I am analyzing data from a cluster randomized cross over trial. There are 9 clusters and 3 periods in the data. The clusters are dental clinics and in each period different patients are sampled. I have three interventions (control, treatment1, treatment2).
Each cluster receives all three interventions but in random order. My question is how to analyze the data. Obviously there is some clustering going on in the dataset. Patients are clustered in clinics and also clustered in periods within clinics (sometimes referred to as "cluster-periods").
The whole point of the design is to estimate treatment effects within clinics as opposed to between clinics in a standard cluster randomized trial. Because of practical limitations i had no data on the clinics when the randomization was done and the clinics are very geographically spread out. This eliminated any stratification or matching strategies and there are too few clusters (clinics) to ensure balance between experimental groups. To eliminate imbalances between experimental groups aswell as eliminate sampling variability due to between cluster variation a cross over design was used.
The literature seem somewhat confused about how to correctly analyze data from such a trial. Here are my thoughts on how to do is so far:
Since i observe each cluster more than once i can effectively estimate treatment effects within clusters. Does this mean i can run a regression with a fixed effect for cluster (cluster dummies) to eliminate all between cluster variation? Will this also take account of the unobserved cluster-specific component in the error term causing individuals within clusters to be correlated?
Can i include dummies for time periods? Will this correct for common period effects?
How do i correctly account for clustering? I cannot used cluster corrected standard errors clustering patient on the cluster level with so few clusters. Clustering by cluster-period I also have very few clusters.
I am sorry if it is a little confusing.