# Several logistic regressions vs multinomial regression

Is it viable to do several binary logistic regressions instead of doing a multinomial regression? From this question: Multinomial logistic regression vs one-vs-rest binary logistic regression I see that the multinomial regression might have lower standard errors.

However, the package I would like to utilize has not been generalized to multinomial regression (ncvreg: http://cran.r-project.org/web/packages/ncvreg/ncvreg.pdf) and so I was wondering if I could simply do several binary logistic regressions instead.

With a multinomial logit model you impose the constraint that all the predicted probabilities add up to 1. When you use separate binary logit model you can no longer impose that constraint, they are estimated in seperate models after all. So that would be the main difference between these two models.

As you can see in the example below (In Stata, as that is the program I know best), the models tend to be similar but not the same. I would be especially careful about extrapolating predicted probabilities.

// some data preparation
. sysuse nlsw88, clear
(NLSW, 1988 extract)

.
. gen byte occat = cond(occupation < 3                 , 1,      ///
>                  cond(inlist(occupation, 5, 6, 8, 13), 2, 3))  ///
>                  if !missing(occupation)
(9 missing values generated)

. label variable occat "occupation in categories"

. label define occat 1 "high"   ///
>                    2 "middle" ///
>                    3 "low"

. label value occat occat

.
. gen byte middle = (occat == 2) if occat !=1 & !missing(occat)
(590 missing values generated)

. gen byte high   = (occat == 1) if occat !=2 & !missing(occat)
(781 missing values generated)

// a multinomial logit model
. mlogit occat i.race i.collgrad , base(3) nolog

Multinomial logistic regression                   Number of obs   =       2237
LR chi2(6)      =     218.82
Prob > chi2     =     0.0000
Log likelihood = -2315.9312                       Pseudo R2       =     0.0451

-------------------------------------------------------------------------------
occat |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
high          |
race |
black  |  -.4005801   .1421777    -2.82   0.005    -.6792433    -.121917
other  |   .4588831   .4962591     0.92   0.355    -.5137668    1.431533
|
college grad  |   1.495019   .1341625    11.14   0.000     1.232065    1.757972
_cons |  -.7010308   .0705042    -9.94   0.000    -.8392165   -.5628451
--------------+----------------------------------------------------------------
middle        |
race |
black  |   .6728568   .1106792     6.08   0.000     .4559296     .889784
other  |   .2678372    .509735     0.53   0.599    -.7312251    1.266899
|
college grad  |    .976244   .1334458     7.32   0.000      .714695    1.237793
_cons |   -.517313   .0662238    -7.81   0.000    -.6471092   -.3875168
--------------+----------------------------------------------------------------
low           |  (base outcome)
-------------------------------------------------------------------------------

// separate logits:
. logit high   i.race i.collgrad , nolog

Logistic regression                               Number of obs   =       1465
LR chi2(3)      =     154.21
Prob > chi2     =     0.0000
Log likelihood = -906.79453                       Pseudo R2       =     0.0784

-------------------------------------------------------------------------------
high |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
race |
black  |  -.5309439   .1463507    -3.63   0.000     -.817786   -.2441017
other  |   .2670161   .5116686     0.52   0.602     -.735836    1.269868
|
college grad  |   1.525834   .1347081    11.33   0.000     1.261811    1.789857
_cons |  -.6808361   .0694323    -9.81   0.000     -.816921   -.5447512
-------------------------------------------------------------------------------

. logit middle i.race i.collgrad , nolog

Logistic regression                               Number of obs   =       1656
LR chi2(3)      =      90.13
Prob > chi2     =     0.0000
Log likelihood = -1098.9988                       Pseudo R2       =     0.0394

-------------------------------------------------------------------------------
middle |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
race |
black  |   .6942945   .1114418     6.23   0.000     .4758725    .9127164
other  |   .3492788   .5125802     0.68   0.496    -.6553598    1.353918
|