What is/are the difference(s) between 2x2 factorial design and 2-way ANOVA? Do they have the same assumptions?
A 2x2 factorial design is a trial design meant to be able to more efficiently test two interventions in one sample. For instance, testing aspirin versus placebo and clonidine versus placebo in a randomized trial (the POISE-2 trial is doing this). Each patient is randomized to (clonidine or placebo) and (aspirin or placebo). The main effect of aspirin and the main effect of clonidine on the outcome of interest can be assessed using a two-way ANOVA.
This trial design is useful to detect an interaction (this is where the effect on the outcome of one factor (e.g. aspirin) depends on the level of the other factor (i.e. whether or not the person gets clonidine)), but one must be careful, as many factorial trials are not powered to detect an interaction. Therefore, one runs the risk of falsely declaring that there is no interaction, when in fact there is one (a type II error).
Therefore, I wouldn't say the two have the same assumptions, as one is a design and one is a statistical method. That being said, the two-way ANOVA is a great way of analyzing a 2x2 factorial design, since you will get results on the main effects as well as any interaction between the effects.
See http://udel.edu/~mcdonald/stattwoway.html for more information.