I have a simple Pre-Test Post-Test Control design like this:
where, $R$ = Random Assignment to groups; $O$ = Observation; $T$ = Treatment; $C$ = Control.
A colleague suggested that by doing and independent-samples t-test between the Gain Scores (i.e., $Posttest - Pretest$) of the two groups, I am actually looking at the interaction effect in a two-way design.
I am wondering how there could exist an interaction effect in such a design?
My colleague says: "You start with two groups, one of which will receive treatment. This is the first factor with means $\mu_1$ and $\mu_2$. Then you measure them again, with one group receiving treatment, the second factor with means $\mu_3$ and $\mu_4$. No interaction is $\mu_1-\mu_3=\mu_2-\mu_4$.
You can plot this by putting: x-axis first two groups, y-axis means, the two lines corresponding to the second factor."