Ratios (e.g. $Z$=$Y$/$X$) are frequently used (e.g. fold-changes in mRNA or protein expression, body mass index [BMI], etc.). Many people advise that variables coded as ratios (e.g. fold-change) should be log transformed because they are heavily skewed to the right. However, ratios ($Y$/$X$) are relative changes and ratio distributions are not normal (en.wikipedia.org/wiki/Ratio\_distribution). If both $X$ and $Y$ are lognormal, then log($Y$/$X$) is normal (is $Y$/$X$ lognormal after taking retransformation bias into account?)

The comparisons between the log transformed ratios are relative changes of the relative changes (i.e. the ratios). Moreover, the necessity of log transformation for right-skewed variables ($Y$) has been questioned. For example, a recent paper (<http://www.ncbi.nlm.nih.gov/pubmed/22806695>) cautions about the misuses of log transformation for a variable. Some of the advices were that log($Y$) guarantees normal distribution only if $Y$ is lognormal. Namely, it does not guarantee normality even for right-skewed variables. Moreover, the anti-log of E(log($Y$)) is the geometric mean (GM) of $Y$, which is always less than E($Y$) and the tests of the differences of E($Y$) and the GM are different. Finally, the GM is neither more robust nor less likely to be affected by the outliers.

Another paper (<http://econtent.hogrefe.com/doi/10.1027/1614-2241/a000110>) showed that t-tests on the raw variables performs well even for lognormally distributed variables. A 3rd paper (<http://link.springer.com/article/10.1023%2FB%3AEEST.0000011364.71236.f8>) showed that the performance of t-test on the ratios and t-test on the log-transformed ratios are similar.

Thus, the question becomes which is the outcome of interest. Because log($Z$) has to be back-transformed to the original units to be meaningful and because of the retransformation bias, I think that the tests of E($Z$) are more meaningful.

Fortunately, parametric tests (e.g. t-tests) are robust to the violation of normality assumption once heteroscedasticity is accounted for (e.g. Welch's t-test). For example, this paper (<http://www.ncbi.nlm.nih.gov/pubmed/24738055>) advises to use ANOVA to test the differences among raw fold-changes in immunoblotting.

So my question is: If my goal is to test the absolute change of the ratios, can I compare the ratios directly without log transformation?

Reference:
[In linear regression, when is it appropriate to use the log of an independent variable instead of the actual values?](https://stats.stackexchange.com/questions/298/in-linear-regression-when-is-it-appropriate-to-use-the-log-of-an-independent-va)