How do I test for a "location shift" in something like the mean or median if the shape of the distribution has changed quite significantly between groups?
Often an experimental intervention seems to completely change the distribution of the data. For example, If I measure limb length in control and toxin-exposed animals, the control group sample might appear to be drawn from a symmetric, normal distribution but the experimental group might be heavy-tailed and have much more dispersion.
All the statistical tests I am familiar with either assume that the data is drawn from normally-distributed populations or that the data is drawn from two populations with non-normal but identical distribution shapes.
I can do something like a two-sample Kolmogorov-Smirnov test which tells me that my groups differ. But this isn't telling me anything about location, right?