Hypothesis testing a location shift in heteroscedastic and non-normal data 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?
 A: You are correct that a K-S test deals with the shape of the populations and not the location.
Before I suggest a strategy to try, consider this: if there is a substantial change in the shape of the distribution then is the location really what you want to know about? Is a  test of a hypothesis relating to the location of any inferential utility?
If you answer yes to those questions, then if you have lots of data, current and past, try a couple of simple transformations to see if you can make the likely population shapes similar. (I always like a log transformation for heteroscedastic data.) Once the populations are tolerably similar use either a permutations test or a Student's t-test on the location and trust that the robustness of those tests will deal with any non-identicalness and non-normality respectively.
Your conclusions should be carefully calibrated taking the nature of the data in mind. A significance test approach where the P-value is interpreted as an index of the strength of evidence against the null is much more useful in such circumstances than a Neyman-Pearson hypothesis test which yields a binary outcome.
A: The modified Mathisen test (Hettmansperger & McKean (1998), Robust Nonparametric Statistical Methods, Ch. 2.11) tests for equal medians under the null hypothesis while making no assumptions about the distributions of either sample other than that they're continuous.
But do bear in mind what @Michael said—a difference in medians is usually interesting only when you think there's a location parameter shifting. More often in your situation people use the Mann–Whitney–Wilcoxon test to assess stochastic dominance: that's whether one group has a lower or higher cumulative distribution function of the response for all its values & would correspond to the idea of seeing whether the toxin has a good, a bad, or not much effect. The assumption here is that the population cdfs don't cross, & is usually checked by examining a plot of the sample cdfs.
