1. Count Models 2 Sociology 8811 Lecture 13
Copyright © 2007 by Evan Schofer
Do not copy or distribute without permission

2. Announcements Paper #1 deadline coming up: March 8 Class schedule
You should have a dataset by now
You should have some simple models by now
If not, you need to do something right away!!!
Class schedule
Today:
Talk a bit about papers
Wrap up count models
Thursday: New topic – Event History Analysis

3. Review: Count Models
Many dependent variables are counts: Non-negative integers
OLS is inappropriate: linearity and normality assumptions are violated
Solution: Poisson & Negative Binomial models
Coefficient interpretation = similar to logit
Exponentiated coefficients show multiplicative effect on rate
Poisson assumes there is no overdispersion
Skewed variables may lead to overdispersion
If overdispersion is identified, use neg binomial model
Neg binomial model offers chi-square test to identify overdispersion!

4. Negative Binomial Example: Web Use
Note: Info on overdispersion is provided
Negative binomial regression Number of obs =
LR chi2(5) =
Prob > chi2 =
Log likelihood = Pseudo R =
wwwhr | Coef. Std. Err z P>|z| [95% Conf. Interval]
male |
age |
educ |
lowincome |
babies |
_cons |
/lnalpha |
alpha |
Likelihood-ratio test of alpha=0: chibar2(01) = Prob>=chibar2 = 0.000
Alpha is clearly > 0! Overdispersion is evident; LR test p<.05
You should not use Poisson Regression in this case

5. General Remarks
It is often useful to try both Poisson and Negative Binomial models
The latter allows you to test for overdispersion
Use LRtest on alpha (a) to guide model choice
If you don’t suspect dispersion and alpha appears to be zero, use Poission Regression
It makes fewer assumptions
Such as gamma-distributed error.

6. Example: Labor Militancy
Isaac & Christiansen 2002
Note: Results are presented as % change

7. Zero-Inflated Poisson & NB Reg
If outcome variable has many zero values it tends to be highly skewed
Under those circumstances, NBREG works better than ordinary Poisson due to overdispersion
But, sometimes you have LOTS of zeros. Even nbreg isn’t sufficient
Model under-predicts zeros, doesn’t fit well
Examples:
# violent crimes committed by a person in a year
# of wars a country fights per year
# of foreign subsidiaries of firms.

8. Zero-Inflated Poisson & NB Reg
Logic of zero-inflated models: Assume two types of groups in your sample
Type A: Always zero – no probability of non-zero value
Type ~A: Non-zero chance of positive count value
Probability is variable, but not zero
1. Use logit to model group membership
2. Use poisson or nbreg to model counts for those in group ~A
3. Compute probabilities based on those results.

9. Zero-Inflated Poisson & NB Reg
Example: Web usage at work
More skewed than overall web usage. Why?
Many people don’t have computers at work!
So, web usage is zero for many

10. Zero-Inflated Poisson & NB Reg
Zero-inflated models in Stata
“zip” = Poisson, zinb = negative binomial
Commands accept two separate variable lists
Variables that affect counts
For those with non-zero counts
Modeled with Poisson or NB regression
Variables that predict membership in “zero” group
Modeled with logit
Ex: zinb webatwork male age educ lowincome babies, inflate(male age educ lowincome babies)

11. ZINB Example: Web Hrs at Work
“Inflate” output = logit for group membership
Zero-inflated negative binomial regression Number of obs =
Nonzero obs =
Zero obs =
Inflation model = logit LR chi2(5) =
Log likelihood = Prob > chi2 =
| Coef. Std. Err z P>|z| [95% Conf. Interval]
webatwork |
male |
age |
educ |
lowincome |
babies |
_cons |
inflate |
male |
age |
educ |
lowincome |
babies |
_cons |
Education reduces odds of zero value
But doesn’t have an effect on count for those that are non-zero
Model predicting zero group

12. Zero-Inflated Poisson & NB Reg
Remarks
ZINB produces estimate of alpha
Helps choose between zip & zinb
Long and Freese (2006) have helpful tool to compare fit of count models: countfit
See textbook
Zero-inflated models seem very useful
Count variables often have many zeros
It is often reasonable to assume a “always zero” group
But, they are fairly new
Not many examples in the literature
Haven’t been widely scrutinized.

13. Zero-truncated Poisson & NB reg
Truncation – the absence of information about cases in some range of a variable
Example: Suppose we study income based on data from tax returns…
Cases with income below a certain value are not required to submit a tax return… so data is missing
Example: Data on # crimes committed, taken from legal records
Individuals with zero crimes are not evident in data
Example: An on-line survey of web use
Individuals with zero web use are not in data
Poisson & NB have been adapted to address truncated data:
Zero-truncated Poisson & Zero-trunciated NB reg.

14. Example: Zero-truncated NB Reg
Web use (zeros removed)
Zero-truncated negative binomial regression Number of obs =
LR chi2(5) =
Dispersion = mean Prob > chi2 =
Log likelihood = Pseudo R =
wwwhr | Coef. Std. Err z P>|z| [95% Conf. Interval]
male |
age |
educ |
lowincome |
babies |
_cons |
/lnalpha |
alpha |
Likelihood-ratio test of alpha=0: chibar2(01) = Prob>=chibar2 = 0.000
Coefficient interpretation works just like ordinary poisson or NB regression.

15. Empirical Example 2
Example: Haynie, Dana L “Delinquent Peers Revisited: Does Network Structure Matter?” American Journal of Sociology, 106, 4: