interpretation of multinominal logit regression coefficient

[Jiong Yao]

Hi Guys. OK, i finally figure out how to make post organized!
i have one question regarding the multinomial logit regression. Below you can first find my model and then the empirical result. you can see for i.wsd variable (the dummy variable with 0 and 1), the coefficient for outcome 3 is negative and it is significant at 90% level (some may say it is not a significant level anymore). this means (if i am right) compared with outcome 3, with the wsd changes from 0 to 1, the probability of outcome 1 becomes bigger while the probability of outcome 3 becomes smaller. However, when i use the "mtabel" command to calculate the predicted probability, i find the probability of outcome 3 increase, though only a little bit. I am really confused about how to explain this. Is there anything wrong? i find this post but don't think it is very relevant to the question here: https://www.statalist.org/forums/for...etation-method. Thanks in advance.

mlogit pi ternorv ternorc cumu10mid politydif i.wsd powdif salience leaage oppleaage i.milnoncombat
/// i.combat i.rebel bsuc_sum rsuc_sum tsuc_sum bsuc_sum_opp rsuc_sum_opp tsuc_sum_opp, cluster(ccode) base(1)
Multinomial logistic regression                 Number of obs     =        261
                                                Wald chi2(36)     =     270.86
                                                Prob > chi2       =     0.0000
Log pseudolikelihood = -248.98639               Pseudo R2         =     0.1185

                                   (Std. Err. adjusted for 78 clusters in ccode)
--------------------------------------------------------------------------------
               |               Robust
            pi |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------+----------------------------------------------------------------
1              |  (base outcome)
---------------+----------------------------------------------------------------
2              |
       ternorv |   .2401438   .5514606     0.44   0.663    -.8406992    1.320987
       ternorc |   .1888044   .4593973     0.41   0.681    -.7115979    1.089207
     cumu10mid |  -.0344206   .0909131    -0.38   0.705    -.2126071    .1437658
     politydif |   .0150453   .0309368     0.49   0.627    -.0455897    .0756802
         1.wsd |  -1.230385    .400907    -3.07   0.002    -2.016149    -.444622
        powdif |  -.5573728   5.237646    -0.11   0.915    -10.82297    9.708224
      salience |  -.0212136    .131619    -0.16   0.872     -.279182    .2367548
        leaage |  -.0000551   .0000854    -0.64   0.519    -.0002225    .0001124
     oppleaage |  -.0000478   .0000865    -0.55   0.580    -.0002173    .0001216
1.milnoncombat |  -.7916929   .5950595    -1.33   0.183    -1.957988    .3746023
      1.combat |  -.3480807   .4898449    -0.71   0.477    -1.308159    .6119975
       1.rebel |  -1.026594   .3696071    -2.78   0.005    -1.751011   -.3021775
      bsuc_sum |  -.4019039   .1352459    -2.97   0.003     -.666981   -.1368268
      rsuc_sum |  -.1221266   .2577125    -0.47   0.636    -.6272338    .3829806
      tsuc_sum |   .1642505    .361909     0.45   0.650    -.5450782    .8735791
  bsuc_sum_opp |  -.7478829   .5317541    -1.41   0.160    -1.790102    .2943359
  rsuc_sum_opp |   .6970344   .4731081     1.47   0.141    -.2302404    1.624309
  tsuc_sum_opp |   .2248461   .6275107     0.36   0.720    -1.005052    1.454744
         _cons |   1.987506   1.528682     1.30   0.194    -1.008656    4.983668
---------------+----------------------------------------------------------------
3              |
       ternorv |    .568666   .5244716     1.08   0.278    -.4592795    1.596611
       ternorc |   .0261033   .3941644     0.07   0.947    -.7464448    .7986513
     cumu10mid |   .1036039   .1016358     1.02   0.308    -.0955985    .3028063
     politydif |  -.0107027   .0412228    -0.26   0.795    -.0914979    .0700925
         1.wsd |  -.7415701   .4303823    -1.72   0.085    -1.585104    .1019637
        powdif |  -2.601234   6.200299    -0.42   0.675     -14.7536    9.551129
      salience |   .0068077   .1127426     0.06   0.952    -.2141637    .2277792
        leaage |  -.0001791   .0001116    -1.61   0.108    -.0003978    .0000396
     oppleaage |    .000015   .0000939     0.16   0.873     -.000169    .0001989
1.milnoncombat |  -1.273212   .5989447    -2.13   0.034    -2.447122   -.0993023
      1.combat |  -.2542086   .4880725    -0.52   0.602    -1.210813    .7023959
       1.rebel |  -.9614701   .4152078    -2.32   0.021    -1.775262   -.1476779
      bsuc_sum |  -.9121091   .3425758    -2.66   0.008    -1.583545   -.2406729
      rsuc_sum |   .3445298   .3724609     0.93   0.355    -.3854802     1.07454
      tsuc_sum |   .2300749   .4050782     0.57   0.570    -.5638637    1.024014
  bsuc_sum_opp |   .2071136   .5183295     0.40   0.689    -.8087934    1.223021
  rsuc_sum_opp |  -.0990741    .539271    -0.18   0.854    -1.156026    .9578777
  tsuc_sum_opp |  -.3641798   .6610856    -0.55   0.582    -1.659884    .9315241
         _cons |   1.100974   1.507687     0.73   0.465    -1.854039    4.055986
--------------------------------------------------------------------------------
mtable, at(wsd=(0 1) milnoncombat=0 combat=0 rebel=0 ) atmeans outcome (1 3) statistics(ci) dec(2)

Expression: Pr(pi), predict(outcome())

           |      wsd         1         3
 ----------+-----------------------------
     Pr(y) |        0      0.08      0.36
        ll |        0      0.02      0.24
        ul |        0      0.14      0.49
     Pr(y) |        1      0.19      0.42
        ll |        1      0.08      0.26
        ul |        1      0.29      0.58

[Clyde Schechter]

this means (if i am right) compared with outcome 3, with the wsd changes from 0 to 1, the probability of outcome 1 becomes bigger while the probability of outcome 3 becomes smaller.

No, it doesn't mean that. In an ordinary logistic regression it would mean that. But in multinomial logistic regression it is essentially impossible to interpret any coefficient in isolation: it can only be interpreted in the context of the coefficients at all the other levels as well.

The reason is that the sum of the probabilities at each level must be 1. So, while the negative coefficient for level three "wants" to decrease the predicted probability at that level, if some other level's coefficient decreases it's levels probability by an even greater amount (in your case, level 2) then the level 3 probability may find itself forced to increase in order to fit into the amount of probability that is left over.

Because of this, I usually advise my students not to even look at the coefficients coming out of -mlogit- but to go immediately to predicted probabilities (for which, by the way, I recommend the -margins- command) and do all interpretation from there.

[Jiong Yao]

No, it doesn't mean that. In an ordinary logistic regression it would mean that. But in multinomial logistic regression it is essentially impossible to interpret any coefficient in isolation: it can only be interpreted in the context of the coefficients at all the other levels as well.

The reason is that the sum of the probabilities at each level must be 1. So, while the negative coefficient for level three "wants" to decrease the predicted probability at that level, if some other level's coefficient decreases it's levels probability by an even greater amount (in your case, level 2) then the level 3 probability may find itself forced to increase in order to fit into the amount of probability that is left over.

Because of this, I usually advise my students not to even look at the coefficients coming out of -mlogit- but to go immediately to predicted probabilities (for which, by the way, I recommend the -margins- command) and do all interpretation from there.

thank you very much Clyde. You are absolutely right. actually, when outcome 2 is set as the reference group, both probabilities for outcome=1 and outcome=3 increase. this explains what you were suggesting and the confusing result i have. Thanks again!

but one more question. though those coefficients are not that "reliable", the significance level of those coefficients still matter, right? I mean if the coefficient is not significant, then its effect is not significant. is this interpretation right? Thanks.