Version info: Code for this page was tested in Stata 17.
Zero-inflated negative binomial regression is for modeling count variables
with excessive zeros and it is usually for
over-dispersed count outcome variables. Furthermore, theory suggests that the
excess zeros are generated by a separate process from the count values and that
the excess zeros can be modeled independently.
Please note: The purpose of this page is to show how to use various data
analysis commands. It does not cover all aspects of the research process which
researchers are expected to do. In particular, it does not cover data
cleaning and checking, verification of assumptions, model diagnostics or
potential follow-up analyses.
Examples of zero-inflated negative binomial regression
Example 1.
School administrators study the attendance behavior of high school juniors at two schools.
Predictors of the number of days of absence include gender of the student and standardized
test scores in math and language arts.
Example 2.
The state wildlife biologists want to model how many fish are being caught by fishermen
at a state park. Visitors are asked how long they stayed, how many
people were in the group, were there children in the group and how many fish were caught.
Some visitors do not fish, but there is no data on whether a person fished or not. Some
visitors who did fish did not catch any fish so there are excess zeros in the data because
of the people that did not fish.
Description of the data
Let’s pursue Example 2 from above. The data set used in this example is
from Stata.
We have data on 250 groups that went to a park. Each group was questioned
before leaving the park about how many fish they caught (count), how many children were in the
group (child), how many people were in the group (persons), and
whether or not they brought a camper to the park (camper). The outcome
variable of interest will be the number of fish caught. Even though the
question about the number of fish caught was asked to everyone, it does not mean
that everyone went fishing. What would be the reason for someone to report a zero
count? Was it because this person was unlucky and didn’t catch any fish, or was
it because this person didn’t go fishing at all? If a person didn’t go fishing,
the outcome would be always zero. Otherwise, if a person went to fishing, the
count could be zero or non-zero. So we
can see that there seemed to be two processes that would generate zero counts:
unlucky in fishing or didn’t go fishing.
Let’s first look at the data. We will start with reading in the data and the
descriptive statistics and plots. This helps us understand the data and give us
some hint on how we should model the data.
webuse fish summarize count child persons camperVariable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- count | 250 3.296 11.63503 0 149 child | 250 .684 .8503153 0 3 persons | 250 2.528 1.11273 1 4 camper | 250 .588 .4931824 0 1 histogram count, discrete freqtab1 child persons camper -> tabulation of child child | Freq. Percent Cum. ------------+----------------------------------- 0 | 132 52.80 52.80 1 | 75 30.00 82.80 2 | 33 13.20 96.00 3 | 10 4.00 100.00 ------------+----------------------------------- Total | 250 100.00 -> tabulation of persons persons | Freq. Percent Cum. ------------+----------------------------------- 1 | 57 22.80 22.80 2 | 70 28.00 50.80 3 | 57 22.80 73.60 4 | 66 26.40 100.00 ------------+----------------------------------- Total | 250 100.00 -> tabulation of camper camper | Freq. Percent Cum. ------------+----------------------------------- 0 | 103 41.20 41.20 1 | 147 58.80 100.00 ------------+----------------------------------- Total | 250 100.00
tabstat count, by(camper) stats(mean v n) Summary for variables: count by categories of: camper camper | mean variance N ---------+------------------------------ 0 | 1.524272 21.05578 103 1 | 4.537415 212.401 147 ---------+------------------------------ Total | 3.296 135.3739 250 ----------------------------------------
We can see from the table of descriptive statistics above that the variance
of the outcome variable is quite large relative to the means. This might be an indication of over-dispersion.
Analysis methods you might consider
Before we show how you can analyze this with a zero-inflated negative binomial analysis, let’s
consider some other methods that you might use.
- OLS Regression – You could try to analyze these data using OLS regression. However, count
data are highly non-normal and are not well estimated by OLS regression. - Zero-inflated Poisson Regression – Zero-inflated Poisson regression does better when
the data are not over-dispersed, i.e. when variance is not much larger than the mean. - Ordinary Count Models – Poisson or negative binomial models might be more
appropriate if there are not excess zeros.
Zero-inflated negative binomial regression
A zero-inflated model assumes that zero outcome is due to two different
processes. For instance, in the example of fishing presented here, the two
processes are that a subject has gone fishing vs. not gone fishing. If not gone fishing, the only
outcome possible is zero. If gone fishing, it is then a count
process. The two parts of the a zero-inflated model are a binary model, usually
a logit model to model which of the two processes the zero outcome is associated
with and a count model, in this case, a negative binomial model, to model the
count process. The expected count is expressed as a combination of the two
processes. Taking the example of fishing again, E(#of fish caught=k) = prob(not
gone fishing )*0 + prob(gone fishing)*E(y=k|gone fishing).
Now let’s build up our model. We are going to use the variables child and camper to model the
count in the part of negative binomial model and the variable persons
in the logit part of the model. The Stata command is shown below. We treat variable camper as a categorical variable by putting
a prefix “-i.-” in front of the variable name. This will make the post
estimations easier. We have included the zip option, which provides a likelihood ratio test of
alpha=0 (basically zinb versus zip).
zinb count child i.camper, inflate(persons) zip
Fitting constant-only model:
Iteration 0: log likelihood = -519.33992
Iteration 1: log likelihood = -471.96077
Iteration 2: log likelihood = -465.38193
Iteration 3: log likelihood = -464.39882
Iteration 4: log likelihood = -463.92704
Iteration 5: log likelihood = -463.79248
Iteration 6: log likelihood = -463.75773
Iteration 7: log likelihood = -463.7518
Iteration 8: log likelihood = -463.75119
Iteration 9: log likelihood = -463.75118
Fitting full model:
Iteration 0: log likelihood = -463.75118 (not concave)
Iteration 1: log likelihood = -440.43162
Iteration 2: log likelihood = -434.96651
Iteration 3: log likelihood = -433.49903
Iteration 4: log likelihood = -432.89949
Iteration 5: log likelihood = -432.89091
Iteration 6: log likelihood = -432.89091
Zero-inflated negative binomial regression Number of obs = 250
Inflation model: logit Nonzero obs = 108
Zero obs = 142
LR chi2(2) = 61.72
Log likelihood = -432.8909 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
count | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
count |
child | -1.515255 .1955912 -7.75 0.000 -1.898606 -1.131903
1.camper | .8790514 .2692731 3.26 0.001 .3512857 1.406817
_cons | 1.371048 .2561131 5.35 0.000 .8690758 1.873021
-------------+----------------------------------------------------------------
inflate |
persons | -1.666563 .6792833 -2.45 0.014 -2.997934 -.3351922
_cons | 1.603104 .8365065 1.92 0.055 -.036419 3.242626
-------------+----------------------------------------------------------------
/lnalpha | .9853533 .17595 5.60 0.000 .6404975 1.330209
-------------+----------------------------------------------------------------
alpha | 2.678758 .4713275 1.897425 3.781834
------------------------------------------------------------------------------
Likelihood-ratio test of alpha=0: chibar2(01) = 1197.43 Pr>=chibar2 = 0.0000
The output has a few components which are explained below.
- It begins with the iteration log giving the values of the log
likelihoods starting with a model that has no predictors and intercept set to zero for the inflated model. - The last value in the log is the final value of the log likelihood for
the full model and is repeated below. - Next comes the header information. On the right-hand side the number of
observations used (316) is given along with the likelihood ratio
chi-squared. This compares the full model to a model without count
predictors, giving a difference of two degrees of freedom. - This is followed by the p-value for the chi-square. The model, as a
whole, is statistically significant. - Below the header, you will find the negative binomial regression
coefficients for each of the variables along with standard errors, z-scores,
p-values and 95% confidence intervals for the coefficients. - Following these are logit coefficients for predicting excess zeros along
with their standard errors, z-scores, p-values and confidence intervals. - Additionally, there will be an estimate of the natural log of the over
dispersion coefficient, alpha, along with the untransformed value. If the
alpha coefficient is zero then the model is better estimated using an
Poisson regression model. - Below the various coefficients you will find the results of the zip
option.- The zip option tests the zero-inflated negative binomial
model versus the zero-inflated Poisson model. A significant likelihood
ratio test for alpha=0 indicates that the zinb model is preferred
to the zip model.
- The zip option tests the zero-inflated negative binomial
Looking through the results of regression parameters we see the following:
- The predictors child and camper in the part of the negative binomial regression model predicting number
of fish caught (count) are both significant predictors. - The predictor person in the part of the logit model predicting
excessive zeros is statistically significant. - For these data, the expected change in log(count)
for a one-unit increase in child is -1.515255 holding other variables constant. - A camper (camper = 1) has an expected log(count)
of 0.879051 higher than that of a non-camper (camper = 0) holding
other variables constant. - The log odds of being an excessive zero would decrease by 1.67 for every
additional person in the group. In other words, the more people in the group the less likely that the zero would be due to not gone fishing. Put it
plainly, the larger the group the person was in, the more likely that the
person went fishing. - We can see at the bottom of our model that the likelihood ratio test
that alpha = 0 is significantly different from zero. This suggests
that our data are over-dispersed and that a zero-inflated negative binomial
model is more appropriate than a zero-inflated Poisson model.
Now, just to be on the safe side, let’s rerun the zinb command with the robust
option in order to obtain robust standard errors for the Poisson regression
coefficients.
zinb count child i.camper, inflate(persons) robust
Fitting constant-only model:
Iteration 0: log pseudolikelihood = -519.33992
Iteration 1: log pseudolikelihood = -471.96077
Iteration 2: log pseudolikelihood = -465.38193
Iteration 3: log pseudolikelihood = -464.39882
Iteration 4: log pseudolikelihood = -463.92704
Iteration 5: log pseudolikelihood = -463.79248
Iteration 6: log pseudolikelihood = -463.75773
Iteration 7: log pseudolikelihood = -463.7518
Iteration 8: log pseudolikelihood = -463.75119
Iteration 9: log pseudolikelihood = -463.75118
Fitting full model:
Iteration 0: log pseudolikelihood = -463.75118 (not concave)
Iteration 1: log pseudolikelihood = -440.43162
Iteration 2: log pseudolikelihood = -434.96651
Iteration 3: log pseudolikelihood = -433.49903
Iteration 4: log pseudolikelihood = -432.89949
Iteration 5: log pseudolikelihood = -432.89091
Iteration 6: log pseudolikelihood = -432.89091
Zero-inflated negative binomial regression Number of obs = 250
Nonzero obs = 108
Zero obs = 142
Inflation model = logit Wald chi2(2) = 40.16
Log pseudolikelihood = -432.8909 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
| Robust
count | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
count |
child | -1.515255 .2417504 -6.27 0.000 -1.989077 -1.041432
1.camper | .8790514 .471303 1.87 0.062 -.0446855 1.802788
_cons | 1.371048 .3902521 3.51 0.000 .6061682 2.135928
-------------+----------------------------------------------------------------
inflate |
persons | -1.666563 .4314861 -3.86 0.000 -2.51226 -.8208658
_cons | 1.603104 .6665327 2.41 0.016 .2967236 2.909484
-------------+----------------------------------------------------------------
/lnalpha | .9853533 .2157394 4.57 0.000 .5625119 1.408195
-------------+----------------------------------------------------------------
alpha | 2.678758 .5779135 1.755075 4.088567
------------------------------------------------------------------------------
Using the robust option has resulted in some change in the model chi-square,
which is now a Wald chi-square. This statistic is based on log
pseudo-likelihoods instead of log-likelihoods. The model is still statistically significant.
The robust
standard errors attempt to adjust for heterogeneity in the model.
Now, let’s try to understand the model better by using some of the post
estimation commands. First off, we use the predict command with the pr
option to get the
predicted probability of being “an excessive zero” due to not having gone
fishing. We then look the
distribution of the predicted probability by the number of persons in the group. We can see that the larger
the group, the smaller the probability, meaning the more likely that the person
went fishing.
predict p, pr table persons, con(mean p)---------------------- persons | mean(p) ----------+----------- 1 | .4841405 2 | .1505847 3 | .0324023 4 | .0062859 ----------------------
Finally, we will use the margins command to get the predicted
number of fish caught, comparing campers with non-campers given different number
of children and marginsplot to visualize the information produced by the
margins command.
margins camper, at(child=(0(1)3))Predictive margins Number of obs = 250 Model VCE : Robust Expression : Predicted number of events, predict() 1._at : child = 0 2._at : child = 1 3._at : child = 2 4._at : child = 3 ------------------------------------------------------------------------------ | Delta-method | Margin Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- _at#camper | 1 0 | 3.302878 1.294607 2.55 0.011 .7654961 5.840261 1 1 | 7.955358 2.056003 3.87 0.000 3.925667 11.98505 2 0 | .7258149 .3452292 2.10 0.036 .049178 1.402452 2 1 | 1.748208 .3534415 4.95 0.000 1.055475 2.44094 3 0 | .1594994 .1028401 1.55 0.121 -.0420634 .3610623 3 1 | .3841725 .1394934 2.75 0.006 .1107704 .6575747 4 0 | .0350504 .0297846 1.18 0.239 -.0233263 .093427 4 1 | .0844228 .0492046 1.72 0.086 -.0120164 .180862 ------------------------------------------------------------------------------ marginsplot, noci scheme(s1mono) legend(position(1) ring(0))
Notice that by default the margins command fixed the expected
predicted probability of being an excessive zero at its mean. For instance, here
is an alternative way for producing the same predicted count given camper = 0 /1
and child = 0.
sum pr local mean_pr = r(mean) margins camper, at(child=0) expression(exp(predict(xb))*(1-`mean_pr'))Predictive margins Number of obs = 250 Model VCE : RobustExpression : exp(predict(xb))*(1-.1615949432440102) at : child = 0------------------------------------------------------------------------------ | Delta-method | Margin Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- camper | 0 | 3.302879 1.288955 2.56 0.010 .7765726 5.829184 1 | 7.955358 2.180409 3.65 0.000 3.681835 12.22888 ------------------------------------------------------------------------------
Things to consider
Here are some issues that you may want to consider in the course of your
research analysis.
- Question about the over-dispersion parameter is in general a tricky one.
A large over-dispersion parameter could be due to a miss-specified model or
could be due to a real process with over-dispersion. Adding an
over-dispersion problem does not necessarily improve a miss-specified model. - The zinb model has two parts, a negative binomial count model and
the logit model for predicting excess zeros, so you might want to review
these Data Analysis Example pages,
Negative Binomial
Regression and
Logit Regression. - Since zinb has both a count model and a logit model, each of the
two models should have good predictors. The two models do not necessarily
need to use the same predictors. - Problems of perfect prediction, separation or partial separation can
occur in the logistic part of the zero-inflated model. - Count data often use exposure variable to indicate the number of times
the event could have happened. You can incorporate exposure into your model
by using the exposure() option. - It is not recommended that zero-inflated negative binomial models be
applied to small samples. What constitutes a small sample does not seem to
be clearly defined in the literature. - Pseudo-R-squared values differ from OLS R-squareds, please see
FAQ: What are pseudo R-squareds? for a discussion on this issue. - In times past, the Vuong test had been used to test whether a zero-inflated negative binomial model or a negative binomial model (without the zero-inflation) was a better fit for the data. However, this test is no longer considered valid. Please see The Misuse of The Vuong Test For Non-Nested Models to Test for Zero-Inflation by Paul Wilson for further information.
See Also
References
- Cameron, A. Colin and Trivedi, P.K. (2009) Microeconometrics using Stata.
College Station, TX: Stata Press. - Long, J. Scott, & Freese, Jeremy (2006). Regression Models for Categorical
Dependent Variables Using Stata (Second Edition). College Station, TX: Stata
Press. - Long, J. Scott (1997). Regression Models for Categorical and Limited
Dependent Variables. Thousand Oaks, CA: Sage Publications.
Last updated on October 12, 2011