Augmentation of Design

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    I would like to augment a fractional factorial design to a central composite design. The fractional factorial design consists of 2^(5-1) number of runs plus 3 centre points (n=19). Please suggest how do I go about augmenting the design to central composite design.</p>


    Robert Butler

    To do things like this you really need to spend the money, purchase some good books on experimental design, and read them and keep them close by as a reference.

    The two I would recommend are:

    1. Understanding Industrial Designed Experiments by Schmidt and Launsby

    2. The Design and Analysis of Industrial Experiments by Davies

    To your point – all composite designs have as their core construct a two level factorial design. This factorial design can be fractionated, as you have done, so the only remaining questions are:

    1. How many center points?

    2. What is the level of the alpha of the star points?

    From the books we have the following:

    To retain orthogonality the number of center points = Nc = 4*(sqrt(Nf) +1) -2k

    Where Nc = number of center points,

    Nf = number of experiments in the factorial = 16 – since you are running a fractional factorial

    k = number of factors in the design = 5

    Therefore Nc = 4*(sqrt(16) +1) – 2*5 = 10

    The level of alpha for the star points of a rotatable composite is

    alpha = (Nf)**(1/4) = (16)**(1/4) = 2.

    Thus, the star points are located at +-2 for each of the factors which means you will have 10 additional runs where the minimum and maximum alphas are two times the minimum/maximum levels for each of the factors in the fractional factorial.

    So, if you want a perfectly orthogonal rotatable composite design with a 2**(5-1) fractional factorial core you will need 16 +10 + 10 design points.  That is a lot of experimentation and if you are pressed for time and/or the cost per experiment is high it may be far more than you can run.

    A possible alternative:

    1. If your situation is that of just starting out then instead of trying to do everything at once (all mains, all two ways, all curvilinear) a better approach from the standpoint of time/money/effort would be to run a near saturated design (5 factors in 8 experiments) with two center points for replication and analyze the results of this design and see what you see.

    This approach buys you a couple of things:

    a. First – it gives you a check to see if there is any point in investigating all 5 factors and if there is any need to spend time looking for curvilinear behavior.  If you run this smaller design and find a) all five factors matter and/or b) there is serious curvilinear behavior present then, you can augment your basic design with additional points and fill out the full composite design.

    b. It provides a reality check with respect to minimum and maximum settings for the design space.  I could bore you senseless with story after story about how everyone KNEW (I mean they REALLY KNEW) we could run the process at the specified minimum and maximum levels for all of the variables of interest in the design. Unfortunately, when we went to do this, Physical Reality raised its ugly head and said, “No way – go back and try again.”  In short, running the smaller design is great way to make sure you can really investigate the things that are of interest.  If the smaller design works with the levels of interest, all well and good and if it doesn’t you haven’t wasted valuable time/money/effort. Also if things don’t work, you can regroup, use what you have learned, and build a design with new minima and maxima for each variable that will permit an investigation.

    2. Perfect orthogonality is a nice idea but, in practice, even with the number of runs listed above, you won’t achieve it for the simple reason that it is highly unlikely you will run each and every design point at the exact specified minimum or maximum settings for every experimental condition.  In those situations where I have actually run a composite design I’ve cut the number of center points to 3-4.  The design is a touch non-orthogonal but the degree of non-orthogonality is such that it doesn’t have any major impact on the results of the analysis.






    @irfan.ahanger01 – I’m certainly not going to disagree with @rbutler (good to see you’re still here, trying to educate the masses!), but I would take a more pragmatic approach.  You already have a design with the center points, so you need to add the Star Points.  For each factor, you will need two – a high and a low, set at an alpha level – to augment the data you already have.  Each of the other factors should be set at their center point when paired with the Star Point factor.  Five factors, times 2 new levels each, is an additional 10 runs.

    If you set up a Central Composite Design in Minitab, they also up the overall number of true center points to six total for a total of 32 runs.

    I’m just a neanderthal compared to @rbutler, but that’s how I would do it.  Just my humble opinion.

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