Ordinal or multinomial regression

[Sherine Maui]

A bit of a rudimentary question, but would like to check I am using the correct regression. I have a dependant variable that measures ones readiness to go to war. There are three categories (1) High or very high, (2) Moderate or low, (3) No intention to go to war at all.

I would think we need to run an ordinal regression since the dependant variable is ordered categories (high - mod - 0 intentions at all), whereas my colleague insists we should use multinomial regression, which I understand is used when the dependant variable is categorical but no link between the categories. So I am not sure which is the correct method to use with our dependant variable?

[Jared Greathouse]

In my opinion, you want to use mlogit because there's no inherent reason as to why 1 should be very high and 0 should be anti-war. The levels are arbitrary, you see.

The issue isn't if the categories are linked, the issue is whether there's an order that the levels must follow (i.e., 12 years of education is fundamentally different from 6 or 18 years).

[Sherine Maui]

Ah, I see what you mean now. mlogit it is. - Thank you, Jared.

[Andrew Musau]

Aside from Jared's useful note, the usual way to choose between ordered logit and multinomial logit is to compare the log-likelihood for the former with that of the latter within which it is nested. A likelihood ratio test will therefore tell you whether simplification from multinomial logit to ordered logit is justified. For example, using the Mroz(1987) data, the test is distributed \(\chi^2(6)\) and yields a tail probability of 0.6%. Therefore, the simplification to an ordered model is rejected.

use https://www3.nd.edu/~rwilliam/statafiles/mroz.dta, clear
*CATEGORIZE ANNUAL WORKING HOURS (NON-PARTICIPATION, PART-TIME, FULL-TIME)
gen lfstatus= cond(hours==0, 0, cond(inrange(hours, 1, 1249), 1, 2))
lab def lfstatus 0 "non-participation" 1 "part-time work" 2 "full-time work"
lab values lfstatus lfstatus
mlogit lfstatus kidslt6 kidsge6 age educ exper nwifeinc
est sto mlogit
ologit lfstatus kidslt6 kidsge6 age educ exper nwifeinc
est sto ologit
lrtest mlogit ologit, force

Res.:

. mlogit lfstatus kidslt6 kidsge6 age educ exper nwifeinc

Iteration 0:   log likelihood = -809.85106  
Iteration 1:   log likelihood = -682.09452  
Iteration 2:   log likelihood = -676.45369  
Iteration 3:   log likelihood = -676.35678  
Iteration 4:   log likelihood = -676.35676  

Multinomial logistic regression                 Number of obs     =        753
                                                LR chi2(12)       =     266.99
                                                Prob > chi2       =     0.0000
Log likelihood = -676.35676                     Pseudo R2         =     0.1648

-----------------------------------------------------------------------------------
         lfstatus |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
------------------+----------------------------------------------------------------
non_participation |  (base outcome)
------------------+----------------------------------------------------------------
part_time_work    |
          kidslt6 |  -1.029752   .2192135    -4.70   0.000    -1.459402   -.6001012
          kidsge6 |   .1452962   .0810486     1.79   0.073    -.0135561    .3041485
              age |   -.061935   .0161806    -3.83   0.000    -.0936485   -.0302215
             educ |   .2352844   .0489108     4.81   0.000      .139421    .3311478
            exper |   .0836159   .0155026     5.39   0.000     .0532314    .1140004
         nwifeinc |  -.0191471   .0093588    -2.05   0.041    -.0374899   -.0008043
            _cons |  -1.051627   .9599877    -1.10   0.273    -2.933168    .8299147
------------------+----------------------------------------------------------------
full_time_work    |
          kidslt6 |   -2.04806   .2883306    -7.10   0.000    -2.613177   -1.482942
          kidsge6 |  -.0562924    .089552    -0.63   0.530    -.2318111    .1192262
              age |  -.1267562   .0173295    -7.31   0.000    -.1607214   -.0927911
             educ |   .2225451   .0508362     4.38   0.000     .1229081    .3221822
            exper |   .1554865   .0162169     9.59   0.000      .123702     .187271
         nwifeinc |  -.0218055   .0102905    -2.12   0.034    -.0419746   -.0016364
            _cons |   1.552764   .9850566     1.58   0.115    -.3779109     3.48344
-----------------------------------------------------------------------------------

.
. est sto mlogit

.
. ologit lfstatus kidslt6 kidsge6 age educ exper nwifeinc

Iteration 0:   log likelihood = -809.85106  
Iteration 1:   log likelihood = -686.68524  
Iteration 2:   log likelihood = -685.50088  
Iteration 3:   log likelihood = -685.49686  
Iteration 4:   log likelihood = -685.49686  

Ordered logistic regression                     Number of obs     =        753
                                                LR chi2(6)        =     248.71
                                                Prob > chi2       =     0.0000
Log likelihood = -685.49686                     Pseudo R2         =     0.1536

------------------------------------------------------------------------------
    lfstatus |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     kidslt6 |  -1.390614   .1813509    -7.67   0.000    -1.746055   -1.035172
     kidsge6 |  -.0341089   .0623172    -0.55   0.584    -.1562484    .0880307
         age |  -.0916151   .0121459    -7.54   0.000    -.1154207   -.0678096
        educ |    .158401   .0356408     4.44   0.000     .0885464    .2282557
       exper |   .1159833   .0112204    10.34   0.000     .0939917     .137975
    nwifeinc |  -.0153582   .0073408    -2.09   0.036    -.0297459   -.0009705
-------------+----------------------------------------------------------------
       /cut1 |   -1.75244   .7084357                     -3.140949   -.3639319
       /cut2 |  -.3338748   .7054785                     -1.716587    1.048838
------------------------------------------------------------------------------

.
. est sto ologit

.
. lrtest mlogit ologit, force

Likelihood-ratio test                                 LR chi2(6)  =     18.28
(Assumption: ologit nested in mlogit)                 Prob > chi2 =    0.0056

.

[Sherine Maui]

That's very helpful to know for the future, thanks Andrew. Testing it on our data also shows we should go mlogit.

[Jared Greathouse]

If your methods teacher is anything like my current one, you'll also wanna report it something like "The lrtest tests the null hypothesis that the nested and single models give the same results. If the null were true, the test statistic would come from a chi squared distribution with x degrees of freedom".

Hopefully I got that right, I think this exact concept is on my midterm🤣