Back-transformation of log-transformed mean and variance
Converts a log-mean and log-variance to the original scale and calculates confidence intervals
sample mean of natural log-transformed values
sample standard deviation of natural log-transformed values
alpha-level used to estimate confidence intervals
There are two methods of calcuating the bias-corrected mean on the original scale.
bt.mean is calculated following equation 14 (the infinite series estimation)
in Finney (1941).
approx.bt.mean is calculated using the commonly known approximation
from Finney (1941):
mean=exp(meanlog+sdlog^2/2). The variance is var=exp(2*meanlog)*(Gn(2*sdlog^2)-Gn((n-2)/(n-1)*sdlog^2) and standard deviation is sqrt(var) where Gn is the infinite series function (equation 10). The variance and standard deviation of the back-transformed mean are var.mean=var/n; sd.mean=sqrt(var.mean). The median is calculated as exp(meanlog). Confidence intervals for the back-transformed mean are from the Cox method (Zhou and Gao, 1997) modified by substituting the z distribution with the t distribution as recommended by Olsson (2005):
A vector containing
median, LCI (lower confidence interval), and UCI (upper confidence interval).
Gary A. Nelson, Massachusetts Division of Marine Fisheries firstname.lastname@example.org
Finney, D. J. 1941. On the distribution of a variate whose logarithm is normally distributed. Journal of the Royal Statistical Society Supplement 7: 155-161.
Zhou, X-H., and Gao, S. 1997. Confidence intervals for the log-normal mean. Statistics in Medicine 16:783-790.
Olsson, F. 2005. Confidence intervals for the mean of a log-normal distribution. Journal of Statistics Education 13(1). www.amstat.org/publications/jse/v13n1/olsson.html
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## The example below shows accuracy of the back-transformation y<-rlnorm(100,meanlog=0.7,sdlog=0.2) known<-unlist(list(known.mean=mean(y),var=var(y),sd=sd(y), var.mean=var(y)/length(y),sd.mean=sqrt(var(y)/length(y)))) est<-bt.log(meanlog=mean(log(y)),sdlog=sd(log(y)),n=length(y))[c(1,3,4,5,6)] known;est
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