While Poisson regression assumes a (known) average, Negative Binomial regression is implemented using what is referred to as maximum likelihood estimation.
Remember that, although Poisson distribution assumes that the mean and variance are the same, sometimes data will show greater variability or extra variation that is greater than the mean. When this occurs, Negative Binomial regression is a better choice because of its greater flexibility in that regard.
To illustrate, what if we consider that a university wants to predict the average number of days a student athlete may miss each year. Predictors (of the number of days of absence from class) include the type of sport the student athlete is a member of and their average GPA score. The variable sport is a four-level nominal variable indicating which sport the athlete participates in (in this case it's either Football
, Track
, Field
Hockey
, or Volleyball
).
If we profile our data, suppose we find the following statistics...