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 at http://r-forge.r-project.org/projects/lognorm/ 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). Regards, Tobias ______________________________________________ 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.
