# Cell sizes for a fractional factorial design that ignores some main effects

We're designing a trial to test three behavioral interventions (call them A, B, and C) for a health condition. Treatment A is an established intervention and B and C are considered add-ons to A.

We want to test the following contrasts:

1. Outcomes of packages involving Treatment A vs. No treatment.
2. Outcomes of adding B, C, or B+C to A.

Because we're not trying to estimate effects of B or C in the absence of A, we came up with a fractional factorial design implementing the following 5 cells:

Cell    A/B/C    included?
--------------------------
1       + + +    included
2       + + -    included
3       + - +    included
4       + - -    included
5       - + +    not included
6       - + -    not included
7       - - +    not included
8       - - -    included


How many participants should we have in cell #8? (the "no treatment" cell)

Normally factorial designs put the same number of participants in each cell, but they also have equal numbers of cells with each treatment present vs. absent. Since we have only one cell without treatment A, should this cell be 4x the size of the others? Even if we're comfortable adjusting for unequal cell sizes, should the "no treatment" cell be substantially larger than the other cells?

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I guess that if you are not interested in the effect of B and C alone, this is because you expect strong interaction with A. So you will probably go for an additive model plus interactions. $\beta_0 + \beta_A A + \beta_B B + \beta_C C + \beta_{AB} A*B + \beta_{AC} A*C + \beta_{BC} B*C + \beta_{ABC} A*B*C$.

Because you do not investigate the role of $B$ and $C$ alone, you actually study an equivalent model which is $\beta_0 + \beta_A A + \beta_{AB} A*B + \beta_{AC} A*C + \beta_{ABC} A*B*C$ which happens to have... 5 parameters. Each of your cell estimates one of them in particular, so I think this is a good reason to put exactly the same number of people in cell #8 as in the others.

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Sorry, what does D represent in this model? I'd expect the 5th term to be βabc(A*B*C), for the unique 3-way interaction when all treatments are present. –  octern Jun 1 '12 at 23:13
I meant B*C and not C*D, thanks for spotting. And actually, you are right, that it should be A*B*C. I edit my answer. –  gui11aume Jun 1 '12 at 23:20
Your equation describes the full model we're able to test, and I think that's what we'd use for goal #2. I'd like to know whether for goal #1, there's any potential to do a more powerful but less specific contrast: "any treatment involving A" vs. "no treatment". Is it possible to pool data from our four cells containing A like that, and does that affect the n we'd want for the no-treatment cell? –  octern Jun 1 '12 at 23:34
Yes, that's a good point. To me it looks like you cannot pool the first 4 cells for the reason that they have received different treatments. The contrast of goal #1 is the difference of cells 4 and 8. So I don't think having more in 8 specifically would help, unless you already know that the variance there is going to be much higher than in the other cells, as @Michael Chernick suggets. –  gui11aume Jun 1 '12 at 23:45
Nice answer. I didn't factor into my answer all the main and interaction effects that might be of interest. But those hypothesis testscould affect the choice of cell size. –  Michael Chernick Jun 2 '12 at 0:10