# How to analyze fractional crossover design (factional survey experiment) with 3 factors in R?

I have the following experimental design: my treatment is composed out of three factors (factor1, factor2, factor3 ) each having two levels. So far I have eight groups and I assigned subjects randomly to these groups.

To raise the power of the study, I thought it was a good idea to test each subject twice. After testing the first treatment, there is a second treatment to the same person, this second treatment is composed out of all the opposite factor levels.

I don't know how exactly this design is called, looks like a fractional crossover design (or some kind of factional survey experiment) to me.

The response variable is continuous, so I am going for an ANOVA (or should I give the nlme package a try?). Now I am struggling with the right way to do it regarding the following four points.

1. Does this model formula fit to the design?

fit = aov(response ~ factor1 * factor2 * factor3 + Error(subject / (factor1 * factor2 * factor3))

2. And what is the difference to the following formula?

fit = aov(response ~ (factor1 * factor2 * factor3) + Error(subject))

3. What if I am only interested in the following effects?

• Main effect of factor1
• Main effect of factor2
• Interaction of factor1 x factor3
• Interaction of factor2 x factor3
4. And how do I interpret the output given the above formulas?