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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.

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  • $\begingroup$ Was there any reason why you transformed the data in the first place? $\endgroup$ Jul 29 '15 at 22:42
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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.

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  • $\begingroup$ Thank you very much for your quick response!!! So you suggest me continue using the transformed variable even for the pairwise analyses? But may be in this case I should interprete only the P vales (Sig.) i.e. if the effects are significant but not the exact values given in the table about the Mean differences? P.S. The condition of equality of variences was violated, that is why I used to the lg10 transformation $\endgroup$ Jul 29 '15 at 23:27
  • $\begingroup$ Yes. Do all of your statistical analyses (including pairwise analyses) in the log-transformed scale, because you presumably want equality of variances for the pairwise tests too. If your readers are expecting data in the original scale, then transform back to that scale for display purposes only. Make what you did clear when describing the methods in your report or publication. $\endgroup$
    – EdM
    Jul 29 '15 at 23:32
  • $\begingroup$ Perfect!! But what do you mean by transforming back to the scale for display purposes? It means that I can use the original data before the transformations or I should make a second transformation of the already transformed variable but this time backwords transformation? $\endgroup$ Jul 29 '15 at 23:56
  • $\begingroup$ if you're reporting individual values, both methods ive the same result, as you note. If you want to report mean values and confidence limits in the original scale, take the means and the confidence limits in the transformed scale and back transform them to the original scale when you construct your summary tables. $\endgroup$
    – EdM
    Jul 30 '15 at 1:14
  • $\begingroup$ You also said earlier that "confidence intervals on the original scale will no longer be symmetric about the mean values" if I use the orgonal data but: when I run the pairwise comparisons as you suggested (with the transformed variable) since I am using Bonferroni test do I have the right to interprete the 95% Confidence Interval for Differences (Lower & Upper Bound) ??? Probably not! $\endgroup$ Jul 30 '15 at 2:12

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