I have more than five hundred thousands samples of a continuous variable measured in two groups: a treatment and a control one. I would like to decide whether the measurements follow the same (unknown) distribution in the two groups.
Graphically this seems to be the case: the density functions (obtained using R) for the two groups overlap entirely.
I tried to asses this in a more formal way using the Kolmogorov-Smirnov test. However I obtain a p.value of 2e-13 which suggests that the the two groups do not follow the same distribution.
I am not convinced though and think that this is due to the fact that I have so many measurements that even the slightest difference leads to reject the null hypothesis.
I tried to check if the mean is the same (using Wilcoxon-Manney test) and again the p.value is very low p < 1e-6. However the difference between the mean values in the two groups is quite low (0.006 and the values can range from 0 to 1) which is, for each practical purpose, identical.
Am I using the wrong statistical tests? How can I assess in a formal way whether the two distributions are the same or not?