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What is/are the difference(s) between 2x2 factorial design and 2-way ANOVA? Do they have the same assumptions?

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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.

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    $\begingroup$ I like the answer except for the point about the interaction. A 2x2 factorial design tested via a 2-way ANOVA is actually an excellent way to test for an interaction. $\endgroup$ – rolando2 Dec 6 '11 at 13:17
  • $\begingroup$ @rolando2 is exactly right. Would you mind editing your comment about interactions? Very often the primary motivation for running a 2x2 experiment is to see if there is an interaction. $\endgroup$ – gung Dec 1 '12 at 1:06
  • $\begingroup$ @rolando2 I have amended the answer to address interactions. Thanks. $\endgroup$ – pmgjones Dec 4 '12 at 21:48
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    $\begingroup$ +1, thanks for the edit. Note that there is nothing in nature that requires interactions interactions to be harder to detect than main effects. The power of the analysis to detect an interaction depends on its magnitude & the N used for your study. It may well be, however, that the interactions that people care about in your field tend to be smaller than the main effects, & this makes them seem less tractable. $\endgroup$ – gung Dec 4 '12 at 21:53
  • $\begingroup$ @gung do I read this comment to say that in order to determine power (and sample size) for a factorial, one only needs to look at the smallest effect size (which could be a main effect or an interaction) and find power for that comparison - thus setting the total sample size to the total needed to find the smallest comparison of interest (ignoring multiple testing?). I had just asked a similiar question but I think this response suggests it is correct: stats.stackexchange.com/questions/253281/… $\endgroup$ – B_Miner Dec 26 '16 at 20:59

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