Suppose a dermatologist wants to study the effectiveness of two different preparations of a skin lotion using two different forms of application, such as one versus two applications per day. She has $12$ patients with a certain skin disease and can apply one form of medication to each arm of each patient. Even though the patients have the same disease, there exists considerable variation among them. The two arms of a patient are expected to respond similarly.
What type of experimental design would be appropriate for this study?
What is the experimental unit?
Give a suitable experimental plan for this study and describe how you obtain this plan.
For that design, outline the ANOVA table, giving sources of variation and degrees of freedom.
My Attempt:
I think this is a crossed design since each subject sees each level of the treatment conditions.
More specifically, this is a $2^2$ factorial design in three replicates. i.e., the model is
$$y_{ijk}=\mu+\tau_i+\beta_j+(\tau\beta)_{ij}+\epsilon_{ijk};\quad i=1,2; j=1,2;k=1,2,3 $$
where, $\tau_i$ is the $ith$ level of one form of the skin lotion
and $\beta_j$ is the $jth$ level of another form of the skin lotion
here experimental unit is arm of each patient.
Is that the case?