Re: [R] Confidence intervals of log transformed data

From: Rubén Roa-Ureta <rroa_at_udec.cl>
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.
HTH
Rubén

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