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?

  • $\begingroup$ Am I missing something? If $X$ and $Y$ are log-normal, then surely $X/Y$ is also log-normal...? $\endgroup$ – M Turgeon Jun 29 '16 at 16:15
  • $\begingroup$ @Turgeon: Yes, log(Y/X) is normal. But I am not sure if Y/X is lognormal when retransformation bias is taken into account. I think that the Wikipedia page has to elaborate on this idea. $\endgroup$ – KuJ Jun 29 '16 at 17:04
  • $\begingroup$ can I compare the ratios directly without log transformation? In at least the following case and implicitly, you are doing the "comparison" of ratios. It is when you compute $\chi^2$ statistic of a contingency table. One way to put its formula is $\sum_{rc}[O_{ij}G_{ij}] - N$, where $O_{ij}$ is the obs. freq. in the cell and $G_{ij}$ is the ratio of it to the expected frequency there. And, therefore, when you compute the (squared) chi-square distance between rows i and i' in the table you are computing differences between the ratios: $d_{ii'}=1/N \sum_c[O_{.j} (G_{ij}-G_{i'j})^2]$. $\endgroup$ – ttnphns Jun 30 '16 at 12:15

Not only do distributions of untransformed ratios have odd shapes not matching the assumptions of traditional statistical analysis, but there is no good interpretation of a difference in two ratios. As an aside if you can find an example where the difference in two ratios is meaningful, when the ratios do not represent proportions of a whole, please describe such a situation.

As a variable used in statistical analysis, ratios have the significant problem of being asymmetric measures, i.e., it matters greatly which value is in the denominator. This asymmetry makes it almost meaningless to add or subtract ratios. Log ratios are symmetric, and can be added and subtracted.

One can spend a good deal of time worrying about what distribution a test statistic has or correcting for the distribution's "strangeness" but it is important to first choose an effect measure that has the right mathematical and practical properties. Ratios are almost always meant to be compared by taking the ratio of ratios, or its log (i.e., double difference in logs of original measurements).

  • $\begingroup$ Dear Professor Frank Harrell: Thank you for your kind answer. I have revised the question. Two examples: 1. BMI is not normal (ncbi.nlm.nih.gov/pubmed/26973438) and may or may not be lognormal. If it is, then log(BMI) is normal. If it is not, then log(BMI) is not normal. However, log(BMI) is rarely used. 2. Fold-changes (Y1/X, Y2/X) of proteins or genes of two experimental groups (Y1, Y2) are compared to a control group (X). Thus, the difference in two ratios are meaningful but the relative difference is not because both experimental groups are normalized by a common control. $\endgroup$ – KuJ Jun 28 '16 at 11:18
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    $\begingroup$ That logic is not correct. Assuming BMI is the dependent variable, it will behave better when logged than when not logged, with respect to linear model assumptions, although better would be to model weight adjusted for height and initial weight. The fact that fold changes in protein expression are compared doesn't mean you subtract two fold changes. The more proper measure would be to take the ratio of fold changes. Normalization is another issue altogether. The practice of separate normalization steps is not good statistically because it assumes controls are measured without error. $\endgroup$ – Frank Harrell Jun 28 '16 at 13:38
  • $\begingroup$ 1. Do you mean that the GM is better than the arithmetic mean of the ratios or the fold-changes (even though the ratios or fold-changes may not be lognormal and the cautions issued by the 1st paper)? 2. This paper (link.springer.com/article/…) showed that the performance of t-test on the ratios and t-test on the log-transformed ratios are similar. Thank you. $\endgroup$ – KuJ Jun 28 '16 at 15:34
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    $\begingroup$ That paper is drammatically incorrect then. log(Y/X) is mathematically a symmetric function and the distribution of log ratios is much more symmetric than the distribution of ratios. $\endgroup$ – Frank Harrell Jun 29 '16 at 4:11
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    $\begingroup$ I would underline what is not part of this excellent advice. Whether ratios are exactly or even approximately lognormal before and exactly or even approximately normal after transformation can't be predicted in advance or in general, and fortunately is quite secondary. The key point is that ratios of positive numbers are often so extraordinarily skewed given that $X < Y$ maps to $0 < X / Y < 1$ and $X > Y$ maps to $X / Y > 1$ that plotting untransformed data, using them in models and thinking about them is almost always much more awkward than working with their logarithms. $\endgroup$ – Nick Cox Jun 30 '16 at 11:46

The answer from @FrankHarrell, and associated comments from him and @NickCox, answer the question admirably. I would add that the implicit focus on the shape of raw distributions of predictors and outcome variables is misplaced; in linear modeling, what's important is linearity of relations of predictors to outcome and the distribution of residuals.

I also wish to add information on two articles cited in the original question that might explain some sources of the difficulty sensed by the OP. It's important to evaluate articles critically, not just accept them because they happen to have been published.

The cited paper on misuses of log transformations by Feng et al rightly notes some abuses that are possible with log transformations, but tends to leave the impression that log transformations should be avoided rather than used intelligently. For example, the paper says:

using transformations in general and log transformation in particular can be quite problematic in practice to achieve desired objectives

with alleged difficulties noted such as:

there is no one-to-one relation between the original mean and the mean of the log-transformed data...it is not conceptually sensible to compare the variability of the data with its transformed counterpart ... comparing the means of two samples is not the same as comparing the means of their transformed versions

and concluding:

rather than trying to find an appropriate distribution and/or transformation to fit the data, one may consider abandoning this classic paradigm altogether...

I don't see that the alleged difficulties noted in that paper provide reasons to avoid informed use of logarithmic or other transformations. Others have noted more serious deficiencies in that paper. Bland, Altman and Rohlf wrote a direct response, In defence of logarithmic transformations. The full response is apparently behind a paywall, but I believe the following quotes would constitute fair use:

They do not illustrate their article with any real data, however, and appear largely to ignore the context in which log transformations are applied...They also quote out of context the people they criticise...Feng et al. also say ‘Although well-defined statistically, the quantity Exp(E(log X)) has no intuitive and biological interpretation.’ We find no problem in intuition concerning it. Although the expression looks complicated, it is simply the geometric mean.

Bland, Altman and Rohlf conclude:

Log transformation is a valuable tool in the analysis of biological and clinical data. We do not think anyone should be discouraged from using it by this ill-argued and misleading paper.

The paper that "advises to use ANOVA to test the differences among raw fold differences (FD) in immunoblotting" deals nicely with some of the technical difficulties in performing densitometry of what are called "western blots" (difficulties of which I am painfully aware), yet the almost off-hand suggestion at the end of the paper to "Determine the average FD and associated P values for the biological replicates by importing the FD from step (2) above into a statistical analysis software package such a PRISM or Analyze IT" does not seem to have received very critical review. (It also does not rule out the possibility of log-transforming the FD values in the statistical analysis.)

A suggestion to use raw FD actually contradicts the idea presented earlier in that paper that this analysis is "a very similar methodology to qPCR," or the quantitative polymerase chain reaction. Statistical analysis of qPCR is best done on the values of "cycles to threshold" or $C_t$ values. These $C_t$ values have direct $\log_2$ relations to the original amounts of the nucleic-acid sequence being analyzed. Of further note in nucleic-acid quantification, the MA plot widely used in microarray analysis is a Bland-Altman plot on logarithmic transformations of expression data. When errors are proportional to values of interest, the logarithmic transformation can make lots of sense.

  • $\begingroup$ Thank you for your detailed explanations. Your answer helps me to understand this problem more clearly. $\endgroup$ – KuJ Jul 2 '16 at 15:47
  • $\begingroup$ Excellent analysis. It's astonishing that the paper you dissect got published. It's full of poor arguments and bad advice. How to work on log scale is utterly basic and belongs in every introductory text. $\endgroup$ – Nick Cox Jul 4 '16 at 8:37

If both $X$ and $Y$ are normal with zero mean, then the ratio $X/Y$ follows a Cauchy distribution with density

$p(x) = \frac{1}{\pi \gamma} \frac{\gamma^2}{(x-x_0)^2 + \gamma^2}$

where $x_0$ is the location parameter, which is kind of a measure of the centrality of the mass, and $\gamma$ the half-width, which is kind of the standard deviation for Cauchy. It has no mean, no variance and no higher moments.

  • $\begingroup$ If X is zero, then Y/X is undefined. $\endgroup$ – KuJ Jun 29 '16 at 23:09
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    $\begingroup$ @KuJ Since $X$ and $Y$ are random variable, the probability that either one would exactly equal to zero is zero. $\endgroup$ – horaceT Jun 30 '16 at 0:01
  • $\begingroup$ If it has no mean, no variance and no higher moments, how can we do statistical tests? $\endgroup$ – KuJ Jun 30 '16 at 3:28
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    $\begingroup$ @KuJ Exactly the same question has been asked. See stats.stackexchange.com/questions/172101/… $\endgroup$ – horaceT Jun 30 '16 at 4:41

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