Poisson regression is used to model count variables.

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.

This page is done using SPSS 19.

Examples of Poisson regression

Example 1.  The number of persons killed by mule or horse kicks in the
Prussian army per year. Ladislaus Bortkiewicz collected data from 20 volumes of
Preussischen Statistik.  These data were collected on 10 corps of the
Prussian army in the late 1800s over the course of 20 years.

Example 2.  The number of people in line in front of you at the grocery
store.  Predictors may include the number of items currently offered at a
special discounted price and whether a special event (e.g., a holiday, a big
sporting event) is three or fewer days away.

Example 3.  The number of awards earned by students at one high school.
Predictors of the number of awards earned include the type of program in which
the student was enrolled (e.g., vocational, general or academic) and the score
on their final exam in math.

Description of the Data

For the purpose of illustration, we have simulated a data set for Example 3
above: https://stats.idre.ucla.edu/wp-content/uploads/2016/02/poisson_sim.sav. In this example,
num_awards is the outcome variable and indicates the number of awards earned
by students at a high school in a year, math is a continuous predictor
variable and represents students’ scores on their math final exam, and prog
is a categorical predictor variable with three levels indicating the type of
program in which the students were enrolled.

Let’s start with loading the data and looking at some descriptive statistics.

GET
  FILE='D:datapoisson_sim.sav'.
  
  DESCRIPTIVES
  VARIABLES=math num_awards
  /STATISTICS=MEAN STDDEV VAR MIN MAX .

Each variable has 200 valid observations and their distributions seem quite
reasonable. The unconditional mean and variance of our outcome variable
are not extremely different. Our model assumes that these values, conditioned on
the predictor variables, will be equal (or at least roughly so).

Let’s continue with our description of the variables in this dataset. The
table below shows the average numbers of awards by program type and seems to
suggest that program type is a good candidate for predicting the number of
awards, our outcome variable, because the mean value of the outcome appears to
vary by prog. Additionally, the means and variances within each level of
prog–the conditional means and variances–are similar.

MEANS tables = num_awards by prog.
GRAPH
  /HISTOGRAM=num_awards.

Analysis methods you might consider

Below is a list of some analysis methods you may have
encountered.  Some of the methods listed are quite reasonable, while others have
either fallen out of favor or have limitations.

Poisson regression

Below we use the genlin command to estimate a Poisson regression
model. We have one continuous predictor and one categorical predictor. In the
genlin line, we list our categorical predictor prog after “by” and
our continuous predictor math after “with”. Both appear in the model
line. We use the covb=robust option in the criteria line to obtain robust standard errors
for the parameter estimates as recommended by Cameron and Trivedi (2009) to
control for mild violation of the distribution assumption that the variance
equals the mean. Finally, we ask SPSS to print out the model fit statistics, the
summary of the model effects, and the parameter estimates.

GENLIN num_awards BY prog WITH math
  /MODEL prog math INTERCEPT=YES
   DISTRIBUTION=POISSON LINK=LOG
  /CRITERIA COVB=ROBUST
  /PRINT FIT SUMMARY SOLUTION.

Sometimes, we might want to present the regression results as incident rate
ratios. These IRR values are equal to our
coefficients from the output above exponentiated and we can ask SPSS to printsolution(exponentiated).

GENLIN num_awards BY prog WITH math
  /MODEL prog math INTERCEPT=YES
  DISTRIBUTION=POISSON LINK=LOG
  /CRITERIA METHOD=FISHER(1) SCALE=1 COVB=ROBUST
  /PRINT SOLUTION (EXPONENTIATED).

The output above indicates that the incident rate for [prog=2] is 2.042
times the incident rate for the reference group, [prog=3].  Likewise, the
incident rate for [prog=1] is 0.691 times the incident rate for the
reference group holding the other variables at constant.  The percent change in
the incident rate of num_awards is an increase of 7% for every unit increase in
math.

Recall the form of our model equation:

log(num_awards) = Intercept + b₁(prog=1) + b₂(prog=2)
+ b₃math.

This implies:

num_awards = exp(Intercept + b₁(prog=1) + b₂(prog=2)+
b₃math) = exp(Intercept) * exp(b₁(prog=1)) * exp(b₂(prog=2))
* exp(b₃math)

The coefficients have an additive effect in the log(y) scale and the
IRR have a multiplicative effect in the y scale.

For additional information on the various metrics in which the results can be
presented, and the interpretation of such, please see Regression Models for
Categorical Dependent Variables Using Stata, Second Edition by J. Scott Long
and Jeremy Freese (2006).

To understand the model better, we can use the
emmeans command to calculate the predicted counts at each
level of prog, holding all other variables (in this example,
math)
in the model at their means.

GENLIN num_awards BY prog WITH math
  /MODEL prog math INTERCEPT=YES
  DISTRIBUTION=POISSON LINK=LOG
  /CRITERIA METHOD=FISHER(1) SCALE=1 COVB=ROBUST
  /PRINT NONE
  /EMMEANS TABLES=prog SCALE=ORIGINAL.

In the output above, we see that the predicted number of events for level 1
of prog is about .21, holding math at its mean.  The predicted
number of events for level 2 of prog is higher at .62, and the predicted
number of events for level 3 of prog is about .31. Note that the
predicted count of level 2 of prog is (.62/.31) = 2.0 times
higher than the predicted count for level 3 of prog. This matches what we
saw in the IRR output table.

Below we will obtain the predicted counts for each value of prog at
two set values of math: 35 and 75.

GENLIN num_awards BY prog WITH math
  /MODEL prog math INTERCEPT=YES
  DISTRIBUTION=POISSON LINK=LOG
  /PRINT NONE
  /EMMEANS TABLES=prog CONTROL =math(35) SCALE=ORIGINAL.
    

GENLIN num_awards BY prog WITH math
  /MODEL prog math INTERCEPT=YES
  DISTRIBUTION=POISSON LINK=LOG
  /PRINT NONE
  /EMMEANS TABLES=prog CONTROL =math(75) SCALE=ORIGINAL.

  

The table above shows that with prog=1 and 
math held at 35, the average predicted count (or
average number of awards) is about .06; when math = 75, the average
predicted count for prog=1 is about 1.01. If we look at these predicted counts at math = 35
and math = 75, we can see that the ratio is (1.01/0.06) = 16.8. This
matches (within rounding error) the IRR of 1.0727 for a 40 unit change: 1.0727^40 =
16.1.

Things to consider

See also

Regression Models for Categorical and Limited Dependent Variables.

References