We have delineated 4 groups into a population and have a theory according to which the distributions of a certain characteristic of this popular are ordered in the sense that $\mu_i > \mu_{i+1}$ for all $i$ ($\mu_i$ being the mean of the characteristic for population $i$). We thus acquired data for the 4 groups (with $N=20,25,30,100$ respectively) and observe the four densities : each have a quasi gaussian shape, each group with its proper variance.
We now would like to test our hypothesis that $\mu_1>\mu_2>\mu_3>\mu_4$. To do that I would like to test $H_1: \mu_1>\mu_2>\mu_3>\mu_4$ against it complementary $H_2$ using Bayes factor (I have another question Bayes factor from posterior odds for this particular point...). My question here is : are there more "standard" test to achieve my purpose ?
