[R-sig-ME] Variable transformation and back transformation

Christina Bogner christina.bogner at uni-bayreuth.de
Mon Mar 23 11:58:36 CET 2009

ONKELINX, Thierry schrieb:
> Dear all,
Dear Dr. Onkelinx, dear Dr. Steibel, dear list,

thanks a lot for your help!
> Christina's question will be a bit more clear with some extra background
> information on the topic. When interpolating concentrations in the soil
> with kriging, a log-transformation is often used. The predicted value
> for a location in the log-scale is a distribution with mean mu(s) and
> standard deviation sigma(s). Mu(s) is a function which yields the mean
> value depending on location s.
> A simple form of backtransformation would be exp(mu(s)). That will no
> longer be the mean of the distribution (in the original scale) at
> location s, but rather its median. The mean of the distribution in the
> original scale is exp(mu(s) + 0.5 * sigma(s) ^ 2). That is the reason
> why Christina includes the variance in the backtransformation.
> This is the backtransformation one would use with a linear model. So I
> suppose that Christina's question is how the deal with the variance from
> the random effect in such a backtransformation.
Indeed, I am unsure how to treat the random-effects. On one side, a 
random-effect is a random variable like the within-group error. On the 
other side, a mixed-effects model provides an estimate of random-effects 
for different experimental units. So what should one to do when 
backtransforming: taking 0.5*random-variance plus 0.5*within-group error 
or the estimate of the random-effect for the respective experimental 
unit and 0.5*within-group variance?
The latter approach follows the logic of fitted values in nlme: we have 
estimates on population level and to get estimates on the level of 
experimental units we add the estimates of random-effects (realisations 
of random variables). So, for backtransformation, I used the fitted 
values on the level of experimental units and added 0.5*within 
group-variance. So, somehow, I treated the fixed and the random-effects 
equally and it confuses me!.
> Personally I would settle with confidence intervals. The nice thing
> about them is that they are based on the order of values. Since a
> monotone transformations (like exp() and log()) don't change the order
> of values, the backtransformation is straightforward: exp([LCL, UCL]).
> That requires two maps to depict the information.
> If you have some important treshold (e.g. some legal maximum
> concentration) you could create a map with the probability of exeeding
> that treshold.
> HTH,
> Thierry
Thanks again!

> ------------------------------------------------------------------------
> ----
> ir. Thierry Onkelinx
> Instituut voor natuur- en bosonderzoek / Research Institute for Nature
> and Forest
> Cel biometrie, methodologie en kwaliteitszorg / Section biometrics,
> methodology and quality assurance
> Gaverstraat 4
> 9500 Geraardsbergen
> Belgium 
> tel. + 32 54/436 185
> Thierry.Onkelinx at inbo.be 
> www.inbo.be 
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