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Re: st: Use of the term 'relative risk ratio'


From   rgutierrez@stata.com (Roberto G. Gutierrez, StataCorp)
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Use of the term 'relative risk ratio'
Date   Thu, 21 Apr 2005 09:09:01 -0500

There has been some discussion about the use of the term 'relative risk ratio'
to describe the exponentiated coefficients from an -mlogit- model.  To start,
I agree that the issue can be confusing and I am open as to how the confusion
can be mitigated.

Just so that we are all on the same page, 

Some background
---------------

Suppose we have a 4-category -mlogit- model.  Define

     p1 = P(Y=1 | x) 		q1 = p(Y=1 | x+1)
     p2 = P(Y=2 | x)		q2 = p(Y=2 | x+1)
     p3 = P(Y=3 | x)		q3 = p(Y=3 | x+1)
     p4 = P(Y=4 | x)		q4 = p(Y=4 | x+1)

If category 3 is of the one of interest, then 

   odds ratio for category 3 = q3/(1-q3)
                               ---------         (OR)
                               p3/(1-p3)

   risk ratio for category 3 = q3
                               --                (RR)
                               p3

Is either of these equivalent to the exponentiated coefficient from -mlogit-
for category 3 on variable -x-?  The answer is no.  That is because everything
in an -mlogit- is model is stated relative to a base category.  Suppose that
here the base category is category 1, in which case we now define the
'relative risk' to be

             P(Y=3)/P(Y=1)

that is, is the risk relative to the base category.  As such, the
exponentiated coefficient in -mlogit- is the ratio of two relative risks, the
one given x+1 to the one given x.  Hence, we call it a relative risk ratio.

   relative risk ratio for 3 = q3/q1
                               -----             (RRR)
                               p3/p1

and this is not (in general) the same as an odds ratio or a risk ratio.

What makes this all confusing?
------------------------------

(1) When you have only two categories, the RRR is equivalent to the OR.  
    This degeneracy, however, does not make an adequate case for calling 
    them odds ratios all the time.  

(2) Risk ratios are sometimes called relative risks and vice versa, which is
    fine since both are ratios of probabilities (risks).  As I have defined
    them they are distinct ratios, but why this would be confusing is easy to
    understand.  Others treat risk ratios and relative risks as the same
    thing.  Fair enough.

(3) Even though I've been clear distinguishing between a relative risk and
    a risk ratio, the term 'relative risk ratio' used to mean a ratio of 
    relative risks has the term 'risk ratio' imbedded within it.  Quite 
    confusing since we have established that it is not a risk ratio.

(4) I've heard that our RRR's are called odds ratios in other software.
    There's not much I can do about that.

What do we do about it?
-----------------------

It seems clear that if we had a less confusing term than 'relative risk ratio'
for the calculation, we would use it.

Ronan Conroy <rconroy@rcsi.ie> has a good point when he states:

> I would like to preserve the distinction between vanilla-flavour odds ratios
> and relative risk ratios, but am a little unhappy that the term is causing
> more heat than light.

Roger Newson <roger.newson@kcl.ac.uk> suggests 'multinomial odds ratio',
Michael Ingre <Michael.Ingre@ipm.ki.se> suggests 'relative odds ratio'.  I am
leaning towards 'multinomial odds ratios', yet it is early in the day here in
the U.S.  If anyone would like to suggest an alternative, I am all ears.

When this is resolved, we will post an FAQ on all of this so that Stata user's
may more easily handle questions from reviewers.

--Bobby
rgutierrez@stata.com
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