This paper assesses the application of regression methods to analyse count data. R-Code and Data are available from the author!
While the common multiple regression method has a wide range of applicability, and can be accommodated to various different kinds of regressor variables, its application is limited to the modelling of response variables from the space of real numbers. For the analysis of other kinds of responses, such as counts, a more generalised set of tools is needed. This toolset is given by the generalised linear model framework and maximum likelihood estimation.
For the specific purpose of this paper, the count data analysis methods of Poisson, Negative-Binomial, Hurdle and Zero-Inflation models are considered. This paper explains their theoretical background and applies them to a unique dataset that motivates their respective use. It is structured as follows: section 2 describes the applied dataset and section 3 the generalised linear model framework. In section 4 and section 5 the basic count data models and their results are discussed, while section 6 and section 7 explain the more advanced methods and their results. section 8 concludes.
Table of Contents
1 Introduction
2 Data
3 Generalised Linear Model Framework
4 Count Data Models I
4.1 The Poisson Regression
4.2 The Negative Binomial Regression
4.3 Quasi-Poisson
5 Estimation Results I
5.1 Standard Errors
5.2 The Equidispersion Assumption
5.3 Count Frequency Prediction
6 Count Data Models II
6.1 The Hurdle Regression
6.2 The Zero Inflated Regression
7 Estimation Results II
8 Conclusion
Research Objectives and Core Topics
This seminar paper evaluates the application of regression methods designed for count data, specifically addressing the limitations of standard multiple regression models when dealing with non-real response variables. The central research question focuses on how different modeling frameworks—such as Poisson, Negative Binomial, Hurdle, and Zero-Inflation models—can effectively analyze recreational boating trip data, characterized by overdispersion and an excess of zero counts.
- Theoretical foundations of Generalised Linear Models (GLM).
- Comparative analysis of basic count data models (Poisson, Negative Binomial, Quasi-Poisson).
- Advanced modeling techniques for excess zeros (Hurdle and Zero-Inflated models).
- Empirical application using a unique 1980 survey dataset of lake recreation.
- Evaluation of model performance through frequency prediction and statistical tests.
Excerpt from the Book
4.2 The Negative Binomial Regression
The crucial assumption of the Poisson model is the equality of variance and expected value, called the equidispersion assumption. This assumption might not hold, if there is unobserved heterogeneity, i.e. variables that are not or can not be included in the model. This can be parameterised by including an error term in the linear predictor
λ(xi; i, β) = exp (xT i β + i) = λ(xi; β)ui with ui = exp(i),
which changes the conditional expectation and variance respectively into:
E[yi|xi] = λ(xi; β)
V[yi|xi] = λ(xi; β) + σ2 uλ(xi; β) 2
with σ2 u = V[ui|xi].
One way to model this is the Negative Binomial (Negbin) class of models. It assumes that the response variable has the probability distribution
f(yi|xi) = ∫∞ 0 f(yi|xi, ui)g(ui|xi)dui
f(yi|xi, ui) ∼ Poisson(λ(xi; i, β))
g(ui|xi) ∼ Gamma(θ,θ). (5)
Fully written, the conditional distribution function of the negative binomial regression is given by
f(yi; xi, β, θ) = Γ(yi + θ) / (Γ(yi + 1)Γ(θ)) * (λ(xi; β) / (λ(xi; β) + θ))yi * (θ / (λ(xi; β) + θ))θ .
Summary of Chapters
1 Introduction: Introduces the scope of the paper, focusing on the transition from standard linear regression to GLM frameworks for count data analysis.
2 Data: Describes the survey dataset on recreational boating trips at lake Somerville, Texas, and outlines the key variables of interest.
3 Generalised Linear Model Framework: Establishes the theoretical basis for linear models and the exponential family distribution within the GLM context.
4 Count Data Models I: Details the theoretical implementation of Poisson, Negative Binomial, and Quasi-Poisson regression models.
5 Estimation Results I: Discusses model estimation, standard error calculations, the equidispersion assumption, and count frequency prediction.
6 Count Data Models II: Explores advanced regression models specifically designed to handle excess zeros, namely Hurdle and Zero-Inflated models.
7 Estimation Results II: Analyzes the output of Hurdle and Zero-Inflated models and compares their effectiveness based on the provided dataset.
8 Conclusion: Summarizes the findings, noting that while Hurdle models provide the best fit for the training data, model selection depends on the specific goals of the analysis.
Keywords
Count Data Analysis, Generalised Linear Models, Maximum Likelihood, Poisson Regression, Negative Binomial Regression, Hurdle Models, Zero-Inflated Models, Equidispersion, Overdispersion, Recreational Boating, Estimation, Bootstrap Simulation, Standard Errors, Heteroscedasticity, Statistical Modeling.
Frequently Asked Questions
What is the primary objective of this paper?
The paper assesses the application of various regression methods to analyze count data, specifically for instances where the response variable represents non-negative integer counts rather than real numbers.
Which statistical frameworks are compared in the study?
The study compares the Poisson model, Negative Binomial regression, Quasi-Poisson models, as well as advanced Hurdle and Zero-Inflated regression models.
What is the main limitation of the standard Poisson regression?
The primary limitation is the equidispersion assumption, which requires the variance and the mean of the distribution to be equal; this is often violated by real-world data containing unobserved heterogeneity.
Why are Hurdle and Zero-Inflated models introduced?
These models are introduced to address the problem of "excess zeros," where the number of observed zero-value instances in a dataset is significantly higher than what a standard Poisson distribution predicts.
What kind of dataset is used for the empirical analysis?
The paper uses a 1980 survey dataset concerning recreational boating trips at lake Somerville, Texas, including variables like trip counts, facility quality, income, and various travel costs.
Which criteria are used to evaluate model performance?
Performance is evaluated through likelihood-ratio tests, residual analysis, count frequency predictions, and the Bayesian Information Criterion (BIC).
How do the authors define the link function in this context?
The link function is used to accommodate the linear predictor, which resides in the space of real numbers, to the conditional expectation of the dependent variable in a limited number space.
What makes the Negbin II model superior to the standard Poisson model for this dataset?
The Negbin II model performs better because it accounts for the overdispersion present in the boating trip data, leading to a better statistical fit than the Poisson approach.
What does the "sandwich estimator" signify in the context of standard errors?
The sandwich estimator is a robust method used to calculate standard errors that are resistant to heteroscedasticity in the model residuals.
What is the key takeaway regarding the use of Hurdle models?
The paper notes that while Hurdle models provide excellent in-sample fit for specific datasets, they carry a risk of overfitting and may exhibit poor out-of-sample performance.
- Quote paper
- Martin Georg Haas (Author), 2019, Regression methods for the analysis of count data. Generalised linear models for limited dependent variables, Munich, GRIN Verlag, https://www.grin.com/document/1003006
