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[分享] Stata的mlogit中为什么用相对风险比RRR（relative risk ratio），而不是比值比OR

alian 发表于 2019-9-27 11:07:34 | 显示全部楼层 |阅读模式

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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.1 x; ^) P. q4 X' J) p\$ K; O

Just so that we are all on the same page, . y2 T, [% H5 e4 p! a! ]

Some background
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Suppose we have a 4-category -mlogit- model.  Define
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p1 = P(Y=1 | x)                 q1 = p(Y=1 | x+1)
p2 = P(Y=2 | x)                q2 = p(Y=2 | x+1)  U& h: X. V% `# L0 V# X  C
p3 = P(Y=3 | x)                q3 = p(Y=3 | x+1)# u0 [" J) A/ ]. W/ _+ R* A1 o
p4 = P(Y=4 | x)                q4 = p(Y=4 | x+1)4 n0 ]. p& K( b7 o* G

If category 3 is of the one of interest, then

odds ratio for category 3 = q3/(1-q3)7 k2 C/ ]  g. Y0 f\$ Y, |
---------     (OR)
p3/(1-p3)- Z% t; r& d8 B- O
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risk ratio for category 3 = q3- |4 \/ u9 r5 F# U# U+ J( j
--                (RR)
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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# H( P6 D' Y\$ M0 I  a0 a

P(Y=3)/P(Y=1)2 c7 t" z8 _& Q3 a

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.9 t9 M: j" J9 T( W
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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.
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What makes this all confusing?
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(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 5 W+ I5 C+ E( d9 I8 M
them odds ratios all the time.
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(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. ; |, S% O) X+ ^; T  Y
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.7 g/ z5 p9 [  r4 U( e
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(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?
-----------------------
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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- R" T5 }3 @) Q8 q' {) S\$ o

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 College Station.  If anyone would like to suggest an alternative, I am all
ears/eyes.
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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./ B4 z" j  W. a- r" X+ \

Finally, given the timing of all this it may be a while before you see the resulting changes to software and documentation.
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Roberto G. Gutierrez, StataCorp
rgutierrez@stata.com* }; Z# H, [* p9 B) J
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In models where only two categories are considered, the mlogit model reduces to standard logit. Consequently the exponentiated regression coefficients, labeled asRRR within mlogit, are equal to the odds ratios as given when theor option is specified under logit; see [R] logit.

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As such, always referring to mlogit’s exponentiated coefficients as odds ratios may be tempting. However, the discussion in example 3 demonstrates that doing so would be incorrect. In general mlogit models, the exponentiated coefficients are ratios of relative risks, not ratios of odds.
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Why does Stata’s -mlogit- command describe Exp(B) as the Relative Risk Ratio (RRR)?4 I% ^7 D0 h! P- w. g( ]6 R

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