Count
Data

**Definition**

Count data is a
statistical data type that the observed values are non-negative integers and
are in the form of counts. The count variable is a random variable where
Poisson, binomial and negative binomial distributions are commonly used to
describe its distribution. Poisson regression can be used to model count
variables.

**Poisson Regression**

The Poisson
distribution describes the probability that a random event will occur in a time
or space interval when the probability of the event occurring is very small,
but the number of trials is very large.

Model

Poisson regression
models are generalized linear models with the Poisson distribution function.
The log link function is commonly used in the models.

The Poisson
probability distribution:

The Poisson response
variable may be modeled as:

Sometimes, the count
responses will pertain to unequal units of time or space. In such cases, we
let . Then we have:

Using the Log Link,
we obtain:

Overdispersion

A characteristic of
the Poisson distribution is that its mean is equal to its variance. If we see
that the observed variance is greater than the mean - this is known as
overdispersion. It tells us that the model is not appropriate.

A common reason for
overdispersion is the exclusion of relevant explanatory variables.

**Example Code in SAS**

data insure;

input
n c car$ age;

ln
= log(n);

datalines;

500 42 small 1

1200 37 medium 1

100 1 large 1

400 101 small 2

500 73 medium 2

300 14 large 2

;

proc genmod data=insure;

class
car age;

model
c = car age / dist = poisson

link = log

offset = ln;

run;

**Example Code in R**

*# input
data*

n <- c(500, 1200, 100, 400, 500, 300)

c <- c(42, 37, 1, 101, 73, 14)

car <- c("small", "medium", "large", "small", "medium", "large")

age <- c(1, 1, 1, 2, 2, 2)

*# identify
car and age as categorical variables*

car <- factor(car)

age <- factor(age)

*# create
the dataframe*

df <- data.frame(n, c, car, age)

*# Poisson regression*

glm(formula = c ~ car +
age + n, data = df, family = poisson(link="log"))

References

1. Cameron,
A. C.; Trivedi, P. K. (2013). Regression Analysis of Count Data Book (Second
ed.). Cambridge University Press. ISBN 978-1-107-66727-3.

2. *Poisson
Regression*. (2019, December 13). SAS® Help Center.
https://documentation.sas.com/doc/en/pgmsascdc/9.4_3.3/statug/statug_ttest_syntax01.htm