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 ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
