Equivalency of multinomial and ordinary logistic regression

[Darina Peycheva]

Dear all,

I hope you can help me with this.

To analyse the association between a three-category outcome with no natural ordering and a binary exposure, I used multinomial logistic regression. I have been advised to present the results, relative risk ratios, as there is an equivalence between relative risk ratios and conditional odds ratios for audience more familiar with odds than risk. I was referred to a Stata Technical Bulletin (stb53.dvi) detailing this equivalence.

I have only used conditional logistic regression for analysing matched case-control studies, and I understand it is also used for repeated measures. However, neither of these scenarios apply to my research/dataset - each case has only one observation, there's no matching/group variable, and the outcome has multiple categories.

Given this, is the equivalence between RRR and COR only valid in the context of matched case-control studies and repeated measures, or does it extend beyond these study designs?

Many thanks for your help,
Darina

[Rich Goldstein]

before there was widespread software for multinomial regression, there were article detailing how to get the "same" result via binary logistic regression - you could look those up if needed

but, here is a simple example going the other way:

. sysuse auto
(1978 automobile data)
r; t=0.00 9:47:09

. logistic foreign price

Logistic regression                                     Number of obs =     74
                                                        LR chi2(1)    =   0.17
                                                        Prob > chi2   = 0.6784
Log likelihood = -44.94724                              Pseudo R2     = 0.0019

------------------------------------------------------------------------------
     foreign | Odds ratio   Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
       price |   1.000035   .0000844     0.42   0.676     .9998699    1.000201
       _cons |    .339666   .1996674    -1.84   0.066     .1073214    1.075023
------------------------------------------------------------------------------
Note: _cons estimates baseline odds.
r; t=0.35 9:47:19

. mlogit foreign price, nolog rrr

Multinomial logistic regression                         Number of obs =     74
                                                        LR chi2(1)    =   0.17
                                                        Prob > chi2   = 0.6784
Log likelihood = -44.94724                              Pseudo R2     = 0.0019

------------------------------------------------------------------------------
     foreign |        RRR   Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
Domestic     |  (base outcome)
-------------+----------------------------------------------------------------
Foreign      |
       price |   1.000035   .0000844     0.42   0.676     .9998699    1.000201
       _cons |    .339666   .1996674    -1.84   0.066     .1073214    1.075023
------------------------------------------------------------------------------
Note: _cons estimates baseline relative risk for each outcome.

note, however, that this may not be enough to satisfy whoever needs to be convinced, but, as I say, there is old lit on this

[Darina Peycheva]

Thanks. In this example the outcome is binary. With binary and rare outcome it is possible that the odds ratio approximates the risk ratio (though exactly same here), but my case concerns multiple category non-rare outcome and conditional logistic regression (clogit). In the reference above (stb53.dvi), p28-29, example is given regarding equivalence between RRR and COR, but COR is derived using a formula rather than obtained using clogit; and there's no example with binary exposure. Also the dataset used for this example, the auto dataset, doesn't seem to be set-up for clogit. My understanding is that clogit has certain requirements for the data (e.g., multiple observations per case, matching/grouping variable etc).

Many thanks,
Darina

[Maarten Buis]

Rich is correct that a multinomial logit on a dependent variable with two outcomes is exactly the same as a logit. More generally, as he hinted, you can approximate a multinomial logistic regression with a set of logistic regressions by being creative with how you create your variables. I find showing how this can be done to be actually quite pedagogically valuable. However, I don't think that this is what Darina actually needs.

My interpretation of Darina's question is that she is confused by the label Relative Risk Ratio and how those coefficients relate to odds ratios. Some of this is a discipline specific thing: different disciplines use the exact same term (in this case, odds ratio) for subtly different things (do the two outcomes that are being compared have to be mutually exclusive or not), and as a result everybody gets confused as soon as they come across someone from a different discipline. Since StataCorp tries to reach multiple disciplines, but can only use one name for a thing, so they just have to use terminology that makes more sense in one discipline than in others. There is a very old post here, and some links in it to even older posts, that tries to explain the relationship (equivalence) of relative risk ratios and odds ratios: https://www.stata.com/statalist/arch.../msg00085.html

[Maarten Buis]

T With binary and rare outcome it is possible that the odds ratio approximates the risk ratio

Beware: a risk ratio is not the same thing as a relative risk ratio. The whole discussion about risk ratios being approximated by odds ratios is completely irrelevant to your problem.

[Bruce Weaver]

If the issue is confusion about the Relative Risk Ratio label in the output from -mlogit-, then perhaps the attached notes will help.

Stata-mlogit-RRR.pdf

[Maarten Buis]

Also note that the conditional odds ratio mentioned in the article you mentioned has nothing to do with the conditional logit model.

[Darina Peycheva]

Thanks, Maarten, all, I've understood wrong what I'm asked to do, and have mistakenly related conditional odds ratios to conditional logit model. Thanks for pointing this out, Maarten!