# Design of Experiments

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?

The experimental unit is one arm. You'll have to make some decisions to narrow in on a design and a model.

As your question is tagged as self-study, I'll give some hints and I think you'll be able to fill in the blanks. First of all, what are ALL the factors that could be included? I can find two that you haven't mentioned at all. The text in your question tells you which one to include and which one not to. How could one of them be included?

Furthermore, each subject only has two arms, hence, cannot try all treatments. This leads to something very similar to a split-plot design, if you're familiar with that term.

Finally, you model specification only speaks of 12 observations but you have 24?

• Is that "form of application" whole plot treatments as the text mentioned one versus two applications per day. – time Sep 17 '14 at 1:10
• Form of application is certainly a factor to consider, but it's not entirely clear if it's actually a whole plot treatment. If the dermatologist does not wish to vary form of application within subject then it's a whole plot treatment. Furthermore, think about how you can include the subject itself. The text hints that this should be done. – swmo Sep 17 '14 at 9:09
• I am not understanding how to randomize the design. If 'preparations of a skin lotion' is one factor which have 2 levels and 'form of application' is another factor which have also 2 levels and also think of the days as block then i need 6 days to include the subject. But it doesn't make any sense. – time Sep 17 '14 at 12:41
• Consider the factors "form of application", "preparation of skin lotion" and "subject/person". As I understand the question, these are the once considered central. There is no factor for days (we do it all in one day). You want to apply the different combinations of factors prep and form an equal number of times and you probably want to think a little about on which people to apply which combination of prep and form, as you don't want the factor subject to be confounded with some effect of interest to you. – swmo Sep 17 '14 at 14:21
• As far i know the necessity of split-plot design occurs when there is at least one factor which is 'hard to change'. And under this hard-to-change factor we randomize the factor which is 'easy to change'. Is that "subjects" can be randomly assigned to both the factors "form of application" and "preparation of skin lotion"? and hence "subjects" are 'easy to change' factor? – time Sep 17 '14 at 14:49