Stata Data Analysis Examples: Negative Binomial Regression



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Stata Data Analysis Examples
Negative Binomial Regression

Examples of Negative Binomial 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. This is the same example that was used in the page on poisson regression.

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 negative binomial regression model, 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.
Poisson Regression - Ordinary poisson regression will have difficulty 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 Negative Binomial Regression Analysis

nbreg daysabs math langarts male

Fitting Poisson model:

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

Fitting constant-only model:

Iteration 0:   log likelihood = -897.78991  
Iteration 1:   log likelihood = -891.24455  
Iteration 2:   log likelihood = -891.24271  
Iteration 3:   log likelihood = -891.24271  

Fitting full model:

Iteration 0:   log likelihood = -881.57337  
Iteration 1:   log likelihood = -880.87788  
Iteration 2:   log likelihood = -880.87312  
Iteration 3:   log likelihood = -880.87312  

Negative binomial regression                      Number of obs   =        316
                                                  LR chi2(3)      =      20.74
Dispersion     = mean                             Prob > chi2     =     0.0001
Log likelihood = -880.87312                       Pseudo R2       =     0.0116

------------------------------------------------------------------------------
     daysabs |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        math |   -.001601     .00485    -0.33   0.741    -.0111067    .0079048
    langarts |  -.0143475   .0055815    -2.57   0.010    -.0252871    -.003408
        male |  -.4311844   .1396656    -3.09   0.002     -.704924   -.1574448
       _cons |   2.716069    .232576    11.68   0.000     2.260229     3.17191
-------------+----------------------------------------------------------------
    /lnalpha |   .2533877   .0955362                      .0661402    .4406351
-------------+----------------------------------------------------------------
       alpha |   1.288383   .1230871                      1.068377    1.553694
------------------------------------------------------------------------------
Likelihood-ratio test of alpha=0:  chibar2(01) = 1334.20 Prob>=chibar2 = 0.000

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 negative binomial regression coefficients for each of the variables along with standard errors, z-scores, p-values and 95% confidence intervals for the coefficients. Additionally, there will be an estimate of the natural log of the over dispersion coefficient, alpha, along with the transformed value. If the alpha coefficient is zero then the model is better estimated using an ordinary poisson regression model.

Below, the coefficients you will find a likelihood ratio test that alpha equals zero. In this example the associated chi-squared value is 1334.2 with one degree of freedom. These results strongly suggest that the negative binomial model is better than the poisson regression model. Now, just to be on the safe side, let's rerun the nbreg command with the robust option in order to obtain robust standard errors for the negative binomial regression coefficients.

nbreg daysabs math langarts male, robust

Fitting constant-only model:

Iteration 0:   log pseudolikelihood = -897.78991  
Iteration 1:   log pseudolikelihood = -891.24455  
Iteration 2:   log pseudolikelihood = -891.24271  
Iteration 3:   log pseudolikelihood = -891.24271  

Fitting full model:

Iteration 0:   log pseudolikelihood = -881.57337  
Iteration 1:   log pseudolikelihood = -880.87788  
Iteration 2:   log pseudolikelihood = -880.87312  
Iteration 3:   log pseudolikelihood = -880.87312  

Negative binomial regression                      Number of obs   =        316
Dispersion           = mean                       Wald chi2(3)    =      25.78
Log pseudolikelihood = -880.87312                 Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |               Robust
     daysabs |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        math |   -.001601   .0062675    -0.26   0.798    -.0138851    .0106832
    langarts |  -.0143475   .0053859    -2.66   0.008    -.0249038   -.0037913
        male |  -.4311844   .1403611    -3.07   0.002    -.7062871   -.1560817
       _cons |   2.716069   .2138534    12.70   0.000     2.296924    3.135214
-------------+----------------------------------------------------------------
    /lnalpha |   .2533877    .092161                      .0727553      .43402
-------------+----------------------------------------------------------------
       alpha |   1.288383   .1187387                      1.075467     1.54345
------------------------------------------------------------------------------

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 negative binomial 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.

Please note that with the robust standard errors option there is no likelihood ratio test for alpha equal to zero.

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

nbreg daysabs langarts male, robust

Fitting Poisson model:

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

Fitting constant-only model:

Iteration 0:   log pseudolikelihood = -897.78991  
Iteration 1:   log pseudolikelihood = -891.24455  
Iteration 2:   log pseudolikelihood = -891.24271  
Iteration 3:   log pseudolikelihood = -891.24271  

Fitting full model:

Iteration 0:   log pseudolikelihood = -881.55269  
Iteration 1:   log pseudolikelihood =  -880.9306  
Iteration 2:   log pseudolikelihood =  -880.9274  
Iteration 3:   log pseudolikelihood =  -880.9274  

Negative binomial regression                      Number of obs   =        316
Dispersion           = mean                       Wald chi2(2)    =      25.52
Log pseudolikelihood = -880.9274                  Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |               Robust
     daysabs |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    langarts |  -.0156493   .0036558    -4.28   0.000    -.0228145   -.0084841
        male |  -.4312069   .1402156    -3.08   0.002    -.7060245   -.1563893
       _cons |    2.70344    .201678    13.40   0.000     2.308158    3.098722
-------------+----------------------------------------------------------------
    /lnalpha |     .25394   .0915708                      .0744646    .4334154
-------------+----------------------------------------------------------------
       alpha |   1.289094   .1180434                      1.077307    1.542517
------------------------------------------------------------------------------

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

nbreg: Changes in Rate for daysabs

          min->max      0->1     -+1/2    -+sd/2  MargEfct
langarts   -9.3413   -0.1879   -0.0865   -1.5570   -0.0865
    male   -2.3892   -2.3892   -2.4022   -1.1957   -2.3837

exp(xb):   5.5280

        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        -0.02***
        (0.00)
male    -0.43**
        (0.14)
_cons   2.70***
        (0.20)
lnalpha 
_cons   0.25**
        (0.09)
ll      -880.93
chi2    25.52

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

model        
language arts     -0.02***
                  (0.00)
male              -0.43**
                  (0.14)
constant           2.70***
                  (0.20)            
log psuedo-
   likelihood   -880.93
chi-squared       25.52
legend: coefficient/(standard error)  ** p<0.01 *** p<0.001

The negative binomial regression model predicting days absent from school stay from language arts and gender was statistically significant (chi-squared = 25.52, 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 -0.02. This translates to a decrease of about 1.56 days absent for a one standard deviation increase in language arts when gender is held constant. Male students had an expected log count -0.43 less than female students which amounts to about 2.39 fewer days absent than females while holding language arts constant.

Cautions, Flies in the Ointment

It is not recommended that negative binomial 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 Negative Binomial Regression