Stata Data Analysis Examples: Poisson Regression



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Stata Data Analysis Examples
Poisson Regression

Examples of Poisson Regression

Example 1. School administrators study the attendance behavior of high school juniors at two schools. Predictors of the number of days of absence include gender of the student and standardized test scores in math and language arts.

Description of the Data

Let's pursue Example 1 from above.

We have attendance data on 316 high school juniors from two urban high schools in the file poissonreg.dta . The response variable of interest is days absent, daysabs. The variables math and langarts give the standardized test scores for math and language arts respectively. The variable male is a binary indicator of student gender.

Let's look at the data.

use http://www.ats.ucla.edu/stat/stata/dae/poissonreg, clear

summarize daysabs math langarts male

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
     daysabs |       316    5.810127    7.449003          0         45
        math |       316    48.75101    17.88076   1.007114   98.99289
    langarts |       316    50.06379    17.93921   1.007114   98.99289
        male |       316    .4873418    .5006325          0          1

tabstat daysabs, stat(n mean var)

    variable |         N      mean  variance
-------------+------------------------------
     daysabs |       316  5.810127  55.48764
--------------------------------------------

histogram daysabs, discrete freq



tab male

       male |      Freq.     Percent        Cum.
------------+-----------------------------------
          0 |        162       51.27       51.27
          1 |        154       48.73      100.00
------------+-----------------------------------
      Total |        316      100.00

Some Strategies You Might Be Tempted To Try

Before we show how you can analyze this with a Poisson regression analysis, let's consider some other methods that you might use.

OLS Regression - You could try to analyze these data using OLS regression. However, count data are highly non-normal and are not well estimated by OLS regression.
Negative Binomial Regression - Negative binomial regression does better with over dispersed data, i.e. variance much larger than the mean.
Zero-inflated Regression Model - This might be a good if there are more zeros than would be expected by either a poisson or negative binomial model.

Stata Poisson Regression Analysis

poisson daysabs math langarts male

Iteration 0:   log likelihood = -1547.9709  
Iteration 1:   log likelihood = -1547.9709  

Poisson regression                                Number of obs   =        316
                                                  LR chi2(3)      =     175.27
                                                  Prob > chi2     =     0.0000
Log likelihood = -1547.9709                       Pseudo R2       =     0.0536

------------------------------------------------------------------------------
     daysabs |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        math |  -.0035232   .0018213    -1.93   0.053     -.007093    .0000466
    langarts |  -.0121521   .0018348    -6.62   0.000    -.0157483   -.0085559
        male |  -.4009209   .0484122    -8.28   0.000     -.495807   -.3060348
       _cons |   2.687666   .0726512    36.99   0.000     2.545272     2.83006
------------------------------------------------------------------------------

The output looks very much like the output from an OLS regression. The output begins the iteration log giving the values of the log likelihoods starting with a model that has no predictors. The last value in the log is the final value of the log likelihood for the full model and is repeated below.

Next comes the header information. On the right-hand side the number of observations used (316) is given along with the likelihood ratio chi-squared with three degrees of freedom for the full model, followed by the p-value for the chi-square. The model, as a whole, is statistically significant. The header also includes a pseudo-R² which is 0.0536 in this example.

Below the header you will find the poisson regression coefficients for each of the variables along with standard errors, z-scores, p-values and 95% confidence intervals for the coefficients.

Now, just to be on the safe side, let's rerun the poisson command with the robust option in order to obtain robust standard errors for the poisson regression coefficients.

poisson daysabs math langarts male, robust

Iteration 0:   log pseudolikelihood = -1547.9709  
Iteration 1:   log pseudolikelihood = -1547.9709  

Poisson regression                                Number of obs   =        316
                                                  Wald chi2(3)    =      27.62
                                                  Prob > chi2     =     0.0000
Log pseudolikelihood = -1547.9709                 Pseudo R2       =     0.0536

------------------------------------------------------------------------------
             |               Robust
     daysabs |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        math |  -.0035232   .0076373    -0.46   0.645     -.018492    .0114455
    langarts |  -.0121521   .0052951    -2.29   0.022    -.0225304   -.0017739
        male |  -.4009209   .1395915    -2.87   0.004    -.6745153   -.1273266
       _cons |   2.687666   .2181435    12.32   0.000     2.260113    3.115219
------------------------------------------------------------------------------

Using the robust option has resulted in a fairly large change in the model chi-square, which is now a Wald chi-square, based on log pseudolikelihoods, instead of a likelihood ratio chi-square.

In the main body of the output are the poisson coefficients, robust standard errors, z-scores, p-values and 95% confidence intervals for the coefficients. The variable math was border-line significant without the robust option and is clearly not significant with it. The robust standard errors attempt to adjust for heterogeneity in the model.

Since math is not significant in the model with robust standard errors, we will rerun the model dropping that variable.

poisson daysabs langarts male, robust

Iteration 0:   log pseudolikelihood = -1549.8567  
Iteration 1:   log pseudolikelihood = -1549.8567  

Poisson regression                                Number of obs   =        316
                                                  Wald chi2(2)    =      27.31
                                                  Prob > chi2     =     0.0000
Log pseudolikelihood = -1549.8567                 Pseudo R2       =     0.0524

------------------------------------------------------------------------------
             |               Robust
     daysabs |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    langarts |    -.01467   .0034382    -4.27   0.000    -.0214087   -.0079313
        male |  -.4093528   .1354388    -3.02   0.003     -.674808   -.1438975
       _cons |   2.646977   .1826364    14.49   0.000     2.289016    3.004937
------------------------------------------------------------------------------

Finally, we will use the prchange command (findit prchange) by J. Scott Long and Jeremy Freese to get the predicted change in days absent.

prchange

poisson: Changes in Rate for daysabs

          min->max      0->1     -+1/2    -+sd/2  MargEfct
langarts   -8.6843   -0.1683   -0.0814   -1.4637   -0.0814
    male   -2.2743   -2.2743   -2.2861   -1.1385   -2.2702

exp(xb):   5.5458

        langarts      male
    x=   50.0638   .487342
sd(x)=   17.9392   .500633

Sample Write-Up of the Analysis

Before we begin the sample write-up we need to get the output into a form more acceptable for publication. The estout command (findit estout by Ben Jann of ETH Zurich), will get us close to what we want.

estout, cells(b(star fmt(%8.2f)) se(par fmt(%8.2f))) stats(ll chi2, fmt(%8.2f))

        .
        b/se
daysabs 
langarts        -0.01***
        (0.00)
male    -0.41**
        (0.14)
_cons   2.65***
        (0.18)
ll      -1549.86
chi2    27.31

With a little bit of manual editing we can produce an acceptable table of the output.

model        
language arts     -0.01***
                  (0.00)
male              -0.41**
                  (0.14)
constant           2.65***
                  (0.18)            
log psuedo-
   likelihood  -1549.86
chi-squared       27.31
legend: coefficient/(standard error)  ** p<0.01 *** p<0.001

The poisson regression model predicting days absent from school using language arts and gender was statistically significant (chi-squared = 27.31, df = 2, p<.0001). The predictors langarts and male were each statically significant. For these data, the expected log count for a one-unit increase in language arts was -0.0146. Male students had an expected log count -0.41 less than female students.

Cautions, Flies in the Ointment

It is not recommended that poisson models be applied to small samples. What constitutes a small sample does not seem to be clearly defined in the literature.

Stata Data Analysis Examples Poisson Regression