# st: RE: ratios

 From "Nick Cox" <[email protected]> To <[email protected]> Subject st: RE: ratios Date Mon, 24 Nov 2003 20:20:14 -0000

```There are lots of papers on the pathologies
of ratios, going back at least to Karl Pearson

* a debate in Systematic Zoology following a
paper by Atchley et al. 1976 (Statalisters
will especially prize the characteristically
lucid and trenchant comment by our own Michael
Hills in 1978)

* a paper in Water Resources Research by Kenney
in 1982.

* Kronmal in JRSS a few years ago.

However, I'm tempted to say that all this fear
of ratios, while in a sense totally justified,
can be overdone. After all, all rates, percents
and proportions are ratios, at least in principle.
Perhaps most are also measured as ratios. Think
of physics without velocities, demography
without birth or death rates, etc., and you
lose much of the basic vocabulary of many sciences.
Anything based on, or using, differential calculus
is in the same boat. We owe to Galileo and Newton
the interesting ratio

free-fall velocity
------------------
elapsed time

However, many ratios y/x have a most obvious
rationale if y / x = constant, at least
approximately, and many of the problems met
in practice come from (implicitly) assuming
that when it is not a fair assumption, or
at the very least misses interesting fine
structure.

A very simple example is when people
decide to summarize shape by ratios,
say width/length, when in practice shapes
vary more subtly than such a ratio implies.

Much the same can be said about the
coefficient of variation, which is
a natural parameter if indeed

sd / mean

is really constant, but a fine
source of obscurity otherwise.

So I'd rather say, banally, it
all depends. Some ratios are familiar
and we know their vices enough to
work with them; some really can bite us.

Nick
[email protected]

David Airey

> In the biological sciences, ratios are often used, but some
> deprecate
> their use. I have been told the ratio sometimes remains
> correlated with
> the denominator, and so the use of ratios to control for or
> normalize
> to the denominator may not do the trick. The distribution
> of the ratio
> or percent is also sometimes difficult to deal with, and
> may benefit
> transformation. Finally, since there are two hidden components to a
> ratio, one doesn't always know whether an increase in a
> ratio is due to
> an increase in the numerator or a decrease in the denominator. This
> would seem particularly vexing in more complex experimental
> settings
> with interactions.
>
> I recently read a note from Cornell on spurious results from using
> ratios, especially when more than one ratio is used:
>
>
> Now, care of Alex Tsai, I have read:
>
> Goldman DP, Smith JP. Methodological biases in estimating
> the burden of
> out-of-pocket expenses. Health Serv Res. 2001 Feb;35(6):1357-64;
> discussion 1365-70. Comment on: Health Serv Res. 1999 Apr;34(1 Pt
> 2):241-54.
>
> which is a surprisingly recent dispute on the merits and
> pitfalls of
> ratios in statistics between experts, with quite serious
> implications
> for policy change.
>
> So, the question I have, is it not possible to ask all the same
> questions using the numerator and denominator separately in a model
> that are asked by using the ratio? If all questions cannot
> via analysis of the components of the ratio, in case where
> they can, is
> this not preferred?

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