My experiment has a 2x2 design, full between-subjects, so that the resulting four groups are as follows: -Group A-1: Experimental treatment A. -Group A-2: Control for experimental treatment A. -Group B-1: Experimental treatment B. -Group B-2: Control for experimental treatment B.
I am expecting that treatment A works (when compared to their control) while treatment B doesn't. As the interaction is significant, I need to follow up with contrasts. In Jamovi and similar software, this is straightforward with Tukey's procedure.
However, note that not all the contrasts are useful here. In fact it makes no sense to compare the experimental treatment A with the control for the treatment B. Tukey just takes all possible combinations of the 4 means and correct accordingly. Here only two comparisons are of interest: Treatment A vs. Control A, and Treatment B vs. Control B. Tukey corrects for 6 comparisons even if most of them are useless.
I've been reading about other procedures to protect alpha from multiple comparisons. Bonferroni is the easiest to apply but overly conservative (right?). Dunnett allows to reduce the number of contrasts but assumes only one control.
If I am interested in only two contrasts (and the interaction that reveals that treatment A works better than treatment B), which correction procedure can I apply?