Re: [R] Confidence intervals of log transformed data

From: Rubén Roa-Ureta <>
Date: Wed, 16 Apr 2008 13:04:51 -0400

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.
Rubén mailing list PLEASE do read the posting guide and provide commented, minimal, self-contained, reproducible code. Received on Wed 16 Apr 2008 - 17:10:55 GMT

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