Untransforming

There is a major problem in transforming data.  All of your analysis is now on a scale that is different from your original scale!  The units are now transformed units.  And, the nonlinearity of the transformation is going to make trouble for you.  For example, you may want to present means and standard deviations on the original scale, but these are going to seem off compared to results on the transformed scale.

There are a few ways around this.  The first way is a bit disingenuous.  Present your data on the original scale, but perform the statistical analysis on the transformed data.  Be sure to label things appropriately, with some or all of the following:

  • The Statistical Methods section should indicate something like “The  logarithmic transformation was used where appropriate.”
  • The Results section needs to indicate whether data were transformed   or untransformed in the text and any figures or tables.
  • The Results section may need to indicate that the transformed and   untransformed analyses are similar.

There is nothing formally wrong with this.  You have informed the reader of exactly what has been done, and everything is clearly labeled so that there is (in theory) no confusion.  However, it is slightly disingenuous because we all know that most people are going to gloss the text under the implicit assumption that the statistical analysis is on the same scale as the data presentation.

The second way is to look at the results from transformed and untransformed data.  If they are similar in outcome, you can consider simply presenting the untransformed analysis with perhaps a mention of the transformed analysis.  This is methodologically wrong, but may be pragmatic.

The third way is to bite the bullet and do the math necessary to back-transform everything.  This can be a headache sometimes, and you still won’t avoid the problem of reviewers noting the differences between the untransformed and transformed analyses.  You can alleviate this somewhat by very clearly indicating that results have been back-transformed in the Statistical Methods and Results sections.

Really, when you transform the data you enter into a no-win situation many times.  Depending on the whim or character of reviewers, you can get complaints from any of these approaches.  The first way may be criticized as being inconsistent.  The second way may be criticized as wrong.  The third way may be criticized as confusing or perhaps unnecessary.

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