How to Quickly Determine Design Resolution

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    Hello, I’d really like to know if there is an easy method to determine design resolution (I, II, III…) on the basis of number of levels, factor and generators. For example: 2^8-2, 2^9-1, 2^27-1.
    I’m sure there’s a trick or short formula i can use without write down the entire table.
    Thank you!


    Robert Butler

    It’s not a trick or short formula rather it is a matter of modulo arithmetic and defining contrasts it can get gory fast.

    The “trick” is the identification of defining contrast (also called the defining word or the defining relation) associated with principle block (the set of experiments which include the I experiment – the experiment with everything at it low level).

    Once you have that/those contrasts you use modulo arithmetic to quickly work out the confounding structure. For some designs such as a half replicate the defining contrast is a simple matter of setting the identity experimental design (the experiment with everything at it low level) equal to the largest interaction with everything at its high level.

    For example – for a half rep of a 2**4 the contrast is I = ABCD so with modulo arithmetic you get the generator of D = ABC. This means C = ABD, B = ACD, A = BCD. Thus all mains are clear of one another and all two way interactions. However with modulo arithmetic AB = CD, AC = BD, and AD = BC so you have at most 3 two way interactions clear of one another but confounded with other two way interactions.

    On the other hand, if you want to do 5 variables in 8 experiments then you have I = ABCD = -BCE = -ADE so the generator for the design is D = ABC and E = -BC and the modulo arithmetic will show you what is confounded with what. A good book with the details is The Design and Analysis of Industrial Experiments by Davies 2nd edition. If you can get the book through inter-library loan the section to read is the chapter on confounding in experimental design and appendix 9D which goes into more detail.

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