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I have the following:

It has been decided to run an experiment involving 2 replicates of a $2^4$ factorial with factors coded A, B, C & D. However, block size is restricted to 8

I'm not sure what the importance of block size being restricted to 8 is here. For example, what difference would it make if the block size was 6 or 12?

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    $\begingroup$ Is this a randomized multicenter design? $\endgroup$ – Todd D Jan 14 '19 at 2:30
  • $\begingroup$ I don't understand what that means. If that's the default then yes, otherwise no $\endgroup$ – baxx Jan 14 '19 at 2:32
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    $\begingroup$ I don't really understand what you are dealing with, maybe some images could help. But intuitively, since your factor has 4 levels (A, B, C, D) then I guess the 8 was chosen such that each block will have a nice integer rep of the factors. $\endgroup$ – user2974951 Jan 14 '19 at 7:31
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This would be an incomplete block design. If you are conducting a 2^4 experiment, a complete block would need to have 16 observations in each block, but here you are restricted to 8 per block. Traditionally, there are a couple of considerations. First, you want to arrange your treatments in blocks so that they are "balanced". This will allow you to confound the block effect with a certain interaction, so you are intentionally choosing which effect you won't be able to test for. Second, the analysis of variance required "adjusting" the treatment and block effects. That is, using modified calculations from a complete block design. I recommend looking at a good design and analysis of experiments book for details. But a quick introduction can be found here. I don't know if using a mixed model approach makes the analysis easier than in the traditional manner.

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  • $\begingroup$ I was looking in Douglas C. Montgomery, 1997, Design and Analysis of Experiments, 4th Edition. It suggested that a general linear model with type III sums of squares will do the "adjusting" correctly. If so, that would greatly simply the analysis, but still require care in balancing the treatments across blocks. $\endgroup$ – Sal Mangiafico Jan 14 '19 at 17:35

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