Eventually in some studies data transformation is inevitable to use proper statistical test, however when we are going to report our result, we report original data and we use data transformation to judge our results, right?
Here is my question: when we are reporting a bar graph with error bars, how should we calculate Standard Errors (SE)? Using original data, or re-transforming SE using transformed data?
You cannot (re-)transform a standard error. This makes no sense. The standar error is linked to that parameter you estimate (be it from untransformed or transformed data). If you calculate an estimate and its SE on transformed data but you want to show the result and the uncertainty on the "original" scale, you can calculate the limits of a confidence interval on the transformed scale and re-transforme those limits to get the limits on the "original" scale.
Once you transform your data, any and all interpretations of your analyses are based on the data in the "transformed space".
For example, say you use a log-transformation to achieve a normal distribution on the dependent variable "depression", to test for the effect of the independent variable "hours of exercise" your DV. This means that any effect you now find in your analysis, is on depression in the "log-space". What is the log-space? It's an abstract vector that your variable now resides in (the term is as vague as its explanation), and I have yet to find someone who actually knows or is able to explain what this transformed space represents.
What does this mean for your interpretation though? You're no longer interpreting depression as it was measured by your instrument, instead your interpretation is on the logarithmic function of your measurement. This means that when you want to graph something based on your analysis, you will have to use the transformed data. Because in the end, you did not analyze the raw data, so it would make no sense to suddenly start graphing and interpreting it.
"What is the log-space?"
I may help you here: This space maps the proportional changes. If you analyze the data in the log-space you analyse relative (proportional) changes rather than absolute changes. A difference of the logs represents a ratio of the original values. In the log-space you can analyze "fold-changes" (ratios) as if these were simple additive shifts (differences).
In biological systems (and this may extend to areas like economics and social sciences), the log-space it is often much more "representative" for the relevance of effects that are studied!
This is special about the logarithm. Other transformations can be tricky, because the meanings of coefficients in a linear (additive) model change and get obscured so that their interpretation might not be possible. But this depends on the model and on the transformation. For instance, the inverse transformation (1/x) can make the coefficient of a simple model (that has no other coefficients) interpretable as a rate (of some process). However, when the model has several coefficients, this interpretation gets lost (this does not mean that the coefs don't have any interpretation - it just means that it changes, and the correct interpretation may be too complicated to be understood [at least by me]).
Basically, if you have transformed your data using a monotonic transformation Yt = f(Y), and you have mean and standard deviation on anything on Yt, you have two ways to obtain it on Y:
- either the so-called delta method, which basically is using the linear formula on variances, V(aX+b) = a²V(X), applied on the first order Taylor expansion of the reciprocal of f around the mean;
- or, as suggested by Jochen, compute à 95% interval, transform it using the reciprocal of f and assume a roughly Gaussian distribution so that half the width of the interval is about twice the SD.
Note that in both cases, interpretation of this SD can be difficult, because in general, transformation are meant to correct for high asymmetry [log transformation is a typical example], hence confidence intervals will not be centered on the back-transformed estimate, and representing the SD can be quite misleading.
So basically, presenting either confidence intervals or quartile intervals or 5%, 95% percentile intervals or any other interval that respect the distributionnal shape of the original data will be less missleading...
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Article TO DETERMINE SKEWNESS, MEAN AND DEVIATION WITH A NEW APPROAC...
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