From: barnett@mummy.agsm.unsw.OZ.AU (Glen Barnett)
Subject: Re: Inferences from ratios verses differences
Organization: The Australian Graduate School of Management
Date: Mon, 11 Jan 1993 05:19:41 GMT

In article <1992Dec29.161137.20735@news.weeg.uiowa.edu> jfmurray@news.weeg.uiowa.edu (James F Murray) writes:
>I have a question on the use of ratios verses differences for statistical 
>inferences.  The situation is that we have two ordinal measures from patients 
>in several hospitals. For each patient we have a match between score on 
>admission and a score on discharge.  We want to compare the change in patients 
>from admission to discharge across hospitals.  
   [ stuff deleted ]
>Our choices for comparison are to take the ratio of the admission/discharge
>score for each individual patient (the ratio has to be formulated this way 
>because of the possibility of a 0 admission score)
>OR
>to take the difference between the two individual patient scores 
>(admission - discharge).  
>
>I believe that the difference score should be used
>and tested by nonparametric methods.  My colleague resists using
>the difference because of uncertainty about the effect on reliability and 
>power to  detect differences.  

So why should a ratio not suffer from "uncertainty about the effect 
on reliability and power to  detect differences" to at least as bad
an extent?

>
>Any suggestions or pointers to literature on  a comparison between
>difference and ratio measures would be greatly appreciated.
>
It seems to me a matter totally related to measurement scale:

A ratio seems appropriate when the items are measured on a ratio
scale. The ratio of items measured on a ratio scale has a meaning.

A difference seems appropriate when dealing with items on an
interval scale. The difference between items measured on an 
interval scale has a meaning. The ratio of items measured
on an interval scale is totally meaningless, in general.
For example:
If I calculated the ratio of (Farenheit) temperatures 
in Sydney for today and yesterday, and get .7, and take the same ratio
for LA and get .91, what does that tell me about the comparison between
Sydeny and LA? Well, the temperature went down in both cases, but that's
about all I can say.

If I calculated the differences, and got -19 for Sydney and -20 for LA
(perfectly possible with the above ratios!), I can at least say the change
in temperature was about the same for the 2 cities.

Your data is on an *ordinal* scale. So if one person is admitted
with a '3' and discharged with a '4' they get an admission-discharge
of -1. If admitted with a zero and discharged with a '1' they get the
*same* -1. Yet is the difference between a '3' and '4' the same as
between '0' and '1'? I doubt it (if it was, you have an interval
scale, not an ordinal scale). 

So your difference scores may not be totally meaningful, except as far 
as direction. That is, what you can say, is either the result
went up, or went down, or stayed the same. This is another
(coarser) ordinal scale. You could, for example, compare
two treatments with this approach using a sign test, adjusted
for ties (the ties occur when there is no change in category). 

However, if you say the differences are still meaningful (essentially
asserting an interval scale rather than just ordinal), then 
you could use the differences to test and be able to speak 
meaningfully about the results.

However, I can't see how you can ascribe any meaning whatever
to the *ratios*. What is the point of testing anything on
meaningless quantities?

Just as an indication, someone who comes in with a zero
will get a ratio of 0 NO MATTER WHAT SCORE THEY GO OUT WITH.
Is that sensible? Are they really "the same"?

Glen

P.S. Issues of power and reliability seem secondary: If the quantities
are meaningless, what good is power or reliability ? I contend that
unless your test is appropriate to the scale the data were measured
on, concepts like power have no meaning either.