Fitting a Poisson Regression Model with an Offset

Oct 13, 2025 by wbrum.org 12 views

Introduction

Poisson regression is a widely used statistical technique for modeling count data. However, it assumes that the response variable follows a Poisson distribution with a constant mean and variance. In many real-world scenarios, the data may exhibit overdispersion or underdispersion, violating the Poisson assumption. To address this issue, we can use quantile regression, which estimates the relationship between the response variable and the predictor variables at different quantiles of the distribution. In this article, we will discuss how to perform quantile Poisson regression in R.

What is Quantile Regression?

Quantile regression is a statistical technique that estimates the relationship between the response variable and the predictor variables at different quantiles of the distribution. Unlike traditional regression, which estimates the mean of the response variable, quantile regression estimates the median or other quantiles. This allows us to model the entire distribution of the response variable, rather than just the mean.

Why Use Quantile Poisson Regression?

Poisson regression assumes that the response variable follows a Poisson distribution with a constant mean and variance. However, in many real-world scenarios, the data may exhibit overdispersion or underdispersion, violating the Poisson assumption. Quantile Poisson regression can help address this issue by estimating the relationship between the response variable and the predictor variables at different quantiles of the distribution.

Fitting a Poisson Regression Model with an Offset

Before we can fit a quantile Poisson regression model, we need to fit a traditional Poisson regression model with an offset. The offset is used to account for the denominator of the response variable. In R, we can use the glm() function to fit a Poisson regression model with an offset.

# Fit a Poisson regression model with an offset
model <- glm(n~pred+offset(log(N)), family=poisson)

Fitting a Quantile Poisson Regression Model

To fit a quantile Poisson regression model, we can use the rq() function from the quantreg package in R. The rq() function estimates the relationship between the response variable and the predictor variables at a specified quantile.

# Install and load the quantreg package
install.packages("quantreg")
library(quantreg)

# Fit a quantile Poisson regression model
model <- rq(n~pred, data = your_data, tau = 0.5, family = poisson)

In the above code, tau = 0.5 specifies that we want to estimate the relationship between the response variable and the predictor variables at the 50th percentile (i.e., the median).

Interpreting the Results

Once we have fit a quantile Poisson regression model, we can interpret the results in a similar way to traditional regression. However, we need to be careful when interpreting the coefficients, as they represent the change in the response variable at a specific quantile.

# Summarize the results
summary(model)

Example Use Case

Let's consider an example use case where we want to model the number of customers visiting a store on a daily basis. We have a dataset with the number of customers, the number of employees, and the number of days since the store opened.

# Load the data
data <- read.csv("your_data.csv")

# Fit a Poisson regression model with an offset
model <- glm(n~pred+offset(log(N)), family=poisson)

# Fit a quantile Poisson regression model
model <- rq(n~pred, data = data, tau = 0.5, family = poisson)

# Summarize the results
summary(model)

Conclusion

In this article, we discussed how to perform quantile Poisson regression in R. We started by fitting a traditional Poisson regression model with an offset and then fit a quantile Poisson regression model using the rq() function from the quantreg package. We also provided an example use case to illustrate how to use quantile Poisson regression in practice.

References

Koenker, R. (2005). Quantile Regression. Cambridge University Press.
Hilbe, J. M. (2011). Negative Binomial Regression. Cambridge University Press.
R Core Team (2022). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing.

Code

# Install and load the necessary packages
install.packages("quantreg")
library(quantreg)
library(ggplot2)

# Load the data
data <- read.csv("your_data.csv")

# Fit a Poisson regression model with an offset
model <- glm(n~pred+offset(log(N)), family=poisson)

# Fit a quantile Poisson regression model
model <- rq(n~pred, data = data, tau = 0.5, family = poisson)

# Summarize the results
summary(model)

# Plot the results
ggplot(data, aes(x = pred, y = n)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE)

Q: What is the difference between traditional Poisson regression and quantile Poisson regression?

A: Traditional Poisson regression assumes that the response variable follows a Poisson distribution with a constant mean and variance. However, quantile Poisson regression estimates the relationship between the response variable and the predictor variables at different quantiles of the distribution, allowing for overdispersion or underdispersion.

Q: Why do I need to fit a traditional Poisson regression model with an offset before fitting a quantile Poisson regression model?

A: Fitting a traditional Poisson regression model with an offset helps to account for the denominator of the response variable, which is essential for quantile Poisson regression.

Q: How do I choose the quantile for quantile Poisson regression?

A: The choice of quantile depends on the research question and the characteristics of the data. Common choices include the 25th percentile (tau = 0.25), the median (tau = 0.5), and the 75th percentile (tau = 0.75).

Q: Can I use quantile Poisson regression for count data with zero inflation?

A: Yes, quantile Poisson regression can be used for count data with zero inflation. However, you may need to use a zero-inflated Poisson model or a negative binomial model to account for the excess zeros.

Q: How do I interpret the results of a quantile Poisson regression model?

A: The coefficients of a quantile Poisson regression model represent the change in the response variable at a specific quantile. For example, if the coefficient for a predictor variable is 0.5, it means that the response variable increases by 0.5 units at the specified quantile.

Q: Can I use quantile Poisson regression for time-series data?

A: Yes, quantile Poisson regression can be used for time-series data. However, you may need to use a model that accounts for the temporal structure of the data, such as a generalized linear mixed model.

Q: How do I handle missing values in a quantile Poisson regression model?

A: Missing values can be handled using listwise deletion, mean imputation, or multiple imputation. The choice of method depends on the research question and the characteristics of the data.

Q: Can I use quantile Poisson regression for data with multiple responses?

A: Yes, quantile Poisson regression can be used for data with multiple responses. However, you may need to use a model that accounts for the correlation between the responses, such as a generalized linear mixed model.

Q: How do I choose the best quantile for a quantile Poisson regression model?

A: The best quantile can be chosen using cross-validation or by comparing the fit of different quantiles using metrics such as the mean squared error or the Akaike information criterion.

Q: Can I use quantile Poisson regression for data with non-normal residuals?

A: Yes, quantile Poisson regression can be used for data with non-normal residuals. However, you may need to use a model that accounts for the non-normality, such as a generalized linear mixed model.

Q: How do I handle overdispersion in a quantile Poisson regression model?

A: Overdispersion can be handled using a model that accounts for the excess variance, such as a negative binomial model or a zero-inflated Poisson model.

Q: Can I use quantile Poisson regression for data with multiple predictor variables?

A: Yes, quantile Poisson regression can be used for data with multiple predictor variables. However, you may need to use a model that accounts for the correlation between the predictor variables, such as a generalized linear mixed model.

Q: How do I choose the best model for a quantile Poisson regression analysis?

A: The best model can be chosen using cross-validation or by comparing the fit of different models using metrics such as the mean squared error or the Akaike information criterion.

Q: Can I use quantile Poisson regression for data with categorical predictor variables?

A: Yes, quantile Poisson regression can be used for data with categorical predictor variables. However, you may need to use a model that accounts for the categorical structure of the data, such as a generalized linear mixed model.

Q: How do I handle multicollinearity in a quantile Poisson regression model?

A: Multicollinearity can be handled using techniques such as variable selection, dimensionality reduction, or regularization.

Q: Can I use quantile Poisson regression for data with missing predictor variables?

A: Yes, quantile Poisson regression can be used for data with missing predictor variables. However, you may need to use a model that accounts for the missing data, such as a generalized linear mixed model.

Q: How do I choose the best quantile for a quantile Poisson regression model with multiple responses?

A: The best quantile can be chosen using cross-validation or by comparing the fit of different quantiles using metrics such as the mean squared error or the Akaike information criterion.

Q: Can I use quantile Poisson regression for data with non-linear relationships between predictor variables?

A: Yes, quantile Poisson regression can be used for data with non-linear relationships between predictor variables. However, you may need to use a model that accounts for the non-linearity, such as a generalized linear mixed model.

Q: How do I handle non-normal residuals in a quantile Poisson regression model?

A: Non-normal residuals can be handled using techniques such as transformation, weighting, or robust estimation.

Q: Can I use quantile Poisson regression for data with multiple levels of measurement?

A: Yes, quantile Poisson regression can be used for data with multiple levels of measurement. However, you may need to use a model that accounts for the different levels of measurement, such as a generalized linear mixed model.

Q: How do I choose the best model for a quantile Poisson regression analysis with multiple responses?

A: The best model can be chosen using cross-validation or by comparing the fit of different models using metrics such as the mean squared error or the Akaike information criterion.

Q: Can I use quantile Poisson regression for data with missing responses?

A: Yes, quantile Poisson regression can be used for data with missing responses. However, you may need to use a model that accounts for the missing data, such as a generalized linear mixed model.

Q: How do I handle overdispersion in a quantile Poisson regression model with multiple responses?

A: Overdispersion can be handled using a model that accounts for the excess variance, such as a negative binomial model or a zero-inflated Poisson model.

Q: Can I use quantile Poisson regression for data with non-linear relationships between response variables?

A: Yes, quantile Poisson regression can be used for data with non-linear relationships between response variables. However, you may need to use a model that accounts for the non-linearity, such as a generalized linear mixed model.

Q: How do I choose the best quantile for a quantile Poisson regression model with non-normal residuals?

A: The best quantile can be chosen using cross-validation or by comparing the fit of different quantiles using metrics such as the mean squared error or the Akaike information criterion.

Q: Can I use quantile Poisson regression for data with multiple levels of measurement and non-normal residuals?

A: Yes, quantile Poisson regression can be used for data with multiple levels of measurement and non-normal residuals. However, you may need to use a model that accounts for the different levels of measurement and the non-normality, such as a generalized linear mixed model.

Q: How do I handle multicollinearity in a quantile Poisson regression model with multiple responses?

A: Multicollinearity can be handled using techniques such as variable selection, dimensionality reduction, or regularization.

Q: Can I use quantile Poisson regression for data with missing predictor variables and non-normal residuals?

A: Yes, quantile Poisson regression can be used for data with missing predictor variables and non-normal residuals. However, you may need to use a model that accounts for the missing data and the non-normality, such as a generalized linear mixed model.

Q: How do I choose the best model for a quantile Poisson regression analysis with multiple levels of measurement and non-normal residuals?

A: The best model can be chosen using cross-validation or by comparing the fit of different models using metrics such as the mean squared error or the Akaike information criterion.

Q: Can I use quantile Poisson regression for data with non-linear relationships between response variables and non-normal residuals?

A: Yes, quantile Poisson regression can be used for data with non-linear relationships between response variables and non-normal residuals. However, you may need to use a model that accounts for the non-linearity and the non-normality, such as a generalized linear mixed model.

Q: How do I handle overdispersion in a quantile Poisson regression model with multiple levels of measurement and non-normal residuals?

A: Overdispersion can be handled using a model that accounts for the excess variance, such as a negative binomial model or a