# 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?

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.