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DoE-Experts: Why do scaling factors change when choosing a different type of CCD?

Florian_Vogt

Hi all,

I was asked about CCD's. In particular, it was a 'which type when' discussion.

In literature there are generally three characteristics of CCDs: Rotatable, Orthogonal and 'On-Face' (more a setting than a characteristic). In JMP, a user first can choose between different design types. In the next step, they can then modify the axial value and others.

1. Question: If a user for example chose a "CCD-Orthogonal" - why would they then want to choose a 'Rotatable' axial value?

2. Question: Why do the axial values of "Rotatable", "Orthogonal" change, depending on which design type was chosen in the first place (yellow boxes in graph below)?

3. Asked Question: Would the number of factors in the design have any impact on these characteristics (rotatability, orthogonality)?!

Thanks for helping me understand

Best regards from Heidelberg,

Florian

Accepted answers

Dan_Obermiller

I will take a crack at this...

Question 1: If you choose CCD-Orthogonal and then a Rotatable axial distance, JMP will probably warn you that the design may no longer be orthogonal. The axial distance will override the Orthogonal label. Orthogonal designs are really determined by the axial distance. Changing that axial distance may give you other properties (like rotatable), but will likely create some correlations among the model terms.

Question 2: The rotatable axial distance is not changing as you change the design. The different designs are only changing the number of center points. Changing the number of center points will force the orthogonal axial distance to change. It must change to maintain the orthogonality.

Question 3: Yes, changing the number of factors will change the axial distances for both rotatable and orthogonal.

I hope this helps. Let me know if I need to explain anything more clearly.

Dan

DonMcCormack

Orthogonality is a function of the parameter estimate precision, rotatability is a function of prediction variance. The prediction variance in a rotatable design is only a function of how far the design point is from the center, not where it is in the design space.
Orthogonality, rotatability, and uniform precision, are all a function of the design matrix. Without getting into too much mathematical detail, think about trying to create a table of values such that the correlation matrix of some function of that table (specifically, in matrix notation: (X^TX)^-1 ) is zero off the diagonal (orthogonality) and another function of that table (specifically, in matrix notation: x(X^TX)^-1x^T, where x is a row in the table) have the same values for any two rows that have the same distance from the center of the design (rotatability). Because the design is coded, the value for anything on the design corners (a CCD is a factorial design with axial and center points added) is either 1 or -1 and the centerpoints are 0. Your a bit limited in terms of what you can do to affect these characteristics. You can add centerpoints and you can change the values of the axial points. In case you're wondering, in a uniform precision design, the prediction variance at the center and at one unit from the center are equal.
Absolutely. Here's a nice not overly theoretical explanation of how the number of factors affects rotatability and orthogonality.

All comments

Dan_Obermiller

I will take a crack at this...

Question 3: Yes, changing the number of factors will change the axial distances for both rotatable and orthogonal.

I hope this helps. Let me know if I need to explain anything more clearly.

Dan

Florian_Vogt

Thanks a lot @Dan_Obermiller, this helps a lot!

As you say, there is a warning when a CCD-Orthogonal Design is changed to rotatable - so there is no practical application scenario?

Do we have the calculation for the axial value available for customers?

Thanks

Florian

DonMcCormack

Orthogonality is a function of the parameter estimate precision, rotatability is a function of prediction variance. The prediction variance in a rotatable design is only a function of how far the design point is from the center, not where it is in the design space.
Orthogonality, rotatability, and uniform precision, are all a function of the design matrix. Without getting into too much mathematical detail, think about trying to create a table of values such that the correlation matrix of some function of that table (specifically, in matrix notation: (X^TX)^-1 ) is zero off the diagonal (orthogonality) and another function of that table (specifically, in matrix notation: x(X^TX)^-1x^T, where x is a row in the table) have the same values for any two rows that have the same distance from the center of the design (rotatability). Because the design is coded, the value for anything on the design corners (a CCD is a factorial design with axial and center points added) is either 1 or -1 and the centerpoints are 0. Your a bit limited in terms of what you can do to affect these characteristics. You can add centerpoints and you can change the values of the axial points. In case you're wondering, in a uniform precision design, the prediction variance at the center and at one unit from the center are equal.
Absolutely. Here's a nice not overly theoretical explanation of how the number of factors affects rotatability and orthogonality.

Dan_Obermiller

I cannot think of any practical application to choose CCD-Orthogonal and then choose a rotatable axial distance UNLESS you wanted a rotatable design with the number of center points specified by the CCD-Orthogonal.

As for the formula for the rotatable axial distance, I did not find it in the JMP help, but it is (NF)^(1/4) where NF is the number of design points in the factorial portion of the design. So for three factors, it would 8^(1/4) or the 4th root of 8.

For an orthogonal axial distance, the formula is more complicated. I do not recall it right away so some research would be needed to turn up that formula. I know that it will most likely be mentioned in a discussion of blocking as a blocked CCD will use an orthogonal axial distance since blocking is typically done orthogonally.

Dan_Obermiller

Found the formula for orthogonal axial distance.

alpha = (((sqrt(NF + 2*k+c) - sqrt(NF))^2)*(NF/4))^(1/4) where k=number of factors, c = number of center points, and NF = number of runs in the factorial part of the design.

So for 3 factors with 2 center points, alpha = (((sqrt(8+2*3+2) - sqrt(8))^2)*(8/4))^(1/4)

= (((sqrt(16)-sqrt(8))^2)*2)^(1/4)

= ((4-sqrt(8))^2*2)^(1/4)

= (((1.17)^2)*2)^(0.25)

= 2.74^(0.25) = 1.287

Note that the formula for the orthogonal axial point will change if blocking is used.

DonMcCormack

The argument for choosing a rotatable design is that you are unsure where the optimal value is, so this is a way to hedge your bets. That said, the difference between a rotatable and nonrotatable design should be investigated using Compare Design to determine if the extra runs are worth it. I doubt that many folks can even come up with a practical value for how much of change in prediction variance would be worthwhile.

Florian_Vogt

Thanks a lot @DonMcCormack and @Dan_Obermiller - that helps a lot!

Greetings from Heidelberg,

Florian