This page shows an example of a Poisson regression analysis with footnotes explaining the output. The data
collected were academic information on
316
students. The response variable is
days absent during the school year (daysabs), and we explore its relationship with
math standardized tests score (mathnce),
language standardized tests score  (langnce)
and gender (female).

As assumed for a Poisson model, our response variable is a count variable, and
each subject has the same length of observation time. Had the
observation time for subjects varied, the Poisson model would need to be adjusted to account for the varying length of observation time
per subject. This
point is discussed later in the page. Also, the Poisson model, as compared to
other count models (i.e., negative binomial or zero-inflated models), is assumed
to be the appropriate model. In other words, we assume that the dependent variable is
not over-dispersed and does not have an excessive number of zeros.

You can download the data set used in this example by clicking
here .

data preg;
  set "C:temppoisson";
  female = (gender = 1);
run;

proc genmod data = preg;
 model daysabs = mathnce langnce female / link=log dist=Poisson;
run;

The GENMOD Procedure

                     Model Information

Data Set              WORK.PREG
Distribution            Poisson
Link Function               Log
Dependent Variable      DAYSABS    number days absent

Number of Observations Read         316
Number of Observations Used         316


           Criteria For Assessing Goodness Of Fit

Criterion                 DF           Value        Value/DF
Deviance                 312       2234.5462          7.1620
Scaled Deviance          312       2234.5462          7.1620
Pearson Chi-Square       312       2774.4139          8.8924
Scaled Pearson X2        312       2774.4139          8.8924
Log Likelihood                     1482.2670

Algorithm converged.


                            Analysis Of Parameter Estimates

                               Standard     Wald 95% Confidence       Chi-
Parameter    DF    Estimate       Error           Limits            Square    Pr > ChiSq
Intercept     1      2.2867      0.0700      2.1496      2.4239     1068.59       
Model Information
                     Model Information

Data Seta              WORK.PREG
Distributionb            Poisson
Link Functionc               Log
Dependent Variabled      DAYSABS    number days absent

Number of Observations Reade         316
Number of Observations Usede         316
a. Data Set - This is the SAS dataset on which the Poisson regression

was performed.
b. Distribution - This is the distribution of the dependent variable.

Poisson regression is a type of

generalized linear model. As such, we need to specify the distribution of

the dependent variable, dist = Poisson, as

well as the link function, superscript

c.
c. Link Function - This is the link function used for the Poisson

regression. By default, when we specify dist = Poisson, the log link

function is assumed (and does not need to be specified); however, for

pedagogical purposes, we include link = log. When we write our model out, log( μ ) = β0 + β1x1 +

...  + βpxp, where μ is the count we are

modeling, log(  ) defines the link function (i.e., how we transform μ to

write it as a linear combination of the predictor variables).
d. Dependent Variable - This is the dependent variable used in the

Poisson regression.
e. Number of Observations Read and  Number of Observations Used

- This is the number of observations read and the number of observation used in the

Poisson

regression. The Number of Observations Used may be less than the 

Number of Observations Read if there are missing values for any variables

in the equation. By default, SAS does a listwise deletion of incomplete cases.
Criteria For Assessing Goodness Of Fit
           Criteria For Assessing Goodness Of Fit

Criterionf                DFg         Valueg        Value/DFh
Deviance                 312       2234.5462          7.1620
Scaled Deviance          312       2234.5462          7.1620
Pearson Chi-Square       312       2774.4139          8.8924
Scaled Pearson X2        312       2774.4139          8.8924
Log Likelihood                     1482.2670

Algorithm convergedi.
Prior to discussing the Criterion, DF, Value and 

Value/DF, we need to discuss the logic of this section. Attention is

placed on Deviance and Scaled Deviance; the argument naturally

extends to Pearson Chi-Square.
First, note that the Deviance has an approximate chi-square distribution with

n-p

degrees of freedom, where n is the number of observations and p is the

number of predictor variables (including the intercept), and the expected value of a chi-square random variable is equal

to the degrees of freedom. Then, if our model fits the data well, the ratio of the 

Deviance to DF, Value/DF, should be about one. Large ratio

values may indicate model misspecification or an over-dispersed response variable; ratios less than one may

also indicate

model misspecification or an under-dispersed response variable. A consequence of

such dispersion issues is that standard errors are incorrectly estimated,

implying an invalid chi-square test statistic, superscript p. Importantly,

however, assuming our model is correctly specified, the Poisson regression estimates remain unbiased in the presence of over-disperion or under-dispersion.

Two "fixes" are either running the same model as a negative binomial

regression, or correcting the standard errors of the estimates. The standard error

correction corresponds to the approach for the scaled criterion. A naive

explanation when the scale option is specified (scale = dscale), the

Scaled Deviance is forced to equal one. By forcing Value/DF to one (dividing

Value/DF by itself), our model becomes "optimally" dispersed; however,

what actually happens is that the standard errors are adjusted ad hoc. The standard

errors are adjusted by a factor, the square root of Value/DF.
f. Criterion - Below are various measurements used to assess the

model fit.
Deviance - This is the deviance for the

model. The deviance is defined as two times the difference of the log

likelihood

for the maximum achievable model (i.e., each subject's response serves as a unique

estimate of the Poisson parameter), and the log likelihood under the fitted

model.

The difference in the Deviance and degrees of freedom of two nested models

can be used in the likelihood ratio chi-square tests.
Scaled Deviance - This is the scaled deviance.

The scaled deviance is equal to the deviance since we did not

specify the scale=dscale option on the model statement.
Pearson Chi-Square - This is the Pearson chi-square

statistic. The Pearson chi-square is defined as the squared difference between

the observed and predicted values divided by the variance of the predicted value summed over

all observations in the model.
Scaled Pearson X2 - This is the scaled Pearson

chi-square statistic.

The scaled Pearson X2 is equal to the Pearson chi-square since we

did not specify the scale=pscale option on the model statement.
Log Likelihood - This is the log likelihood of

the model. Instead of using the deviance, we can take two times the

difference between the log likelihood for nested models to perform a

chi-square test.
g. DF and Value - These are the degrees of freedom DF

and the respective Value for the Criterion measures. The DF

is equal to n-p, where n is the number of observation used and p is

the number of parameters estimated.
h. Value/DF - This is the ratio of Value to DF given

in superscript g. Refer to the discussion at the beginning of this section for an

interpretation/use of this value.
i. Algorithm Convergered - This is a note indicating that the algorithm for parameter estimates

has converged, implying that a solution was found.
Analysis Of Parameter Estimates
                            Analysis Of Parameter Estimates

                               Standard     Wald 95% Confidence       Chi-
Parameterj    DFk   Estimatel      Errorm           Limitsn          Squareo   Pr > ChiSqo
Intercept     1      2.2867      0.0700      2.1496      2.4239    1068.59        
j. Parameter - Underneath are the predictor
variables and the Scale parameter.
k. DF - These are the degrees of freedom DF spent on each of
the respective parameter estimates. Note that the DF for the Scale
parameter is set to 0. The DF define the distribution used to test 
Chi-Square, superscript o.
l. Estimate -These are the estimated Poisson regression coefficients for the model. Recall that the dependent variable is
a count variable, and Poisson regression models the log of the expected count
as a linear function of the predictor variables. We can interpret the Poisson
regression coefficient as follows: for a one unit change in the predictor variable, the
difference in the logs of expected counts is expected to change by the respective
regression coefficient, given the other predictor variables in the model are held
constant.
Also, note that each subject in our sample was followed for one school year.
If this was not the case (i.e., some subjects were followed for  half a
year, some for a year and the rest for two years) and we were to neglect the
exposure time, our Poisson regression estimates would be biased, since our model
assumes all subjects had the same observation time. If this was an issue, we would
use the offset option, offset=log_timevar,
where log_timevar corresponds to the logged version of the variable specifying length of time an individual was
followed to adjust the Poisson regression estimates.
    Intercept - This is the Poisson regression estimate
when all variables in the model are evaluated at zero. For males (the variable
female evaluated at zero) with zero mathnce and langnce
test scores, the log of the expected count for daysabs is 2.2867 units. Note that evaluating 
mathnce and langnce at zero is out of the range of plausible test
scores. If the test scores were mean-centered, the intercept would have a
natural interpretation: the log of the expected count for males with average 
mathnce and langnce test scores.
    mathnce - This is the Poisson regression estimate for a one unit increase in
math standardized test score, given the other
variables are held constant in the model. If a student
were to increase her mathnce test score by one point, the difference in
the logs of expected counts would be expected to decrease by 0.0035 unit, while holding
the other variables in the model constant.
langnce - This is the Poisson regression estimate
for a one unit increase in language standardized test score, given the other
variables are held constant in the model. If a student
were to increase her langnce test score by one point, the difference in
the logs of expected counts would be expected to decrease by 0.0122 unit while holding
the other variables in the model constant.
female -  This is the estimated Poisson
regression coefficient comparing females to males, given the other variables are
held constant in the model. The difference in the logs of expected counts is
expected to be 0.4010 unit higher for females compared to males, while holding
the other variables constant in the model.
Scale - This is the Scale value for the
Poisson
model. Since our model was not scaled (Scaled Deviance or Scaled
Pearson X2), the default scale for the Poisson model is set to one. This is
noted by the comment at the bottom of the output: NOTE: The scale parameter was held fixed.
m. Standard Error - These are the standard errors of the
individual regression coefficients. They
are used in both the Wald 95% Confidence Limits, superscript n, and the
Chi-Square test
statistic, superscript o.
n. Wald 95% Confidence Limits - This is the Wald Confidence Interval
(CI) of an individual Poisson regression coefficient, given the other predictors are in the
model. For a given predictor variable with a level of 95% confidence, we'd say
that we are 95% confident that upon repeated trials, 95% of the CI's would
include the "true" population Poisson regression coefficient. It is calculated
as Estimate ± (zα/2)*(Standard Error), where zα/2
is a critical value on the standard normal distribution. The CI is equivalent to the
Chi-Square test statistic: if the CI includes zero, we'd fail to
reject the null hypothesis that a particular regression coefficient is zero, given the other predictors are in the model.
An advantage of a CI is that it is illustrative; it provides information on where the "true" parameter may lie
and the precision of the point estimate.
o. Chi-Square and Pr > ChiSq - These are the test statistics
and p-values, respectively, testing the null hypothesis that an individual
predictor's regression coefficient is zero, given that the rest of the
predictors are in the model. The Chi-Square test statistic is the squared ratio of the 
Estimate to the
Standard Error of the respective predictor. The Chi-Square value follows a standard
chi-square distribution with degrees of freedom given by DF, which is used to test against
the alternative hypothesis that the Estimate is not equal to zero. The probability that a particular 
Chi-Square test statistic is as extreme as, or more
so, than what has been observed under the null hypothesis is defined by Pr>ChiSq.