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This page shows some examples on how to generate the predicted count from a zero-inflated Poisson or a zero-inflated negative binomial model based on the parameter estimates. Zero-inflated models allow us to model two processes simultaneously. Let's take ZIP as an example. Basically, zero outcome arises from two different processes. In one process, the outcome is always zero and in the other process, zero outcome, as well as other outcomes obey the Poisson process. With the two parts of the model, how do we generate the predicted count after running the model? The examples demonstrate the steps to this end.
webuse fish, clear
zip count persons livebait, inf(child camper) nolog
Zero-inflated Poisson regression Number of obs = 250
Nonzero obs = 108
Zero obs = 142
Inflation model = logit LR chi2(2) = 506.48
Log likelihood = -850.7014 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
count |
persons | .8068853 .0453288 17.80 0.000 .7180424 .8957281
livebait | 1.757289 .2446082 7.18 0.000 1.277866 2.236713
_cons | -2.178472 .2860289 -7.62 0.000 -2.739078 -1.617865
-------------+----------------------------------------------------------------
inflate |
child | 1.602571 .2797719 5.73 0.000 1.054228 2.150913
camper | -1.015698 .365259 -2.78 0.005 -1.731593 -.2998038
_cons | -.4922872 .3114562 -1.58 0.114 -1.10273 .1181558
------------------------------------------------------------------------------
predict p
The variable p created above is the predicted count based on this model.
Now we show the steps to create the same p using the parameter
estimates. Basically, it has two parts, the model for the usual Poisson
process and the model for the process of zeros. Variable a1 below is
the linear prediction based on the first model and variable a2 is the
linear prediction for the second model which is a logit model by default.
Variable pzero is the predicted probability for being in the first
process which only produces zero count. Variable pcount is then the
predicted count based on the two processes.
gen a1 = -2.178472 + .8068853*persons + 1.757289*livebait
gen a2 = -.4922872 + 1.602571*child -1.015698*camper
gen pzero = exp(a2)/(1+exp(a2))
gen pcount = exp(a1)*(1-pzero) /*for logit model*/
sum p pcount
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
p | 250 2.770999 3.269588 .079269 13.55015
pcount | 250 2.770997 3.269585 .0792689 13.55014
The only difference between this example and the previous one is that the inflation part in this one is modeled by probit model instead of logit model.
webuse fish, clear
zip count persons livebait, inf(child camper) probit nolog
Zero-inflated Poisson regression Number of obs = 250
Nonzero obs = 108
Zero obs = 142
Inflation model = probit LR chi2(2) = 506.29
Log likelihood = -850.3968 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
count |
persons | .8062521 .0453179 17.79 0.000 .7174306 .8950736
livebait | 1.755824 .2444357 7.18 0.000 1.276739 2.234909
_cons | -2.174616 .2858538 -7.61 0.000 -2.734879 -1.614353
-------------+----------------------------------------------------------------
inflate |
child | .9658273 .1576773 6.13 0.000 .6567855 1.274869
camper | -.6112131 .2146819 -2.85 0.004 -1.031982 -.1904442
_cons | -.295569 .1869964 -1.58 0.114 -.6620753 .0709372
------------------------------------------------------------------------------
predict p
gen a1 = -2.174616 + .8062521*persons + 1.755824 *livebait
gen a2 = -.295569 + .9658273*child -.6112131*camper
gen pzero = normal(a2) /*for probit model*/
gen pcount = exp(a1)*(1-pzero)
sum p pcount
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
p | 250 2.754194 3.272803 .0649889 13.53128
pcount | 250 2.754194 3.272803 .0649889 13.53128
Now we switch to zero-inflated negative binomial model. The way to calculate the predicted values is exactly the same as for zero-inflated Poisson models.
webuse fish, clear
zinb count persons livebait, inf(child camper) nolog
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
------------------------------------------------------------------------------
| 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
------------------------------------------------------------------------------
predict p
gen a1 = -2.730064 + .9742984*persons + 1.557523*livebait
gen a2 = -2.695385 + 3.185999*child -2.020951*camper
gen pzero = exp(a2)/(1+exp(a2))
gen pcount = exp(a1)*(1-pzero)
sum p pcount
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
p | 250 3.131795 4.189243 .0159387 15.11586
pcount | 250 3.131795 4.189243 .0159391 15.11586
In previous examples, we have manually generated these variables using the parameter estimates. In this example, we make use of the Stata's stored matrix for parameter coefficients. This is the general and more useful approach in practice.
webuse fish, clear
zip count persons livebait, inf(child camper) nolog
Zero-inflated Poisson regression Number of obs = 250
Nonzero obs = 108
Zero obs = 142
Inflation model = logit LR chi2(2) = 506.48
Log likelihood = -850.7014 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
count |
persons | .8068853 .0453288 17.80 0.000 .7180424 .8957281
livebait | 1.757289 .2446082 7.18 0.000 1.277866 2.236713
_cons | -2.178472 .2860289 -7.62 0.000 -2.739078 -1.617865
-------------+----------------------------------------------------------------
inflate |
child | 1.602571 .2797719 5.73 0.000 1.054228 2.150913
camper | -1.015698 .365259 -2.78 0.005 -1.731593 -.2998038
_cons | -.4922872 .3114562 -1.58 0.114 -1.10273 .1181558
------------------------------------------------------------------------------
predict p
matrix list e(b)
e(b)[1,6]
count: count: count: inflate: inflate: inflate:
persons livebait _cons child camper _cons
y1 .80688527 1.7572894 -2.1784716 1.6025705 -1.0156983 -.49228716
gen a1 = _b[count:_cons] + _b[count:persons]*persons + _b[count:livebait]*livebait
gen a2 = _b[inflate:_cons] + _b[inflate:child]*child +_b[inflate:camper]*camper
gen pzero = exp(a2)/(1+exp(a2))
gen pcount = exp(a1)*(1-pzero) /*for logit model*/
sum p pcount
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
p | 250 2.770999 3.269588 .079269 13.55015
pcount | 250 2.770999 3.269588 .079269 13.55015
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