Every new study presents its own challenges. (I would have to say that one of the great things about being a biostatistician is the immense variety of research questions that I get to wrestle with.) Recently, I was approached by a group of researchers who wanted to evaluate an intervention. Actually, they had two, but the second one was a minor tweak added to the first. They were trying to figure out how to design the study to answer two questions: (1) is intervention \(A\) better than doing nothing and (2) is \(A^+\), the slightly augmented version of \(A\), better than just \(A\)?
It was clear in this context (and it is certainly not usually the case) that exposure to \(A\) on one day would have no effect on the outcome under \(A^+\) the next day (or vice versa). That is, spillover risks were minimal. Given this, the study was an ideal candidate for a cross-over design, where each study participant would receive both versions of the intervention and the control. This design can be much more efficient than a traditional RCT, because we can control for variability across patients.
While a cross-over study is interesting and challenging in its own right, the researchers had a pretty serious constraint: they did not feel they could assign intervention \(A^+\) until \(A\) had been applied, which would be necessary in a proper cross-over design. So, we had to come up with something a little different.
This post takes a look at how to generate data for and analyze data from a more standard cross-over trial, and then presents the solution we came up with for the problem at hand.
Cross-over design with three exposures
If we are free to assign any intervention on any day, one possible randomization scheme involving three interventions could look like this:
Key features of this scheme are: (1) all individuals are exposed to each intervention over three days, (2) on any given day, each intervention is applied to one group of participants (just in case the specific day has an impact on the outcome), and (3) not every permutation is included (for example, \(A\) does not immediately proceed \(Control\) in any sequence), because the relative ordering of interventions in this case is assumed not to matter. (We might need to expand to six groups to rectify this.)
Data simulation
In this simulation, we will assume (1) that the outcome is slightly elevated on days two and three, (2) \(A\) is an improvement over \(Control\), (3) \(A^+\) is an improvement over \(A\), (4) there is strong correlation of outcomes within each individual, and (5) group membership has no bearing on the outcome.
First, I define the data, starting with the different sources of variation. I have specified a fairly high intra-class coefficient (ICC), because it is reasonable to assume that there will be quite a bit of variation across individuals:
vTotal = 1
vAcross