11. Negative Binomial Regression Models

Much like the Poisson distribution, the negative binomial distribution 
works with counts, that is the non-negative integers, 0, 1, 2, 3, 4, .....

One of the ways that counts are generated according to a negative binomial 
distribution is through a mixture of random variables each of which is 
distributed as Poisson, but with different expectations lambda{i}. If the 
expectations follow a gamma distribution, then you have a gamma mixture of 
Poisson random variables which follow the negative binomial distribution.

The expectation of the negative binomial distribution is the same as the 
expectation of the Poisson distribution. However, the variance of the 
negative binomial distribution is (mu + k*mu^2) where k is called the 
dispersion parameter for the negative binomial. This parameter is 
estimated when you fit a model where you assume that the response is 
distributed as Negative Binomial. If k=0, then both the mean and variance 
are mu, recognizable as the mean and variance of a Poisson distribution. 
The Poisson is the negative binomial distribution where k=0.

Overdispersion, as discussed in Section 9, exists when each observation is 
drawn from a Poisson distribution and has its own expectation. Such a 
model adds structure when the condition is imposed that the expectations 
are drawn from a gamma distribution. This feature generates data from the 
negative binomial distribution.

PROC GENMOD will fit data to a negative binomial distribution with the 
folloing MODEL statement.

PROC GENMOD DATA=mydata;
CLASS <cls_vars> ;
MODEL counts = <predictors> / DIST=negbin LINK=log TYPE3;
run;

You may decide to fit independent count data with both the Poisson and 
negative binomial models. How should you evaluate which one is better? It 
depends on the validity of your assumptions.  What would make for a 
'better' model? Is it reasonable to model your data as either one of these 
distributions? What is the underlying theory behind your choice? How are 
you selecting regressors?

To fit independent count data to a negative binomial distribution, here is 
a simple example to estimate the mean and dispersion parameter.

DATA this;
INPUT y @@;
cards;
1 2 2 2 3 3 3 4 4 4 5 6 6 6 7 8 9 11
;

PROC GENMOD data=this;
MODEL y = / DIST=negbin LINK=log;
ESTIMATE 'Mean' intercept 1 / exp;
run;


Criteria For Assessing Goodness Of Fit

Criterion              Value

Log Likelihood       49.1923


NOTE: -2 * Log Likelihood = -98.4 
          (see OUTPUT below and the likehood function in NLMIXED)

Analysis Of Parameter Estimates

                                    Likelihood Ratio
                          Standard   95% Confidence     Chi-
Parameter   DF  Estimate     Error       Limits       Square Pr > ChiSq

Intercept    1    1.5640    0.1295   1.2939    1.8312  145.83    <.0001
Dispersion   1    0.0926    0.1012  -0.0398    0.4147

NOTE: The negative binomial dispersion parameter was estimated by maximum
      likelihood.

Contrast Estimate Results

                     Standard                             Chi-
Label      Estimate     Error  Alpha  Confidence Limits Square Pr > ChiSq

Mean         1.5640    0.1295   0.05   1.3101    1.8178 145.83    <.0001
Exp(Mean)    4.7778    0.6188   0.05   3.7067    6.1584


In addition to estimation of the mean with PROC GENMOD you can compute 
mean and dispersion parameters for the negative binomial from NLMIXED:

PROC NLMIXED DATA=this;
PARMS b0 = -1;
eta =  b0 ;
mean = EXP(eta);

/* loglikelihood of response given mean */

loglike =  y*LOG(k*mean) - (y+(1/k))*LOG(1+k*mean)
           + LGAMMA(y+(1/k))  - LGAMMA(1/k);
          * - LGAMMA(y+1);

* since the last term does not depend on parameters,
  the estimated coefficients are the same with or with out it,
  however, the likelihood statistic changes;

MODEL y ~ general(loglike);
ESTIMATE 'mean' exp(b0);
TITLE1 'Negative Binomial estimation with NLMIXED';
run;

Fit Statistics

-2 Log Likelihood  = -98.4

NOTE: -2*loglikelihood matches GENMOD result when
      last term of the likelihood is omitted;

Parameter Estimates

                   Standard
Parameter Estimate    Error DF  t Value  Pr > |t|  Lower Upper  Gradient

b0          1.5640   0.1295 18    12.08  <.000001  1.2919 1.8361  -1.89E-6
k          0.09261   0.1012 18     0.91  0.372339 -0.1201 0.3053  -4.17E-6

Additional Estimates

                 Standard
Label  Estimate     Error    DF  t Value  Pr > |t|   Alpha     Lower  Upper

mean     4.7778    0.6188    18     7.72  <.000001    0.05    3.4778  6.0778

 
Note that k > 0 which implies the variance of the sample data cannot be 
less than the mean for a given set of counts. If k < 0, then estimation of 
Neg Bin may not be appropriate.