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I am planning on doing a variance components analysis (random effects anova). I'm wondering if anyone has any advice regarding balancing the number of levels to include in the design vs. the type of design. For example, I could do a $2^3$ full factorial design, but if I want to include more levels, I would have to switch to a fractional factorial. All factors are qualitative and can include any number of levels. For example days the experiment is run, or analyst that runs the experiment, etc. I have a stats background but am (obviously) new to DoE.

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This you could investigate, for your situation, using simulation. Are you OK with a main-effects design, or do you need interactions? Since your factors are all qualitative, dropping levels means there are some questions you cannot anymore ask of your data ... so this will entirely depend on your research goals.

But I would probably go for fractioning!

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