Back-transforming a reflected logged variable

Question

In order to prepare variables for multiple imputation, I did some data transformation on skewed variables. Therefore, I reflected them (largest value+1 minus variable) and took the lg10.

After imputation, I want to present some descriptives of my data and therefore "undo" the transformation on the imputed data. Unfortunately, when taking the 10th power of my transformed variable, I do get the original SD, but not the original MEAN and MIN and MAX (tested this in the non-imputed data set). Also, I can find no constant to add or subtract to render MEAN to my non-transformed variable.

I think there must be an easy explanation or solution, but I don't seem to find it right now. I'm grateful for your help!

Answer 1

Here is a sample calculation using Stata's Mata. The only quirk that should need explanation is that the colon prefix : flags elementwise computation. I don't see why you can't retrieve your original maximum and minimum. In general, however, as taking means is linear and taking logarithms is not, means are not reversible unless all values are identical. Reversibility of SD under nonlinear transformation looks like an illusion for similar reasons.

: orig = (0,1,2,3,5,7,11,13)

: trans = max(orig) :- orig  :+ 1

: trans
        1    2    3    4    5    6    7    8
    +-----------------------------------------+
  1 |  14   13   12   11    9    7    3    1  |
    +-----------------------------------------+


: max(exp(log(trans))) :- exp(log(trans))
        1    2    3    4    5    6    7    8
    +-----------------------------------------+
  1 |   0    1    2    3    5    7   11   13  |
    +-----------------------------------------+

If this doesn't answer your question, show results for a simple dataset, like this one, making clear the software instructions or commands you used.

To reverse logarithms to base 10, use a power of 10 instead.