1. Count Models 1 Sociology 8811 Lecture 12
Copyright © 2007 by Evan Schofer
Do not copy or distribute without permission

2. Count Variables
Many dependent variables are counts: Non-negative integers
# Crimes a person has committed in lifetime
# Children living in a household
# new companies founded in a year (in an industry)
# of social protests per month in a city
Can you think of others?

3. Count Variables
Count variables can be modeled with OLS regression… but:
1. Linear models can yield negative predicted values… whereas counts are never negative
Similar to the problem of the Linear Probability Model
2. Count variables are often highly skewed
Ex: # crimes committed this year… most people are zero or very low; a few people are very high
Extreme skew violates the normality assumption of OLS regression.

4. Count Models Two most common count models:
Poisson Regression Model
Negative Binomial Regression Model
Both based on the Poisson distribution:
m = expected count (and variance)
Called lambda (l) in some texts; I rely on Freese & Long 2006
y = observed count

5. Poisson Regression Strategy: Model log of m as a function of Xs
Quite similar to modeling log odds in logit
Again, the log form avoids negative values
Which can be written as:

6. Poisson Regression: Example
Hours per week spent on web

7. Poisson Regression: Web Use
Output = similar to logistic regression
. poisson wwwhr male age educ lowincome babies
Poisson 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 |
Men spend more time on the web than women
Number of young children in household reduces web use

8. Poisson Regression: Stata Output
Stata output yields familiar statistics:
Standard errors, z/t- values, and p-values for coefficient hypothesis tests
Pseudo R-square for model fit
Not a great measure… but gives a crude explained variance
MLE log likelihood
Likelihood ratio test: Chi-square and p-value
Comparing to null model (constant only)
Tests can also be conducted on nested models with stata command “lrtest”.

9. Interpreting Coefficients
In Poisson Regression, Y is typically conceptualized as a rate…
Positive coefficients indicate higher rate; negative = lower rate
Like logit, Poisson models are non-linear
Coefficients don’t have a simple linear interpretation
Like logit, model has a log form; exponentiation aids interpretation
Exponentiated coefficients are multiplicative
Analogous to odds ratios… but called “incidence rate ratios”.

10. Interpreting Coefficients
Exponentiated coefficients: indicate effect of unit change of X on rate
In STATA: “incidence rate ratios”: “poison … , irr”
eb= 2.0 indicates that the rate doubles for each unit change in X
eb= .5 indicates that the rate drops by half for each unit change in X
Recall: Exponentiated coefs are multiplicative
If eb= 5.0, a 2-point change in X isn’t 10; it is 5 * 5 = 25
Also: you must invert to see opposite effects
If eb= 5.0, a 1-point decrease in X isn’t -5, it is 1/5 = .2

11. Interpreting Coefficients
Again, exponentiated coefficients (rate ratios) can be converted to % change
Formula: (eb - 1) * 100%
Ex: (e.5 - 1) * 100% = 50% decrease in rate.

12. Interpreting Coefficients
Exponentiated coefficients yield multiplier:
. poisson wwwhr male age educ lowincome babies
Poisson 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 |
Exponentiation of .359 = 1.43; Rate is 1.43 times higher for men
(1.43-1) * 100 = 43% more
Exp(-.14) = Each baby reduces rate by factor of .87
(.87-1) * 100 = 13% less

13. Predicted Counts
Stata “predict varname, n” computes predicted value for each case
. predict predwww if e(sample), n
. list wwwhr predwww if e(sample)
| wwwhr predwww |
| |
1. | |
2. | |
3. | |
12. | |
13. | |
15. | |
16. | |
19. | |
20. | |
21. | |
23. | |
24. | |
25. | |
27. | |
33. | |
Some of the predictions are close to the observed values…
Many of the predictions are quite bad…
Recall that the model fit was VERY poor!

14. Predicted Probabilities
Stata extension “prcount” can compute probabilities for each possible count outcome
For all cases, of for particular groups
It plugs values (m), Xs, & bs into formula:
Rate: [ , ]
Pr(y=0|x): [ , ]
Pr(y=1|x): [ , ]
Pr(y=2|x): [ , ]
Pr(y=3|x): [ , ]
Pr(y=4|x): [ , ]
Pr(y=5|x): [ , ]
Pr(y=6|x): [ , ]
Pr(y=7|x): [ , ]
Pr(y=8|x): [ , ]
Pr(y=9|x): [ , ]
male age educ lowincome babies x=

15. Issue: Exposure
Poisson outcome variables are typically conceptualized as rates
Web hours per week
Number of crimes committed in past year
Issue: Cases may vary in exposure to “risk” of a given outcome
To properly model rates, we must account for the fact that some cases have greater exposure than others
Ex: # crimes committed in lifetime
Older people have greater opportunity to have higher counts
Alternately, exposure may vary due to research design
Ex: Some cases followed for longer time than others…

16. Issue: Exposure
Poisson (and other count models) can address varying exposure:
Where ti = exposure time for case i
It is easy to incorporate into stata, too:
Ex: poisson NumCrimes SES income, exposure(age)
Note: Also works with other “count” models.

17. Poisson Model Assumptions
Poisson regression makes a big assumption: That variance of m = m (“equidisperson”)
In other words, the mean and variance are the same
This assumption is often not met in real data
Dispersion is often greater than m: overdispersion
Consequence of overdispersion: Standard errors will be underestimated
Potential for overconfidence in results; rejecting H0 when you shouldn’t!
Note: overdispersion doesn’t necessarily affect predicted counts (compared to alternative models).

18. Poisson Model Assumptions
Overdispersion is most often caused by highly skewed dependent variables
Often due to variables with high numbers of zeros
Ex: Number of traffic tickets per year
Most people have zero, some can have 50!
Mean of variable is low, but SD is high
Other examples of skewed outcomes
# of scholarly publications
# cigarettes smoked per day
# riots per year (for sample of cities in US).

19. Negative Binomial Regression
Strategy: Modify the Poisson model to address overdispersion
Add an “error” term to the basic model:
Additional model assumptions:
Expected value of exponentiated error = 1 (ee = 1)
Exponentiated error is Gamma distributed
We hope that these assumptions are more plausible than the equidispersion assumption!

20. Negative Binomial Regression
Full negative biniomial model:
Note that the model incorporates a new parameter: a
Alpha represents the extent of overdispersion
If a = 0 the model reduces to simple poisson regression

21. Negative Binomial Regression
Question: Is alpha (a) = 0?
If so, we can use Poisson regression
If not, overdispersion is present; Poisson is inadequate
Strategy: conduct a statistical test of the hypothesis: H0: a = 0; H1: a > 0
Stata provides this information when you run a negative binomial model:
Likelihood ratio test (G2) for alpha
P-value < .05 indicates that overdispersion is present; negative binomial is preferred
If P>.05, just use Poisson regression
So you don’t have to make assumptions about gamma dist….

22. Negative Binomial Regression
Interpreting coefficients: Identical to poisson regression
Predicted probabilities: Can be done. You must use big Neg Binomial formula
Plugging in observed Xs, estimates of a, Bs…
Probably best to get STATA to do this one…
Long & Freese created command: prvalue

23. Negative Binomial Example: Web Use
Note: Bs are similar but SEs change a lot!
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
Note: Standard Error for education increased from .004 to .012! Effect is no longer statistically significant.

24. 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

25. General Remarks
Poisson & Negative binomial models suffer all the same basic issues as “normal” regression
Model specification / omitted variable bias
Multicollinearity
Outliers/influential cases
Also, it uses Maximum Likelihood
N > 500 = fine; N < 100 can be worrisome
Results aren’t necessarily wrong if N<100;
But it is a possibility; and hard to know when problems crop up
Plus ~10 cases per independent variable.

26. 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.

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