p.adjust is the simplest R function for adjusting p-values, though do you have an assumed ordering of control, low, and high? Also, there are tons of analysis procedures for response times, many of them having their parameters designed to match onto the decision processes (e.g., Ratcliff's 1978 diffusion model, and lots of those developed from it). These usually assume that the RTs are skewed, perhaps log-normal, etc. What were your expectations from theory for these distributions and were the observations similar? Of course transforming them means your are testing different hypotheses. Finally, you say your experiment was designed to provoke outliers. So why would you want to do tests of the center of the distribution moving. If you want to look for shifts in the extreme of the distributions, why aren't you looking at those (and not lessening their impact)?

Thanks for the reply Daniel Wright . I didn’t even know that ordering was a factor to be considered for mixed ANOVAs. Every paper that I have read that has conducted a similar manipulation has used a repeated ANOVA as the main analysis. As I have rarely ventured beyond the basics in statistical analyses before, I thought emulating these was the best option. However, the data for results and time taken are non-normal, even after performing a log transformation. Previous studies make no reference to dealing with outliers in the data. Having read a number of recent papers in my field of study advocating for robust methods, I came across the WRS2 package for R, as my version of SPSS cannot perform robust analysis. I will investigate other procedures for RT analysis, although the fact that I have never encountered these in previous similar studies worries me that I am going down a rabbit hole.

As your data non normally distributed you need to go kraksal wallas test instead of ANOVA. Then for post hoc test you need to make chi square test between each two Levels or groups to get the full results of comparison.

Shane Barry , given you have response time data, could be useful to look at response time models. Ratcliff's and related ones are from cognitive science, so for things like lexical decision tasks in linguistics, but they might be applicable in other areas (and the diffusion models are originally from Brownian motion in physics).