Version info: Code for this page was tested in Mplus version 6.12.

Poisson regression is used to model dependent variables that are counts.

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 Poisson regression

Example 1.  The number of persons killed by mule or horse kicks in the
Prussian army per year.  von 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 a single 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

Let’s pursue Example 3 from above.

The data for this example were simulated and are in the file
https://stats.idre.ucla.edu/wp-content/uploads/2016/02/poisson_sim.dat. 
In this example, num_awards is the outcome variable and indicates the
number of awards earned by students at a single high school in a single 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 look at the data.  It is always a good idea to start with descriptive
statistics.

Data:
File is g:daehttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/poisson_sim.dat;
Variable:
Names are 
id num_awards prog math p1 p2 p3;
Missing are all (-9999); 
usevariables are num_awards prog p1 p2 p3 math;
analysis:
type = basic;
plot: type is plot1; 

RESULTS FOR BASIC ANALYSIS


     ESTIMATED SAMPLE STATISTICS


           Means
              NUM_AWAR      PROG          P1            P2            P3
              ________      ________      ________      ________      ________
      1         0.630         2.025         0.225         0.525         0.250


           Means
              MATH
              ________
      1        52.645


           Covariances
              NUM_AWAR      PROG          P1            P2            P3
              ________      ________      ________      ________      ________
 NUM_AWAR       1.103
 PROG          -0.001         0.474
 P1            -0.097        -0.231         0.174
 P2             0.194        -0.013        -0.118         0.249
 P3            -0.097         0.244        -0.056        -0.131         0.188
 MATH           4.879        -0.966        -0.590         2.146        -1.556


           Covariances
              MATH
              ________
 MATH          87.329


           Correlations
              NUM_AWAR      PROG          P1            P2            P3
              ________      ________      ________      ________      ________
 NUM_AWAR       1.000
 PROG          -0.001         1.000
 P1            -0.221        -0.802         1.000
 P2             0.370        -0.038        -0.566         1.000
 P3            -0.214         0.817        -0.311        -0.607         1.000
 MATH           0.497        -0.150        -0.151         0.460        -0.385


           Correlations
              MATH
              ________
 MATH           1.000
 
 
     MAXIMUM LOG-LIKELIHOOD VALUE FOR THE UNRESTRICTED (H1) MODEL IS     293.292

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 analysis

In the Mplus syntax below, we specify that the variables to be used in the
Poisson regression are num_awards, p2, p3 and math. 
(The variables p2 and p3 are indicator variables for prog.)  We also specify that num_awards is a count variable.  (Because the
variable name num_awards has more than eight characters, we get a warning in the
output that this variable name has been truncated to eight characters.)  By
default, Mplus uses restricted maximum likelihood (MLR), so robust standard
errors are given in the output.  The MLR standard errors are computed using
a sandwich estimator.  These are what we generally call robust standard
errors.  Cameron and Trivedi (2009) recommend the use
of robust standard errors when estimating a Poisson model.  If you do not want robust standard errors, you can use the
analysis: estimator = ml; block. 

Data: 
File is g:daehttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/poisson_sim.dat;
Variable:
Names are 
id num_awards prog math p1 p2 p3;
Missing are all (-9999) ; 
usevariables are num_awards p2 p3 math;
count is num_awards;
model:
num_awards on p2 p3 math;

MODEL FIT INFORMATION

Number of Free Parameters                        4

Loglikelihood

          H0 Value                        -182.752
          H0 Scaling Correction Factor       0.976
            for MLR

Information Criteria

          Akaike (AIC)                     373.505
          Bayesian (BIC)                   386.698
          Sample-Size Adjusted BIC         374.025
            (n* = (n + 2) / 24)



MODEL RESULTS

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 NUM_AWARDS ON
    P2                 1.084      0.321      3.376      0.001
    P3                 0.370      0.400      0.924      0.356
    MATH               0.070      0.010      6.723      0.000

 Intercepts
    NUM_AWARDS        -5.247      0.646     -8.123      0.000

In the syntax below, some of the variables in the model are given labels. These labels must be in parentheses and must be
the last item listed on the line, so the model is broken up over several lines. We have given the label
a2 to the indicator
variable p2, and the label a3 to the indicator variable p3. Once we have assigned labels to the variables, we can use those
labels in the model test block. Setting both a2 and a3 to 0 allows us to get the two degree-of-freedom test of the variable
prog.

Data: 
File is g:daehttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/poisson_sim.dat;
Variable:
Names are 
id num_awards prog math p1 p2 p3;
Missing are all (-9999); 
usevariables are num_awards p2 p3 math;
count is num_awards;
model:
num_awards on 
p2 (a2)
p3 (a3)
math;
model test:
a2 = 0;
a3 = 0;

< - some output omitted - >

MODEL FIT INFORMATION

Number of Free Parameters                        4

Loglikelihood

          H0 Value                        -182.752
          H0 Scaling Correction Factor       0.976
            for MLR

Information Criteria

          Akaike (AIC)                     373.505
          Bayesian (BIC)                   386.698
          Sample-Size Adjusted BIC         374.025
            (n* = (n + 2) / 24)

Wald Test of Parameter Constraints

          Value                             14.838
          Degrees of Freedom                     2
          P-Value                           0.0006

We can see that the variable prog, as a whole, is statistically significant. 
To help assess the fit of the model, we can look at the model fit statistics in the output.  Several measures of goodness of fit
are provided.  For both the AIC and BIC, smaller is better.

To obtain the results as incident rate ratios, we need to use the model
constraint block.  Again, we use labels to refer to the variables
in the model.  In the model constraint block, we use the new
statement to label the new parameters, which will be the exponentiated
parameters from the model. 

Data: 
File is g:daehttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/poisson_sim.dat;
Variable:
Names are 
id num_awards prog math p1 p2 p3;
Missing are all (-9999); 
usevariables are num_awards p2 p3 math;
count is num_awards;
model:
num_awards on 
p2 (a2)
p3 (a3)
math (a1);
model constraint:
new(p2_exp p3_exp math_exp);
p2_exp = exp(a2);
p3_exp = exp(a3);
math_exp = exp(a1);

MODEL FIT INFORMATION

Number of Free Parameters                        4

Loglikelihood

          H0 Value                        -182.752
          H0 Scaling Correction Factor       0.976
            for MLR

Information Criteria

          Akaike (AIC)                     373.505
          Bayesian (BIC)                   386.698
          Sample-Size Adjusted BIC         374.025
            (n* = (n + 2) / 24)



MODEL RESULTS

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 NUM_AWARDS ON
    P2                 1.084      0.321      3.376      0.001
    P3                 0.370      0.400      0.924      0.356
    MATH               0.070      0.010      6.723      0.000

 Intercepts
    NUM_AWARDS        -5.247      0.646     -8.123      0.000

 New/Additional Parameters
    P2_EXP             2.956      0.949      3.115      0.002
    P3_EXP             1.447      0.580      2.497      0.013
    MATH_EXP           1.073      0.011     95.830      0.000

Recall the form of our model equation:

log(num_awards) = Intercept + b₁(prog=2) + b₂(prog=3)
+ b₃math.

This implies:

num_awards = exp(Intercept + b₁(prog=2) + b₂(prog=3)+ b₃math)
= exp(Intercept) * exp(b₁(prog=2)) * exp(b₂(prog=3)) *
exp(b₃math)

Things to consider

See also

References