[R] Confidence intervals of log transformed data

Tobias Verbeke tobias.verbeke at telenet.be
Wed Apr 16 20:00:10 CEST 2008

Rubén Roa-Ureta wrote:
> tom soyer wrote:
>> Hi
>>  I have a general statistics question on calculating confidence interval of
>> log transformed data.
>> I log transformed both x and y, regressed the transformed y on transformed
>> x: lm(log(y)~log(x)), and I get the following relationship:
>> log(y) = alpha + beta * log(x) with se as the standard error of residuals
>> My question is how do I calculate the confidence interval in the original
>> scale of x and y? Should I use
> [...]
> Confidence interval for the mean of Y? If that is the case, when you 
> transformed Y to logY and run a regression assuming normal deviates you 
> were in fact assuming that Y distributes lognormally. Your interval must 
> be assymetric, reflecting the shape of the lognormal.  The lognormal 
> mean is lambda=exp(mu + 0.5*sigma^2), where mu and sigma^2 are the 
> parameters of the normal variate logY. A confidence interval for lambda is
> Lower Bound=exp(mean(logY)+0.5*var(logY)+sd(logY)*H_alpha/sqrt(n-1))
> Upper Bound=exp(mean(logY)+0.5*var(logY)+sd(logY)*H_(1-alpha)/sqrt(n-1))
> where the quantiles H_alpha and H_(1-alpha) are quantiles of the 
> distribution of linear combinations of the normal mean and variance 
> (Land, 1971, Ann. Math. Stat. 42:1187-1205, and Land, 1975, Sel. Tables 
> Math. Stat. 3:385-419).
> Alternatively, you can model directly
> Y=p1*X^p2, p1=exp(your alpha), p1=beta
> with a lognormal likelihood and predict the mean of Y with the fitted 
> model (I'm guessing here).
> It could be useful to check Crow and Shimizu, Lognormal distributions. 
> Theory and practice, 1988, Dekker, NY.

For the record, I'm working on a package to deal with these problems


I uploaded a very first function lnormCI to
the svn repository a few minutes ago;
Be cautious, though: it is pre-alpha and I
know there is a problem with at least one
of the methods implemented (haven't worked
on it since 5 months or so).


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