Negative Binomial Regression

Negative binomial regression analysis is a statistical modeling technique used in the field of regression analysis, particularly for count data. Its primary aim is to model the relationship between a dependent variable representing counts or frequencies and one or more independent variables.

It aims to provide a robust statistical model for count data that overcomes the limitations of the Poisson regression model, which assumes that the mean and variance are equal. It allows for inference on the relationship between the dependent and independent variables while also enabling predictions of future counts based on the model.

• Negative binomial regression is valuable for count data analysis as it accommodates overdispersion. Here, the variance of the count data exceeds the mean.
• It's beneficial for counting data with excessive zeros or higher variability than assumed by a Poisson distribution.
• It provides a more flexible and robust alternative to Poisson Regression for count data analysis. It allows for more accurate estimation and prediction.
• It finds applications in numerous fields, such as epidemiology, social sciences, economics, public health, and more.

Negative binomial regression is a statistical technique that models the relationship between a dependent variable and one or more independent variables, modeling count data. It extends the Poisson regression by addressing overdispersion in data where the variance exceeds the mean count. The model roots in the negative binomial distribution, which allows for more flexibility in handling variability within count data.

Greenwood, Yule, and others first introduced the negative binomial distribution in the early 20th century. It emerged as an extension of the Binomial distribution. It represents the number of successes in a fixed number of independent trials. The negative binomial distribution accommodates scenarios where the number of trials to achieve a specified number of successes is variable. This distribution is the foundation for the negative binomial regression model, which became popular in the mid to late 20th century with advancements in statistical methodologies.

Its emphasis on count data analysis characterizes the negative binomial regression model. It is mainly when the data exhibit more variation than predicted by a simple Poisson model. It assumes that the counts follow a negative binomial distribution, allowing for a flexible framework to handle overdispersion. Overdispersion occurs when there is more variability in the data than the Poisson distribution can account for. It makes the negative binomial regression model an effective tool to address this statistical issue. 

Negative binomial regression, like any statistical model, operates under several vital assumptions:

1. Independence: The observations in the dataset should be independent of each other. Each data point should not be influenced by or related to other data points. It ensures that the model's residuals are not correlated.
2. Linearity: The relationship between the dependent variable (count data) and the independent variables should be linear. The model assumes that the effect of each independent variable on the log count rate is constant across all levels of that variable.
3. No Multicollinearity: Independent variables should not be highly correlated with each other. Multicollinearity can lead to unstable estimates and make it difficult to discern the individual effects of predictors.
4. Correct Specification: The model assumes the correct functional form and appropriate selection of variables. Mis-specification of the model can lead to biased estimates and incorrect inferences.
5. Overdispersion: This is a fundamental characteristic the model addresses, assuming that the variance is greater than the mean in the count data. However, it's essential to confirm that the model adequately handles overdispersion.

The formula for negative binomial regression models the expected (mean) count, denoted as μ, of a dependent variable (usually count data) as a function of one or more predictor variables. The negative binomial regression model assumes that the expected count μ is related to the predictors through a logarithmic link function. The general formula is as follows:

μ = e^(β₀ + β₁x₁ + β₂x₂ + ... + βₖxₖ

Where:

• μ is the expected count or mean of the dependent variable.
• e is the base of the natural logarithm (approximately equal to 2.71828).
• β₀, β₁, β₂, ..., βₖ are the coefficients associated with the predictor variables.
• x₁, x₂, ..., xₖ are the values of the predictor variables.
• In this formula:
• β₀ represents the intercept, which is the expected count when all predictor variables are set to zero.
• β₁, β₂, ..., βₖ are the coefficients that quantify the effect of each predictor variable on the expected count. These coefficients indicate how a one-unit change in each predictor affects the logarithm of the expected count.

The negative binomial regression model estimates the values of the coefficients (β₀, β₁, β₂, ..., βₖ) based on the given data, and these coefficients help explain how the predictor variables influence the count data. This model accounts for overdispersion, making it suitable for data with variance more significant than the mean, which is a common occurrence in count data.

Let us understand it more with the help of examples:

Example #1

Let's consider a scenario where a retail analyst wants to predict the number of daily online customer orders based on different advertising strategies. The analyst collects data on three advertising channels (social media, email campaigns, and search engine ads) and the number of orders received for each day over several weeks.

Using negative binomial regression, the analyst constructs a model to predict the daily order count based on the amount spent on each advertising channel. The model may reveal that, for instance:

• For every $100 increase in spending on social media ads, the expected log count of orders increases by 0.6.
• For every $100 increase in spending on email campaigns, the expected log count of orders increases by 0.8.
• For every $100 increase in spending on search engine ads, the expected log count of orders increases by 0.5.
• This model helps predict the daily order count based on the advertising spending across different channels, allowing the analyst to optimize advertising budgets to maximize order counts. 

Example #2

A recent article published in BMC Medical Research Methodology in 2023 sheds light on the utility of negative binomial regression in health research. The study, conducted by a team of researchers, explores the advantages and practical applications of this statistical technique in the analysis of count data.

The article emphasizes its usefulness in epidemiological and healthcare studies where the count data exhibit variability exceeding the Poisson distribution's assumptions.

The study highlights key takeaways, including the flexibility and robustness of negative binomial regression, its suitability for data with excessive zeros, and its interpretive aspects.

This research offers valuable insights for healthcare analysts, epidemiologists, and researchers, providing a comprehensive understanding of when and how to apply negative binomial regression for improved analysis of count data in health-related studies. 

Following are the advantages and disadvantages of using negative binomial regression:

Below is a comparison between Negative Binomial Regression, Poisson Regression, and Logistic Regression:

This article has been a guide to what is Negative Binomial Regression Analysis. We explain its assumptions, formula, comparison with poison regression, and examples. You may also find some useful articles here -

• P Value Formula
• Maximum Likelihood Estimation
• Basic Statistics Concepts