# Statistical paired test for relative change

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?

• One way to understand the relationship between the log transformation of two values $x_1$ and $x_2$ and their relative error $e=\frac{x_2-x_1}{x_1}$ is to note that $\log(x_2)-\log(x_1) = \log(\frac{x_2}{x_1}) = \log(1 + e) = e + O(e)^2,$ showing that when $|e|\ll 1$ (which frequently is the case unless changes are dramatic), a difference of logs basically is the relative error.
– whuber
May 7, 2014 at 18:23
• That is useful. However the part $O(e)^2$ makes me quite uncomfortable, because I am not sure, how would this affect the test results. So should we test $\log (x_2) - \log(x_1) = 0$ vs. $<>0$? With whatt test can I carry on?
– kuba
May 7, 2014 at 18:43