I wanted to compare the means and variances of these groups. I learnt that a test such as Welch's ANOVA with a Games-Howell posthoc test is useful for comparison of groups with unequal variances. The number of samples in each group are around 4000.
I also wanted to compare the variances to show that the variance of group 2 is significantly different and lower that the variance of group 1 by X%, for example. I read in papers (http://rer.sagepub.com/content/42/3/237) that non-normality doesn't have a major impact in most cases. I also read that large sample sizes ensure that the tests are fairly robust. Also, does non-normality also include multimodal distributions?
The tendency to form multiple modes is probably an aspect that needs to be looked into. However, at this point I am only trying to show the effects of the four "treatments" by comparing the means and variances of the distributions.
Here is one answer that says how means of multimodal distributions could be compared if samples are large: https://stats.stackexchange.com/a/466359/271374
Some more ideas or suggestions would be helpful