I have transformed my quantitative variable by using the log10 function in order to run some parametric tests (ANOVA) but when I want to make pairwise comparisons of the mean effects should I use some back transformation functions? For example I can use the reverse function by taking 10 to the power of the transformed variable values but in this case I receive a variable which is totally the same as the original data (before the first transformation). So I am wondering if it is OK for the pairwise comparisons to use the original data (without any transformations) or I should use some more sophisticated back transformation technique / function.
For me it does not make much sense to transform once and after that when I apply a back transformation to have absolutely the same result (i.e., the original data)? This seems like some kind of contradiction because on the one hand, we transform the data but when we cannot use it we just use the original data.
Best Answer
You presumably log-transformed your data for ANOVA because residuals weren't normally distributed and/or they depended on the magnitudes of the data values. So, for the same reasons, further statistical tests (like pairwise comparisons) should also be performed on the transformed data.
When you write up your results you might back-transform to the original scale to make results easier for a reader who expects to see values in that scale, but note that confidence intervals on the original scale will no longer be symmetric about the mean values.