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:
- Outcomes of packages involving Treatment A vs. No treatment.
- 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?