Do you have any suggestions, which test I should use in order check if the relative change is statistically different than 0. I am sure that I should check the relative (fold) changes.
Specifically, for each observation I have two measurements: first and second. I would like to check if $\rm \frac{measurement_{second} – measurement_{first}}{mesurement_{first}}$ is different than 0.
In this question: The result of absolute change and relative change in t test is different, it is suggested to make use of logarithm of change. How to use these logarithms? Something like that:
I transform data: $\rm \log (mesurement_{second}) – \log (mesurement_{first})$ and use paired t-test? What other tests I can use after such transformation?
Are there tests that tackle my problem directly without transformation of data?
Best Answer
First you should decide if you are interested in the mean of the relative changes or the relative change of the means (or substitute median or other summary for mean). We don't worry about this with the paired t-test since the difference of the means is algebraically equal to the mean of the differences. But this is not the case for relative change and other comparisons.
I would suggest doing a permutation test. Your title says "paired" so you can do a paired test:
The p-value is the proportion of permuted statistics that are as or more extreme than your original.
This can be done without any transformations, with mean or median, etc. Whatever makes the most sense to you.