multinomial logit interpretation: RRR vs COR

[natalia malancu]

hi guys!

i have read through all the old stata list discussions on odds /relative odds/relative risk and came to the understanding that ORs are not the same as RRR when more than 2 categories are involved.
all good and well until i came across the attached stata technical bulletin and CORs (http://www.ats.ucla.edu/Stat/stata/library/sg124.pdf). could someone please explain to me how one would phrase the interpretation of a coefficient in this setup? the odds of being in 1 versus the base (category 3) sounds very much like someone added the versus word to the OR interpretation or simply replaced the words relative risk with the word odds (in RRR lingo you note that the relative risk of being in category 1 vs base increases by a factor of X, in COR is sounds like the odds of being in 1 versus the base (category 3) increase /fall by .... don't see much difference here)

[Maarten Buis]

If you have read the old discusions on this list, then you know that in my discipline (sociology) we hapily call relative risk ratios odds ratios, so I would not agree with your summary of the discussion that RRRs are different from ORs. This is just a matter of different definitions of what an odds is exactly. There are disciplines that define an odds as the expected number of successes per non-success. In that case an odds can refer only to two mutually exclusive categories, and you need a different word for the same thing if you have more than 2 categories. In my discipline an odds is the expected number of observations in group A per observation in group B. A and B could be mutually exclusive but don't have to be, so we don't need different words for the coefficients in logit and mlogit. Neither definition is wrong or right; they are just definitions. The only important thing is that you and your audience share the same definition.

As to CORs think of a case where your dependent/left-hand-side/y-variable has three categories, 1, 2, and 3 and you run separate logistic regressions of 2 versus 1 while ignoring all observations with value 3 and 3 versus 1 while ignoring all observations with value 2. Those separate logistic regressions will give you odds ratios conditional on being either in either the outcome category or the baseline category. This is what a multinomial logit does plus the additional constraint that all predicted probabilities have to add up to 1. So you can interpret the RRRs as odds ratios conditional on not being in another category than the baseline or the category of that equation. As I said above, in my discipline all odds are considered conditional odds, i.e. you have to specify both categories rather than say category versus not category, so we would consider the C in COR superfluous and we just leave that C out and call CORs ORs.

[natalia malancu]

Turns out I was not trained by my own people- good to know (I was taught that the relative risk in a 2x2 is (a/a+b) / (c/c+d) while the odds ratio is ad/bc).
Here is what still puzzles me. When discussing things in terms of RRR one talks about the relative risk of being in category one rather than the base category. Apparently when one discusses them in terms of COR one talks about the odds of being in category one rather than the base category. I can understand why for a sociologist these two statements are synonymous. From an RR-OR distinction perspective I still find it a bit hard to grasp, as though I understand that an mlogit is based on separate logits ( and so in the framework of each we can think in terms of OR where odds are defined as success vs. non-success) there is still that sum condition that ties them together. As such I get the sense that by using this phrasing one looses track of the latter aspect. I'm sure I'm missing something out.

PS: also what do you mean by or the category of that equation when you say "odds ratios conditional on not being in another category than the baseline or the category of that equation"

[natalia malancu]

I guess what you mean to say is that when talking about COR one should states that the odds of one being in category I (rather than the baseline) conditional on not being in any another category except I or the baseline.

[Maarten Buis]

What you have probably been told is the a risk ratio (not relative risk) is (a/a+b) / (c/c+d). a/(a+b) is the risk of a and c/(c+d) is the risk of c, and risk ratio is the ratio of these risks. A relative risk (or odds) would be a/(a+b)/b/(a+b) = a/b, i.e. the risk of a relative to the risk of b. The relative risk ratio (or odds ratio) would be (a/b) / (c/d) = ad/bc

[natalia malancu]

I wish. Everything would have been as clear as it became now. I actually performed a little google search and quite a number of definitions that popped up for relative risk also confused it with risk ratio ( being that at the end of the day it's still a ratio, but of... something else). Thank you

[natalia malancu]

ps: i guess i am to the very least correct when saying that incidence=absolute risk=a/a+b

[natalia malancu]

i traced back one of the many sources that explain relative risk as a/a+b/ c/c+d http://practice.sph.umich.edu/micphp...ative_risk.php

[natalia malancu]

apologizes for coming back on this, but in the Stata Technical Bulletin 53- attached- on page 28 shouldn't the denominator for COR2 say 'odds(y = 2' instead of 'odds(y = 1'

[Maarten Buis]

Natalia asked me privately:

However, I was wondering whether you could provide me with some sociological reference for it. In my interactions in the department I often get told that risk ratio and relative risk are one the same thing and different from odds (I get pointed mostly to epidemiological references such as http://practice.sph.umich.edu/micphp...ative_risk.php or http://ocw.jhsph.edu/courses/FundEpi.../Lecture16.pdf).

A clear bibliographical reference would help me settle this once and for all.

Terminology is not absolute, and it is certainly not fixed in statistics. There is no law that forbids people to use the word relative risk in the way do so, and it kinda makes sense. Just as there is no law that forbids people to use word relative risk as odds, and that too kinda makes sense. So when using the word relative risk you just need to define what you mean by it.


However, the word relative risk as used in Stata (as relative risk ratio) is an odds, the definitative source there is Stata's documentation. Since you asked the question on the Statalist, I used Stata's definition. Outside Stata I would never use the word relative risk and always use the word odds to avoid confusion.

[Maarten Buis]

Natalia asked:

I don't know whether to answer you here or there since I´m less interested in Stata´s choice of terminology and more in yours (and that´s a Stata board). My ask for a bibliographical reference was related to your personal preference.

Statistics is also a valid topic on the statalist.

To repeat my earlier comment: don't expect this to be settled once and for all. A term is just a symbol that stands for a concept. Anything can serve as a symbol. If we define the word "dog poo" to stand for the ratio of odds, then we can without problem write about the "dog poo of education on membership of the higher class". This would be an unusual terminology, but not wrong in a strict sense. It is often (much) more convenient to choose a word whose informal meaning somewhat corresponds with the concept we want to use it for. Here the word "relative risk" is equally appropriate for odds as for risk ratios: they are both the risk of one event divided by the risk of another event, they just differ in what the other event is.

A reference that would work well in sociology would be J. Scott Long (1997) Regression Models for Categorical and Limited Dependent Variables. Thousand Oaks: Sage. in particular section 6.6.4 starting on page 168.

[natalia malancu]

thank you, maarten!