Replicates for a Composite Design DOE

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    Rene A van Leeuwen

    We were having a discussion about the Composite Design DOE, which is a type of Response Surface Design.
    For 2 factors this is based on 4 cornerpoints (2k) expanded with 4 axial points and a centerpoint (which is replicated 5 times).
    Should one replicate this number of runs (13 in total) or is the fact that the centerpoint is run five times enough?
    If not, what could be a proper statistical calculation for the number of replicates? Minitab only provides a Power and Samplesize calculation for Full Factorial. s=0.2 / Base design: 2, 4 / 5 centerpoints / effect=0.3 / Power: 0.9 give a number of replicates of 5, which is unacceptable. Lowering the power to 0.8 will decrease the replicates to 4 (still unacceptable).
    Setting up a Response Surface Design in Minitab, the dialoguebox (or the next ones) doesn’t mention the numer of replicates, so that brought the idea in my mind that replicates are not necessary.
    Has anyone got a smart idea about this?
    René, BB


    Erik L

    If you’re on the verge of a CCD type design I’d question what benefit would be derived through the use of a replicate. To have advanced to this level of process knowledge, and to be mapping the detailed response surface I’d imagine that you already understand set-up to set-up variation and the impact that noise and/or nuisance variables have on your process. Use of blocking and replication at lower-level DOEs should have convinced you in the appropriateness of your factors and to reinforce your belief over a range of potential process conditions. Have you added the axial runs to potentially capture quadratic terms that created lack of fit from a past full factorial approach? By the time you’re using this type of DOE you should intimately know your process and you’re just putting the final touches on pinging in on the optimal solution. If you don’t feel that way, then I’d go back to a full or fractional factorial. 2-5 center points are typically all that’s required to come up with a solid estimate of variability for the experimental area. As an aside question, did you think about face-centered CCDs?


    Rob K

    I would agree with the previous reply from Erik L that a central composite design is not the best place to begin your investigation.  I would recommend using a combination of factorial designs and steepest ascent to locate the region of optimal response before using a response surface design to map out the response surface model equation.
    However, I was curious about the problem as you posed it so I did a bit of investigation into power computations for the 2-factor central composite design that you mentioned.  The power is determined by the standard error of the model term estimates and by the number of degrees of freedom available for error.  It turns out that adding the 4 axial points (at +/- 1.414) in the central composite design is equivalent to replicating the 4 corner points in the 2^2 factorial design.  So, roughly speaking you should need about half as many replicates for the central composite design as you would for the factorial design.  So if you concluded that 4 replicates would be sufficient for the factorial design, you would be okay with only 2 replicates of the central composite design.
    But in general I would still recommend starting with factorial designs and then adding axial points as the last step to estimate quadratic terms.
    Rob K

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