I'm trying to design an experiment where I measure a variable as a function of 5 two-level factors, labelled A, B, C, D and E.

I'm trying to understand how to best design this experiment so I can conduct it in 8 runs. I've tried to follow the guidance given in Box, Hunter & Hunter, and found two experimental $2^{5-2}$ designs that seem to contradict each others.

One is given in the table p. 272, where D=AB and E=AC. That would yield the following design:

`ABCDE +++++ ++-+- +-+-+ +---- -++-- -+--+ --++- ---++ `

The other one is the example whose table is given on p. 236, defined by D=BC and E=ABC:

`ABCDE +++++ ++--- +-+-- +--++ -+++- -+--+ --+-+ ---++ `

On what criteria should one choose one design over another? Sorry if this sounds like a newbie's question, but I'd really like to understand this issue.

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#### Best Answer

Neither one of these designs contradicts each other, they are generated in different ways. A $2^{5-2}_{III}$ is not unique.

Using a design with so many factors and so few runs, necessiates the fact that main effects and two factor interactions will be confounded, the question is how. Since you want to do an experiment with $8 = 2^{3}$ observations, the full factorial that will generate this design will be based on 3 factors; call them A, B, and C. In the 3 factor design, you will have interactions

- AB, AC, BC
- ABC

To expand this design to include two more factors, each of the new factors ;call them D and E; must be confounded with one of the above listed interactions. Two ways to do that without confounding two main effects is to either have

- D = AB and E = AC which means DE = BC
- D = BC and E = ABC which means DE = A

You can try other combinations, and see the results of your confounding structure.

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