Re: st: lnskew question

[Roger Newson <roger.newson@kcl.ac.uk>]

At 13:23 09/12/02 +0000, I wrote:
> A
> The inverse function of
>
> y=ln(exp(x)-k)
>
> is
>
> x=ln(exp(y)+k)
>
> and you can use this to back-transform the confidence interval for the 
> arithmetic mean of y to get a confidence interval for the "lnskewometric 
> mean" of x. (This exercise might be easier if you use my -parmest- 
> package, downloadable from SSC, which saves the output from a model fit as 
> a data set with 1 observartion per parameter and data on estimates, 
> confidence limits and P-values. Type -ssc describe parmest- to find out more.)
Sorry, that was a mistake. (Thanks to Nick Cox for pointing this out to 
me.) I was misreading -[R] lnskew- and assuming that -exp- meant 
"exponentiation". In fact it means "expression". The inverse function of

y=ln(expression(x)-k)

is

x=inverse_expression(exp(y)+k)

where expression(x) is an expression in x and inverse_expression(z) is its 
inverse. Therefore, the "lnskewometric mean" is defined as

lnskewometric_mean(x)=inverse_expression(exp(arithmetic_mean(y))+k)

where y is defined using the first expression, and arithmetic_mean(y) is 
the arithmetic mean of y. In the special case where expression(x) is the 
exponentiation function exp(x), my previous definition was valid.

However, the fact remains that the "lnskewometric mean" in general may be a 
good proxy for the median, but it is not easy to derive CIs for the 
differences and/or ratios between "lnskewometric means". On the other hand, 
it is fairly straightforward to derive CIs for arithmetic, geometric and 
algebraic means and their differences and ratios.

I hope this helps.

Roger


--
Roger Newson
Lecturer in Medical Statistics
Department of Public Health Sciences
King's College London
5th Floor, Capital House
42 Weston Street
London SE1 3QD
United Kingdom

Tel: 020 7848 6648 International +44 20 7848 6648
Fax: 020 7848 6620 International +44 20 7848 6620
  or 020 7848 6605 International +44 20 7848 6605
Email: roger.newson@kcl.ac.uk

Opinions expressed are those of the author, not the institution.

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