15.2 Quasi-Poisson and Negative Binomial Models
3 min read•july 30, 2024
When dealing with count data, can throw a wrench in your analysis. Quasi-Poisson and negative binomial models come to the rescue, allowing for more flexibility than standard Poisson regression. These models tackle the issue of exceeding the in your data.
Both approaches have their strengths. Quasi-Poisson uses a dispersion parameter to adjust for overdispersion, while negative binomial assumes a specific distribution. Your choice depends on your data and research goals. Either way, these models help you make sense of messy count data.
Quasi-Poisson and Negative Binomial Models
Properties and Assumptions
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- Extend the Poisson model to allow for overdispersion, which occurs when the variance of the response variable exceeds its mean
- :
- Assumes the variance is proportional to the mean, with a dispersion parameter (φ) quantifying the degree of overdispersion
- Variance equals φ times the mean
- :
- Assumes the response variable follows a negative binomial distribution, a mixture of Poisson distributions with gamma-distributed rates
- Has an additional parameter (θ) controlling the shape of the distribution and the degree of overdispersion
- Both models assume:
- Observations are independent
- Explanatory variables have a linear relationship with the log of the mean response
Estimation and Interpretation
- Used to model count data with overdispersion, where the response variable is a non-negative integer
- Estimated using maximum likelihood or quasi-likelihood methods, providing estimates of regression coefficients and dispersion or shape parameters
- Regression coefficients represent the change in the log of the mean response for a one-unit change in the corresponding explanatory variable, holding other variables constant
- Exponentiating the coefficients yields incidence rate ratios (IRRs), representing the multiplicative change in the mean response for a one-unit change in the explanatory variable
- Dispersion parameter (φ) in the quasi-Poisson model and shape parameter (θ) in the negative binomial model provide information about the degree of overdispersion
- measures (deviance and Pearson chi-square statistics) assess model adequacy
- Residual diagnostics (deviance and Pearson residuals) identify outliers and assess model assumptions
Modeling Overdispersion
Quasi-Poisson Model
- Quasi-likelihood approach assuming the variance is proportional to the mean
- Dispersion parameter (φ) quantifies the degree of overdispersion
- More flexible and handles a wider range of overdispersion patterns
- Does not provide a full probability distribution for the response variable
Negative Binomial Model
- Fully specified probability model assuming a negative binomial distribution for the response variable
- Shape parameter (θ) controls the shape of the distribution and the degree of overdispersion
- More restrictive in its assumptions but provides a full probability distribution
- Allows for more efficient inference when the assumptions are met
Quasi-Poisson vs Negative Binomial
Comparison of Models
- Both handle overdispersion in count data but differ in assumptions and properties
- Quasi-Poisson model:
- Quasi-likelihood approach
- Assumes variance is proportional to the mean
- Dispersion parameter (φ) quantifies overdispersion
- Negative binomial model:
- Fully specified probability model
- Assumes negative binomial distribution for the response variable
- Shape parameter (θ) controls distribution shape and overdispersion
Choosing Between Models
- Choice depends on data characteristics and research objectives
- Quasi-Poisson model:
- Sufficient for moderate overdispersion and focus on estimating regression coefficients
- Negative binomial model:
- Appropriate for substantial overdispersion and interest in both regression coefficients and probability distribution of the response variable
- Presence of zero-inflation may require models like zero-inflated Poisson (ZIP) or zero-inflated negative binomial (ZINB)
Model Selection for Overdispersion
Factors to Consider
- Degree and pattern of overdispersion
- Presence of zero-inflation
- Research objectives
Selection Criteria
- Goodness-of-fit measures (Akaike Information Criterion or Bayesian Information Criterion) balance model fit and complexity
- Likelihood ratio tests compare nested models (Poisson and negative binomial) to assess the significance of the overdispersion parameter
Guiding Principles
- Research questions
- Data characteristics
- Interpretability of results in the study context
Key Terms to Review (19)
AIC: Akaike Information Criterion (AIC) is a statistical measure used to compare the goodness of fit of different models while penalizing for the number of parameters included. It helps in model selection by providing a balance between model complexity and fit, where lower AIC values indicate a better model fit, accounting for potential overfitting.
BIC: The Bayesian Information Criterion (BIC) is a criterion for model selection among a finite set of models, based on the likelihood of the data and the number of parameters in the model. It helps to balance model fit with complexity, where lower BIC values indicate a better model, making it useful in comparing different statistical models, particularly in regression and generalized linear models.
Ecological studies: Ecological studies are research designs that examine the relationships between exposure and outcome at the group or population level, rather than at the individual level. These studies often use existing data to assess correlations between factors, such as environmental variables and health outcomes, and can help identify potential associations that warrant further investigation through more detailed studies.
Goodness-of-fit: Goodness-of-fit is a statistical measure that evaluates how well a model's predicted values align with observed data. It assesses the discrepancy between the actual data points and the values predicted by the model, helping to determine how well the model explains the data. This concept is essential in selecting appropriate models, particularly when using criteria to compare their performance, understanding overdispersion in certain data types, and fitting non-linear relationships.
Healthcare outcomes: Healthcare outcomes refer to the end results of healthcare services, reflecting the effectiveness, efficiency, and quality of care provided to patients. These outcomes can be measured through various indicators such as patient recovery rates, mortality rates, quality of life, and patient satisfaction, which help evaluate the impact of medical interventions on individual health status and overall public health.
Likelihood Ratio Test: The likelihood ratio test is a statistical method used to compare the goodness-of-fit of two models, one of which is a special case of the other. It assesses whether the additional parameters in a more complex model significantly improve the fit compared to a simpler, nested model. This test is particularly useful for evaluating homogeneity of regression slopes and determining model adequacy across various frameworks.
Log Transformation: Log transformation is a mathematical operation where the logarithm of a variable is taken to stabilize variance and make data more normally distributed. This technique is especially useful in addressing issues of skewness and heteroscedasticity in regression analysis, which ultimately improves the reliability of statistical modeling.
Mean: The mean is a statistical measure that represents the average of a set of values, calculated by summing all the values and dividing by the number of observations. In the context of quasi-Poisson and negative binomial models, the mean plays a critical role in understanding how the data is distributed, especially when dealing with count data that may exhibit overdispersion. This concept is essential for interpreting the parameters of these models and for making predictions based on the observed data.
Negative Binomial Model: The negative binomial model is a statistical distribution used to model count data that exhibit overdispersion, where the variance exceeds the mean. It extends the Poisson distribution by adding a parameter to account for this extra variability, making it particularly useful in scenarios where the occurrence of events is influenced by random effects or latent variables.
Offset: In statistical modeling, an offset is a variable that is added to a model to account for exposure or size differences among observations, allowing for more accurate predictions. This is particularly useful when dealing with count data, where the total counts might vary due to different exposure times or population sizes. By including an offset, the model can adjust for these variations, ensuring that the results are meaningful and interpretable.
Overdispersion: Overdispersion occurs when the observed variance in data is greater than what the statistical model predicts, particularly in count data where Poisson regression is often used. This can signal that the model is not adequately capturing the underlying variability, leading to potential issues in inference and prediction. Recognizing overdispersion is crucial for choosing appropriate models and ensuring accurate results in statistical analyses.
Poisson Distribution: The Poisson distribution is a probability distribution that expresses the likelihood of a given number of events occurring within a fixed interval of time or space, under the condition that these events occur independently of one another. It is closely related to the exponential family of distributions and serves as a foundation for understanding count data, particularly in contexts where the mean and variance are equal. This distribution is especially relevant when exploring link functions, overdispersion, and alternative modeling approaches like quasi-Poisson and negative binomial models.
Python: Python is a high-level programming language known for its readability and versatility, widely used in data analysis, machine learning, and web development. Its simplicity allows for rapid prototyping and efficient coding, making it a popular choice among data scientists and statisticians for performing statistical analysis and creating predictive models.
Quasi-poisson model: A quasi-Poisson model is a type of statistical model used to analyze count data that exhibit overdispersion, which occurs when the variance exceeds the mean. It extends the traditional Poisson regression by incorporating an additional dispersion parameter to better fit the data, making it particularly useful when standard Poisson assumptions do not hold. This model provides a way to address the limitations of Poisson regression in situations where the data are more variable than expected.
R: In statistics, 'r' is the Pearson correlation coefficient, a measure that expresses the strength and direction of a linear relationship between two continuous variables. It ranges from -1 to 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation. This measure is crucial in understanding relationships between variables in various contexts, including prediction, regression analysis, and the evaluation of model assumptions.
Residual Analysis: Residual analysis is a statistical technique used to assess the differences between observed values and the values predicted by a model. It helps in identifying patterns in the residuals, which can indicate whether the model is appropriate for the data or if adjustments are needed to improve accuracy.
Sas: SAS, or Statistical Analysis System, is a software suite used for advanced analytics, business intelligence, and data management. It provides a comprehensive environment for performing statistical analysis and data visualization, making it a valuable tool in the fields of data science and statistical modeling.
Underdispersion: Underdispersion refers to a situation in statistical modeling where the observed variability in the data is less than what the model predicts. This phenomenon often occurs when count data exhibit less variability than expected under a Poisson distribution, which assumes that the mean and variance are equal. In such cases, models like Quasi-Poisson and Negative Binomial can provide a better fit by allowing for greater flexibility in capturing the true distribution of the data.
Variance: Variance is a statistical measurement that describes the extent to which data points in a dataset differ from the mean of that dataset. It provides insight into the spread or dispersion of data, allowing for the evaluation of how much individual values vary from the average. Understanding variance is crucial in various contexts, such as assessing the reliability of estimators, modeling count data, and implementing regularization techniques to avoid overfitting in regression models.