# bt.log: Back-transformation of log-transformed mean and variance In fishmethods: Fishery Science Methods and Models

## Description

Converts a log-mean and log-variance to the original scale and calculates confidence intervals

## Usage

 `1` ```bt.log(meanlog = NULL, sdlog = NULL, n = NULL, alpha = 0.05) ```

## Arguments

 `meanlog` sample mean of natural log-transformed values `sdlog` sample standard deviation of natural log-transformed values `n` sample size `alpha` alpha-level used to estimate confidence intervals

## Details

There are two methods of calcuating the bias-corrected mean on the original scale. The `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):

LCI=exp(meanlog+sdlog^2/2-t(df,1-alpha/2)*sqrt((sdlog^2/n)+(sdlog^4/(2*(n-1)))) and

UCI=exp(meanlog+sdlog^2/2+t(df,1-alpha/2)*sqrt((sdlog^2/n)+(sdlog^4/(2*(n-1))))

where df=n-1.

## Value

A vector containing `bt.mean`, `approx.bt.mean`,`var`, `sd`, `var.mean`,`sd.mean`, `median`, LCI (lower confidence interval), and UCI (upper confidence interval).

## Author(s)

Gary A. Nelson, Massachusetts Division of Marine Fisheries [email protected]

## References

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

## Examples

 ```1 2 3 4 5 6``` ```## 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 ```

fishmethods documentation built on Nov. 21, 2018, 1:04 a.m.