I am interested in identifying factors that influence the proportion of time each of my samples occupy one of two states (imagine it as ON vs OFF) throughout the course of one day. In a series of preliminary experiments, I found that my data can be surprisingly nicely modeled with a beta distribution. Subsequent experiments suggest that I can influence this distribution with specific treatments. I would like to determine if there is a statistically significant difference between these two groups (treated vs untreated). I know that for data that come from normal distributions, I can perform an unpaired t-test or if I'm comparing multiple treatments, an ANOVA followed by Dunnett's but since these data come from a fundamentally different distribution, I'm not sure how to go about this test.
For example, a typical experiment can accommodate anywhere from 24 to 128 samples per treatment depending on the resources available at the time of the experiment. Suppose I simulate an experiment as follows in R (with completely random parameters here since I don't have the parameters from my data on hand):
n = 64 alpha1 = 125 beta1 = 55 alpha2 = 250 beta2 = 25 untreated = rbeta(n, alpha1, beta1) treated = rbeta(n, alpha2, beta2)
What would be an appropriate analysis in order to determine if these two sets of samples are likely to be from different distributions or if what I'm seeing is purely random fluctuation?