Understanding Negative Binomial Regression: An Overview

By Shashikant Nishant Sharma

Negative binomial regression is a type of statistical analysis used for modeling count data, especially in cases where the data exhibits overdispersion relative to a Poisson distribution. Overdispersion occurs when the variance exceeds the mean, which can often be the case in real-world data collections. This article explores the fundamentals of negative binomial regression, its applications, and how it compares to other regression models like Poisson regression.

What is Negative Binomial Regression?

Negative binomial regression is an extension of Poisson regression that adds an extra parameter to model the overdispersion. While Poisson regression assumes that the mean and variance of the distribution are equal, negative binomial regression allows the variance to be greater than the mean, which often provides a better fit for real-world data where the assumption of equal mean and variance does not hold.

Mathematical Foundations

The negative binomial distribution can be understood as a mixture of Poisson distributions, where the mixing distribution is a gamma distribution. The model is typically expressed as:

A random variable X is supposed to follow a negative binomial distribution if its probability mass function is given by:

f(x) = (n + r – 1)C(r – 1) P^rq^x, where x = 0, 1, 2, ….., and p + q = 1.

Here we consider a binomial sequence of trials with the probability of success as p and the probability of failure as q.

Let f(x) be the probability defining the negative binomial distribution, where (n + r) trials are required to produce r successes. Here in (n + r – 1) trials we get (r – 1) successes, and the next (n + r) is a success.

Then f(x) = (n + r – 1)C(r – 1) P^r-1q^n-1.p

f(x) = (n + r – 1)C(r – 1) P^rqⁿ

When to Use Negative Binomial Regression?

Negative binomial regression is particularly useful in scenarios where the count data are skewed, and the variance of the data points is significantly different from the mean. Common fields of application include:

• Healthcare: Modeling the number of hospital visits or disease counts, which can vary significantly among different populations.
• Insurance: Estimating the number of claims or accidents, where the variance is typically higher than the mean.
• Public Policy: Analyzing crime rates or accident counts in different regions, which often show greater variability.

Comparing Poisson and Negative Binomial Regression

While both Poisson and negative binomial regression are used for count data, the choice between the two often depends on the nature of the data’s variance:

• Poisson Regression: Best suited for data where the mean and variance are approximately equal.
• Negative Binomial Regression: More appropriate when the data exhibits overdispersion.

If a Poisson model is fitted to data that is overdispersed, it may underestimate the variance leading to overly optimistic confidence intervals and p-values. Conversely, a negative binomial model can provide more reliable estimates and inference in such cases.

Implementation and Challenges

Implementing negative binomial regression typically involves statistical software such as R, SAS, or Python, all of which have packages or modules designed to fit these models to data efficiently. One challenge in fitting negative binomial models is the estimation of the dispersion parameter, which can sometimes be sensitive to outliers and extreme values.

Conclusion

Negative binomial regression is a robust method for analyzing count data, especially when that data is overdispersed. By providing a framework that accounts for variability beyond what is expected under a Poisson model, it allows researchers and analysts to make more accurate inferences about their data. As with any statistical method, the key to effective application lies in understanding the underlying assumptions and ensuring that the model appropriately reflects the characteristics of the data.

References

Chang, L. Y. (2005). Analysis of freeway accident frequencies: negative binomial regression versus artificial neural network. Safety science, 43(8), 541-557.

Hilbe, J. M. (2011). Negative binomial regression. Cambridge University Press.

Ver Hoef, J. M., & Boveng, P. L. (2007). Quasi‐Poisson vs. negative binomial regression: how should we model overdispersed count data?. Ecology, 88(11), 2766-2772.

Liu, H., Davidson, R. A., Rosowsky, D. V., & Stedinger, J. R. (2005). Negative binomial regression of electric power outages in hurricanes. Journal of infrastructure systems, 11(4), 258-267.

Yang, S., & Berdine, G. (2015). The negative binomial regression. The Southwest respiratory and critical care chronicles, 3(10), 50-54.