# Back-transformation of log-transformed mean and variance

### Description

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

### Usage

1 |

### 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 gary.nelson@state.ma.us

### 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
``` |

Want to suggest features or report bugs for rdrr.io? Use the GitHub issue tracker. Vote for new features on Trello.