 8 factorial runs, 6 centre points, 6 axial runs and 3 blocks):

No matter if Orthogonal or Orthogonal Blocked CCD designs, the number of centre points will impact the axial distance calculation. More infos here Central Composite Design - Orthogonal Axial Values 
Unfortunately I don't have the formula to calculate the axial value for orthogonal blocks CCD.

Hope this answer will help you,