Detecting a difference in the direction of two groups of vectors I have two groups of patients, one group obese, one non-obese. I have vectors representing the longitudinal axis of the kidneys in each patient. I am comparing the orientation of the right kidneys between the two groups.I would like to determine the difference (if any) between the two groups in the direction of these vectors. I am looking for suggestions for statistical tests/modules that might be present in Stata or R, or general advice.
 A: Assuming the sample is representative of the population of interest, I'd advocate the following:
I'd first advocate a test of difference in mean rotation. If adiposity is displacing the kidneys (a theory which has become less popular recently), we can standardize all measurements in terms of a minimal degree of rotation from the origin. Since we haven't measured torsion, we have to assume these kidneys have not come remotely close to spinning a full 180 degrees. I don't know if that's a reasonable assumption. If you have one observation per participant, a t-test is suited to do this. I'd be concern that, due to intraindividual variability, we might be interested in repeated measures within a participant, especially whether or not large changes in BMI result in large shifts of the kidney. Using histograms to inspect distributions should give some insight as to whether or not outliers or highly irregular data (e.g. multimodal values) are problematic in the sample.
Secondarily, I'd advocate a test of variance. These are far less sensitive than a mean test. Nonetheless, we'd be interested if obese participants have a greater variability of kidney rotation, since presumably that's a cause of negative health outcomes (perhaps tissue damage due to torsion or impingement of ureters, I don't know). 
A: If the two distributions match a 'normal' circular distribution (Von Mises distribution), you can use a Watson two test, that is available for R at least (package 'circular' or 'circstats'). If they are not normal, I don't know of a non-parametric alternative, but you can bootstrap a 95% confidence interval of average direction in both groups.
All of this assumes that kidney rotation spans a full 360 degrees, or a substantial part of it. If this is not the case, a simple t-test is just fine, as suggested above.
