ZINB margins, dydx(*)

[Zhongying Gan]

Hello,

I think the coefficients directly from the zinb regression are hard to interpret, as I need to exponentiate it, and talk about how a variable affects the incidence-rate ratios.

. use https://www.stata-press.com/data/r17/fish,clear
(Fictional fishing data)

. zinb count persons livebait, inflate(child camper)

Fitting constant-only model:

Iteration 0:   log likelihood = -519.33992  
Iteration 1:   log likelihood = -451.38662  
Iteration 2:   log likelihood = -444.49118  
Iteration 3:   log likelihood = -442.96272  
Iteration 4:   log likelihood = -442.71065  
Iteration 5:   log likelihood = -442.66718  
Iteration 6:   log likelihood =  -442.6631  
Iteration 7:   log likelihood = -442.66299  
Iteration 8:   log likelihood = -442.66299  

Fitting full model:

Iteration 0:   log likelihood = -442.66299  (not concave)
Iteration 1:   log likelihood = -432.83107  (not concave)
Iteration 2:   log likelihood = -426.32934  
Iteration 3:   log likelihood = -413.75019  
Iteration 4:   log likelihood = -403.09586  
Iteration 5:   log likelihood = -401.56013  
Iteration 6:   log likelihood = -401.54781  
Iteration 7:   log likelihood = -401.54776  
Iteration 8:   log likelihood = -401.54776  

Zero-inflated negative binomial regression      Number of obs     =        250
                                                Nonzero obs       =        108
                                                Zero obs          =        142

Inflation model = logit                         LR chi2(2)        =      82.23
Log likelihood  = -401.5478                     Prob > chi2       =     0.0000

------------------------------------------------------------------------------
       count |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
count        |
     persons |   .9742984   .1034938     9.41   0.000     .7714543    1.177142
    livebait |   1.557523   .4124424     3.78   0.000     .7491503    2.365895
       _cons |  -2.730064    .476953    -5.72   0.000    -3.664874   -1.795253
-------------+----------------------------------------------------------------
inflate      |
       child |   3.185999   .7468551     4.27   0.000      1.72219    4.649808
      camper |  -2.020951    .872054    -2.32   0.020    -3.730146   -.3117567
       _cons |  -2.695385   .8929071    -3.02   0.003     -4.44545   -.9453189
-------------+----------------------------------------------------------------
    /lnalpha |   .5110429   .1816816     2.81   0.005     .1549535    .8671323
-------------+----------------------------------------------------------------
       alpha |   1.667029   .3028685                      1.167604    2.380076
------------------------------------------------------------------------------

Therefore, I decide to use margins, dydx(*), as it will tell me the effects of a variable on the number of counts, which is much easier to interpret.

. margins, dydx(*)

Average marginal effects                        Number of obs     =        250
Model VCE    : OIM

Expression   : Predicted number of events, predict()
dy/dx w.r.t. : persons livebait child camper

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     persons |   3.051303   .6943597     4.39   0.000     1.690383    4.412222
    livebait |   4.877841   1.580321     3.09   0.002     1.780469    7.975213
       child |  -1.506576   .2890582    -5.21   0.000     -2.07312   -.9400329
      camper |   .9556555   .3077272     3.11   0.002     .3525213     1.55879
------------------------------------------------------------------------------

However, I find it very bizarre that margins, dydx(*) even presents the results for child and camper. Child and camper are inside "inflate" only, and are therefore used to predict only the degenerate zeros, and are not used in the second stage negative binomial regression. How is Stata able to estimate the effects of child and camper on the number of counts then?

. zinb count persons livebait child camper, inflate(child camper)

Fitting constant-only model:

Iteration 0:   log likelihood = -519.33992  
Iteration 1:   log likelihood = -451.38662  
Iteration 2:   log likelihood = -444.49118  
Iteration 3:   log likelihood = -442.96272  
Iteration 4:   log likelihood = -442.71065  
Iteration 5:   log likelihood = -442.66718  
Iteration 6:   log likelihood =  -442.6631  
Iteration 7:   log likelihood = -442.66299  
Iteration 8:   log likelihood = -442.66299  

Fitting full model:

Iteration 0:   log likelihood = -442.66299  (not concave)
Iteration 1:   log likelihood =  -431.0508  (not concave)
Iteration 2:   log likelihood = -421.09041  (not concave)
Iteration 3:   log likelihood = -420.10731  (not concave)
Iteration 4:   log likelihood = -414.28162  
Iteration 5:   log likelihood =  -393.6678  
Iteration 6:   log likelihood = -388.95768  
Iteration 7:   log likelihood = -388.84164  
Iteration 8:   log likelihood = -388.82783  
Iteration 9:   log likelihood = -388.82573  
Iteration 10:  log likelihood = -388.82545  
Iteration 11:  log likelihood =  -388.8254  
Iteration 12:  log likelihood = -388.82539  

Zero-inflated negative binomial regression      Number of obs     =        250
                                                Nonzero obs       =        108
                                                Zero obs          =        142

Inflation model = logit                         LR chi2(4)        =     107.68
Log likelihood  = -388.8254                     Prob > chi2       =     0.0000

------------------------------------------------------------------------------
       count |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
count        |
     persons |   1.084071   .1078292    10.05   0.000     .8727292    1.295412
    livebait |   1.556289   .4026417     3.87   0.000     .7671257    2.345452
       child |  -1.287611   .2350382    -5.48   0.000    -1.748277   -.8269444
      camper |   .2635901    .242317     1.09   0.277    -.2113426    .7385228
       _cons |   -2.95122    .468753    -6.30   0.000    -3.869959   -2.032481
-------------+----------------------------------------------------------------
inflate      |
       child |   14.66326   564.7218     0.03   0.979    -1092.171    1121.498
      camper |  -14.47428   564.7236    -0.03   0.980    -1121.312    1092.364
       _cons |  -14.71499    564.723    -0.03   0.979    -1121.552    1092.122
-------------+----------------------------------------------------------------
    /lnalpha |   .4738543   .1625641     2.91   0.004     .1552345    .7924741
-------------+----------------------------------------------------------------
       alpha |   1.606173    .261106                      1.167932    2.208854
------------------------------------------------------------------------------

. margins, dydx(*)

Average marginal effects                        Number of obs     =        250
Model VCE    : OIM

Expression   : Predicted number of events, predict()
dy/dx w.r.t. : persons livebait child camper

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     persons |   3.512221   .8344028     4.21   0.000     1.876821     5.14762
    livebait |   5.042135   1.646428     3.06   0.002     1.815196    8.269075
       child |  -5.274766   42.49499    -0.12   0.901    -88.56342    78.01389
      camper |    1.94288   42.48519     0.05   0.964    -81.32657    85.21233
------------------------------------------------------------------------------

If child and camper are both in the "inflate" part and the negative binomial part, what exactly does the coefficient for them under margins, dydx(*) mean?

Thank you.

[Clyde Schechter]

So suppose we have two observations with different values of, say, child. The expected difference in their values of count comes from two separate parts of the model. First, the difference in values of child is associated with the probability that this observation is in the "always zero" component of the mixture of outcome distributions. In the output you show, the observation with the higher value of child has a greater chance of being in the "always zero" component because its coefficient in the "inflate" output is positive. On the other hand, if it is not in the "always zero" component, then within the outcome distribution of that component, its expected value of count is lower. These two things reinforce each other: that observation is drawn towards zero by a higher probability of being an "always zero" observation, and in the event it isn't an "always zero" observation, its expected value is lower also (because of the negative coefficient in the non-inflate output). -margins- is combining these two effects.

It works similarly if child only appears in the inflate part of the model. In that case, there is only a single effect in operation: the positive inflate coefficient means a greater probability of being a "zero only" observation, and since 0 is lower than any other possible outcome value in zinb, this means that the overall expected value of count will be lower in observations with higher values of child.

[Zhongying Gan]

So suppose we have two observations with different values of, say, child. The expected difference in their values of count comes from two separate parts of the model. First, the difference in values of child is associated with the probability that this observation is in the "always zero" component of the mixture of outcome distributions. In the output you show, the observation with the higher value of child has a greater chance of being in the "always zero" component because its coefficient in the "inflate" output is positive. On the other hand, if it is not in the "always zero" component, then within the outcome distribution of that component, its expected value of count is lower. These two things reinforce each other: that observation is drawn towards zero by a higher probability of being an "always zero" observation, and in the event it isn't an "always zero" observation, its expected value is lower also (because of the negative coefficient in the non-inflate output). -margins- is combining these two effects.

It works similarly if child only appears in the inflate part of the model. In that case, there is only a single effect in operation: the positive inflate coefficient means a greater probability of being a "zero only" observation, and since 0 is lower than any other possible outcome value in zinb, this means that the overall expected value of count will be lower in observations with higher values of child.

This makes a lot of sense. Thanks.