If X and Y are log-normal, then log(X) and log(Y) are normal, hence
log(X)-log(Y)=log(X/Y) is normal too, and so X/Y is log-normal.

Best,
Bendix
----------------------
Bendix Carstensen
Senior Statistician
Steno Diabetes Center
Niels Steensens Vej 2
DK-2820 Gentofte
Denmark
tel: +45 44 43 87 38
mob: +45 30 75 87 38
fax: +45 44 43 07 06
[log in to unmask]
www.biostat.ku.dk/~bxc
----------------------



> -----Original Message-----
> From: A UK-based worldwide e-mail broadcast system mailing 
> list [mailto:[log in to unmask]] On Behalf Of Allan Reese (Cefas)
> Sent: Tuesday, November 15, 2005 12:11 PM
> To: [log in to unmask]
> Subject: Query re Ratio statistics
> 
> 
> I'm working with ratio data.  The statistic of interest is 
> the ratio of two measurements.  Both measurements are counts 
> but are large with proportional accuracy, and each follows a 
> lognormal distribution.  Typical values for the ratio are 
> 10^4 to 10^8.  I'm looking for any theory to support 
> distributional assumptions. 
> 
> I across some SPSS documentation on a procedure I'd not 
> previously noticed.  "Ratio statistics" is in the base 
> module. 
> http://www.rrz.uni-hamburg.de/RRZ/Software/SPSS/Algorith.115/ratio.pdf
> lists the formulae.
> 
> The reason for writing is that the documentation, and a 
> footnote in the output, state the assumption that the ratios 
> follow a *normal* distribution.  The standard I'm working to 
> assumes that ratios will follow a *log-normal* distribution.
> 
> I found plenty of references to the ratio of two *standard* 
> normal variates (Cauchy) and plenty to ratios of quadratic 
> functions of normal variates (F).  Kendall (Advanced Theory 
> of Statistics) gives a general integral for X/Y in terms of 
> distributions and characteristic functions.  Attempting to 
> apply this, the ratio of two independent normal variates 
> seems to integrate back to a normal distribution (dF 
> proportional to exp(-x^2)).  
> 
> However, running simulations (500 repeats of 2000 sample 
> pairs), suggests pretty conclusively that the ratio does not 
> follow a normal distribution but is fully consistent with a 
> lognormal.  From simulations: normal/normal ->  lognormal; 
> lognormal/lognormal -> lognormal.
> 
> I think this happens because both counts (called N and Na) 
> are definitely positive numbers with pretty small CVs.  
> Taking the inverse of a standard normal distribution gives a 
> distribution with an asymptote at zero, but 1/Na gives 
> another simple peaked distribution.  Hence N/Na is very like 
> a product of independent normal variates and tends to a 
> lognormal distribution.
> 
> Would anyone like to comment or point to references on 
> whether it is more correct to assume that the ratio of two 
> variates will follow a normal, lognormal or other distribution?
> 
> Allan
> 
> 
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