Overview
Negative binomial regression is another approach to building regression models for count data. It has many advantages over Poisson regression, especially when there is over-dispersion.
The pdf for the negative binomial distribution is
Let α = 1/ν and p = ν/(ν+μ). Since the mean of a negative binomial distribution is μ = ν (1–p)/p, we can express the variance as ν(1–p)/p2 = μ/p = μ(ν+μ)/ν = μ(1+μ/ν) = μ(1+ αμ); i.e.
var = μ(1 + αμ) = μ + αμ2
From the above, we see that the lower the value of α, the lower the variance. In fact, when α is near zero (i.e. ν is large), then we get the assumption for the Poisson distribution, namely that the mean and the variance are equal.
The resulting regression model is called the negative binomial 2 model (aka the mean dispersion negative binomial model).
There is another negative binomial regression model that is based on the following assumption for the variance
var = μ(1 + α) = μ + αμ
This is called the negative binomial 1 model (aka the constant dispersion negative binomial model), and often this provides a better fit for the data. This is created by setting ν = μ/α instead of ν = 1/α.
Over-dispersion
When the variance is larger than the mean, we have over-dispersion. The Poisson model assumes that we don’t have any over-dispersion. Some possible reasons for over-dispersion are:
- Missing predictors (i.e. independent variables)
- Incorrect model
- Outliers
On this website, we explore the following regression models that can reduce over-dispersion compared to a Poisson regression model:
- Negative binomial model: as described on this webpage
- Zero-truncated model: start counting at 1 and not 0. E.g. # of days in the hospital can’t be 0.
- Zero-inflated and hurdle models: handle excess zeros using logistic regression
In addition to censoring at zero, we can also left-censor more generally (e.g. no count less than 3) and right-censor (e.g. any count over 8 is counted as 8). We won’t explore these further, however.
Negative Binomial Regression Topics
We discuss the following topics about negative binomial regression:
- Creating a Negative Binomial Regression model using Solver
- Real Statistics Data Analysis Tool (Solver option)
- Real Statistics Data Analysis Tool (Newton’s method option)
- Predictions
- Comparisons with a Poisson regression model
- Additional Insights
Links
References
Hilbe, J. M. (2014) Modeling count data. Cambridge University Press
https://www.cambridge.org/core/books/modeling-count-data/BFEB3985905CA70523D9F98DA8E64D08
Hintze, J. L. (2007) Negative binomial regression. NCSS
https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Negative_Binomial_Regression.pdf