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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