What statistical analysis is most appropriate for a cross-over treatment design? How many participants?

Question

I am asking on behalf of a colleague. I was hoping to have an answer for him, but I rather seek some guidance and be a bit more confident. He is designing a treatment study, which I am inventing here for illustrative purposes. The treatment (singing) is supposed to increase mood, while the control (humming) may increase mood, but we are not sure. Mood will be measured by a continuous variable.

N participants will be randomized to one of two groups: baseline, singing, humming control (ABC), or baseline, humming control, singing (ACB). The experimental phase (BC or CB) of each group will consist of three weekly sessions for six weeks. In other words, after baseline, participants will either complete three weeks of singing, followed by three weeks of a humming control, or vice-versa. He expects that the ABC group will show improvements early on (during the singing phase), which will either be maintained or slightly revert toward baseline during the humming control phase. The ACB group is expected to show little change from baseline until they begin to sing.

Questions:

What statistical analysis is most appropriate for the design?

(I was thinking a simple paired samples t-tests design, wherein we would compare A to B, B to C, and A to C for each group. Please correct me.)

How many participants would be needed, assuming a "moderate" effect size? Moderate can perhaps be conceptualized as a change of .5 to .7 standard deviation on the continuous dependent variable. I referred him to two books (DESIGN AND ANALYSIS OF CLINICAL TRIALS and STATISTICS APPLIED TO CLINICAL TRIALS) but they are pretty technical, and a discussion may be more helpful. He cannot have too many participants, and hopes that the power from the cross-over design will be helpful. Hopefully 10 or so is enough, but he is unsure.

Any help or guidance would be appreciated!

Answer 1

I would use a linear mixed model with random slopes and intercepts for each participant and two fixed effects variables (condition) and time (weeks). Basically, you're interested in the slope of change in the mood over time in different conditions -- if there is an effect of humming similar to signing, then these two slopes should be statistically similar while the baseline is flat (i.e., there would be a condition by time interaction). There is a lot of information on mixed models presented in functional form and one example that I like is Baayen (2008) book on analysis of Linguistic Data. In terms of participants, I don't have a suggestion for you. One way is to match it to your previous studies.