Paired t-test of medians I need to run an inferential test on some large data where the individual data points have a heavily skewed distribution. I'm considering doing a paired t-test across a number of days comparing the results of the test group to the control group. But if I take daily means, I'm still subject to outliers influencing those values. What if I take daily medians and run the paired t-test in the same way? Is there anything "wrong" with this approach? Something tells me there is, primarily because I don't know how medians "behave". Does the CLT apply to medians also? Should they be normally distributed?
 A: A t-test of means has a t-distribution because (under the assumptions) the numerator of the statistic is normally distributed, while the square of the denominator is $\sigma^2$ times a chi-square divided by its df, and is independent of the numerator.
This doesn't happen when testing medians, so you don't have an argument for a t-test. If you're worried about the influence of outliers, it will impact the standard deviation more than the mean, so replacing the numerator alone doesn't solve the problem.
If there are robust estimates of scale you might get close to a t-distribution if samples aren't too small, but the d.f. may change and there may be a need to scale the resulting statistic. (See, for example, the development in a similar case here: T-test using only summary data in a box plot)
There are robustified versions of t-tests that (for example) replace the means with trimmed means and variances with Winsorized variances. However, as above, the result doesn't have a t-distribution any more; on the other hand it's not so hard to simulate for some reasonable distributional assumption, and the result won't be substantially impacted by a small proportion of outliers. In large samples, it may often be approximated reasonably by a t-distribution, but again there's typically going to be a scaling factor needed, and suitable df may take some work to figure out.
A: You can use the wilcoxonsigned-rank-test to calculate the statistical significance of paired data given the null hypothesis that the median difference between two paired variables is zero. This is a nonparametric test, so skew isn't a problem. It employs rank transformation, so it sacrifices power, but reduces the influence of outliers.
It doesn't sound like your data are really paired though. If membership was randomly assigned to your test and control groups, you have repeated measures of independent samples. In this case, you may want a nonparametric alternative to a mixed-model. Check out this reference (it includes r code):
Gu, C., & Ma, P. (2005). Generalized nonparametric mixed-effect models: Computation and smoothing parameter selection. Journal of Computational and Graphical Statistics, 14(2), 485–504. Retrieved from http://www.stat.purdue.edu/~chong/ps/guma.pdf.
