## TA: Nicolas Bottan

Welcome to a new issue of e-Tutorial, where we’ll focus on Count Data models, with special focus to Poisson and Negative Binomial regression. 1

# Data

You can download the data set, called health.dta from the Econ 508 web site. Save it in your preferred directory.

The data set used here comes from A. Colin Cameron and Per Johansson, “Count Data Regression Using Series Expansion: With Applications”, Journal of Applied Econometrics, Vol. 12, No. 3, 1997, pp. 203-224.

The data is in STATA format, and you can download it from the Econ 508 web site. According to the authors, the data set is based on the 1977-78 Australian Health Survey. It contains 5190 observations on the following variables:

• NONDOCCO: Number of consultations in the past four week with non-doctor health professionals (chemist, optician, physiotherapist, etc.)
• SEX Gender of patient (female=1)
• AGE Age of patient (in years)
• INCOME Patient’s annual income (in hundreds of dollars)
• LEVYPLUS Dummy for private insurance coverage (=1)
• FREEPOOR Dummy for free government insurance coverage due to low income (=1)
• FREEREPA Dummy for free government insurance coverage due to old age, disability, or veteran status (=1)
• ILLNESS Number of illnesses in past two weeks
• ACTDAYS Number of days of reduced activity in past two weeks due to illness or injury
• HSCORE Health questionnaire score (high score=bad health)
• CHCOND1 Dummy for chronic condition not limiting activity (=1)
• CHCOND2 Dummy for chronic condition limiting activity (=1)

According to the authors, the data is overdispersed (the sample mean of the dependent variable is 0.215 and the sample variance of it is 0.932). This might be an indicator that the Poisoness property (mean equals variance) may be violated, and a Negative Binomial Regression might be necessary.

The aim of the e-TA will be to try to reproduce Cameron and Johansson (1997) main results.

First we load the data. 

   use health.dta, clear
Next we generate the variable age-squared:
   gen AGE2 = AGE^2

Finally you can run the generalized linear models of your choice. Here we will focus on how to run Poisson Regression and Negative Binomial Regression Models

# Poisson Regression

   poisson NONDOCCO SEX AGE AGE2 INCOME LEVYPLUS FREEPOOR FREEREPA ILLNESS ACTDAYS HSCORE CHCOND1 CHCOND2
Iteration 0:   log likelihood = -4205.1938  Iteration 1:   log likelihood = -3382.4299  Iteration 2:   log likelihood = -3112.8639  Iteration 3:   log likelihood = -3109.3816  Iteration 4:   log likelihood = -3109.3722  Iteration 5:   log likelihood = -3109.3722  Poisson regression                                Number of obs   =       5190                                                  LR chi2(12)     =    1086.94                                                  Prob > chi2     =     0.0000Log likelihood = -3109.3722                       Pseudo R2       =     0.1488------------------------------------------------------------------------------    NONDOCCO |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]-------------+----------------------------------------------------------------         SEX |   .3316144   .0696475     4.76   0.000     .1951079    .4681209         AGE |  -3.307805   1.228064    -2.69   0.007    -5.714766   -.9008437        AGE2 |    4.39034    1.30202     3.37   0.001     1.838428    6.942252      INCOME |  -.0352855   .1113467    -0.32   0.751    -.2535211    .1829501    LEVYPLUS |   .3278973    .097685     3.36   0.001     .1364383    .5193564    FREEPOOR |   .0154283   .2110056     0.07   0.942    -.3981352    .4289917    FREEREPA |   .4820755   .1160326     4.15   0.000     .2546559    .7094952     ILLNESS |    .054726   .0215542     2.54   0.011     .0124806    .0969714     ACTDAYS |   .0979188   .0061003    16.05   0.000     .0859625    .1098751      HSCORE |   .0447936   .0116531     3.84   0.000     .0219539    .0676332     CHCOND1 |   .5186225    .087033     5.96   0.000      .348041     .689204     CHCOND2 |   1.078644   .0983912    10.96   0.000     .8858014    1.271488       _cons |  -2.443619   .2401184   -10.18   0.000    -2.914242   -1.972995------------------------------------------------------------------------------
You can check the "Poisoness" property by typing:
   poisgof
         Deviance goodness-of-fit =  5040.935         Prob > chi2(5177)        =    0.9103         Pearson goodness-of-fit  =  15332.47         Prob > chi2(5177)        =    0.0000
The null hypothesis (of Poisoness) can not be rejected in the test above, meaning that a Poisson Regression is fine for this data. Nevertheless, below we explore how to compute the Negative Binomial Regression anyway.

# Negative Binomial Regression

You can run a Negative Binomial Regression as follows:

   nbreg NONDOCCO SEX AGE AGE2 INCOME LEVYPLUS FREEPOOR FREEREPA ILLNESS ACTDAYS HSCORE CHCOND1 CHCOND2 
Fitting Poisson model:Iteration 0:   log likelihood = -4205.1938  Iteration 1:   log likelihood = -3382.4299  Iteration 2:   log likelihood = -3112.8639  Iteration 3:   log likelihood = -3109.3816  Iteration 4:   log likelihood = -3109.3722  Iteration 5:   log likelihood = -3109.3722  Fitting constant-only model:Iteration 0:   log likelihood =  -2940.014  (not concave)Iteration 1:   log likelihood = -2313.0946  Iteration 2:   log likelihood =  -2312.632  Iteration 3:   log likelihood =  -2312.632  Fitting full model:Iteration 0:   log likelihood = -2214.1623  Iteration 1:   log likelihood = -2205.8669  Iteration 2:   log likelihood = -2161.0568  Iteration 3:   log likelihood = -2160.4958  Iteration 4:   log likelihood = -2160.4953  Iteration 5:   log likelihood = -2160.4953  Negative binomial regression                      Number of obs   =       5190                                                  LR chi2(12)     =     304.27Dispersion     = mean                             Prob > chi2     =     0.0000Log likelihood = -2160.4953                       Pseudo R2       =     0.0658------------------------------------------------------------------------------    NONDOCCO |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]-------------+----------------------------------------------------------------         SEX |   .2308069   .1246663     1.85   0.064    -.0135347    .4751484         AGE |  -2.675554   2.430949    -1.10   0.271    -7.440127     2.08902        AGE2 |   3.854298    2.62543     1.47   0.142    -1.291451    9.000047      INCOME |  -.0621359   .1906016    -0.33   0.744    -.4357083    .3114364    LEVYPLUS |   .2986752   .1583588     1.89   0.059    -.0117024    .6090528    FREEPOOR |   -.196811   .3515133    -0.56   0.576    -.8857645    .4921425    FREEREPA |   .5877179   .2185872     2.69   0.007     .1592949    1.016141     ILLNESS |   .1443791   .0467823     3.09   0.002     .0526875    .2360708     ACTDAYS |   .1370558   .0170753     8.03   0.000     .1035888    .1705228      HSCORE |   .0739655   .0279625     2.65   0.008     .0191601    .1287709     CHCOND1 |   .4115285    .142977     2.88   0.004     .1312987    .6917583     CHCOND2 |   1.124148   .1830997     6.14   0.000     .7652796    1.483017       _cons |  -2.783845   .4351314    -6.40   0.000    -3.636687   -1.931003-------------+----------------------------------------------------------------    /lnalpha |   2.187067   .0758164                      2.038469    2.335664-------------+----------------------------------------------------------------       alpha |    8.90904   .6754515                      7.678844    10.33632------------------------------------------------------------------------------Likelihood-ratio test of alpha=0:  chibar2(01) = 1897.75 Prob>=chibar2 = 0.000

As we know the negative binomial models does not assume that conditional means are equal to the conditional variances. This inequality is captured by estimating a dispersion parameter (alpha), where alpha = 0 corresponds to the 'equidispersion' case (i.e. poisson). What does the test suggest?