🥖Linear Modeling Theory Unit 15 – Overdispersion & Quasi-Likelihood Methods

Key Concepts

• Overdispersion occurs when the observed variance in a dataset exceeds the expected variance under a specified statistical model
• Quasi-likelihood methods provide a framework for modeling overdispersed data without requiring a fully specified probability distribution
• Quasi-likelihood estimating equations are derived from the mean and variance functions of the response variable
  • These equations are similar to the score equations in maximum likelihood estimation
• Quasi-likelihood methods allow for the estimation of regression coefficients and their standard errors in the presence of overdispersion
• Dispersion parameter quantifies the degree of overdispersion in the data
  • It is estimated from the data and used to adjust the standard errors of the regression coefficients
• Quasi-AIC (QAIC) and quasi-deviance are used for model selection and goodness-of-fit assessment in quasi-likelihood models
• Quasi-likelihood methods can be applied to various types of data, including count data (Poisson regression) and binary data (logistic regression)

Statistical Foundations

• Overdispersion violates the assumption of equal mean and variance in certain statistical models (Poisson regression)
• Maximum likelihood estimation (MLE) is a common method for estimating model parameters in statistical modeling
  • MLE finds the parameter values that maximize the likelihood function given the observed data
• Likelihood function quantifies the probability of observing the data given the model parameters
• Score equations are derived from the log-likelihood function and are used to find the maximum likelihood estimates
• Fisher information matrix is the negative expectation of the second derivative of the log-likelihood function
  • It is used to calculate the standard errors of the estimated parameters
• Quasi-likelihood methods relax the requirement of a fully specified probability distribution
  • They only require the specification of the mean and variance functions of the response variable
• Quasi-likelihood estimating equations are derived from the mean and variance functions and are solved to obtain the parameter estimates

Overdispersion Explained

• Overdispersion is a common issue in count data and binary data analysis
• It occurs when the observed variance in the data is greater than the expected variance under the assumed statistical model
• In Poisson regression, overdispersion implies that the variance of the response variable exceeds the mean
  • This violates the equidispersion assumption of the Poisson distribution
• Overdispersion can lead to underestimated standard errors and incorrect inference if not accounted for
• Causes of overdispersion include unobserved heterogeneity, clustering, and the presence of outliers or extreme values
• Failing to account for overdispersion can result in overly narrow confidence intervals and inflated Type I error rates
• Overdispersion can be detected using goodness-of-fit tests (chi-square test) or by comparing the residual deviance to the degrees of freedom

Quasi-Likelihood Methods

• Quasi-likelihood methods provide a flexible approach to modeling overdispersed data
• They extend the concept of likelihood to situations where the full probability distribution is not specified
• Quasi-likelihood methods only require the specification of the mean and variance functions of the response variable
  • The mean function relates the expected value of the response to the linear predictor (regression equation)
  • The variance function describes how the variance of the response depends on the mean
• Quasi-likelihood estimating equations are derived from the mean and variance functions
  • They are solved iteratively to obtain the parameter estimates
• Quasi-likelihood methods introduce a dispersion parameter to account for overdispersion
  • The dispersion parameter is estimated from the data and used to adjust the standard errors of the regression coefficients
• Quasi-likelihood methods can be applied to various types of data, including count data (quasi-Poisson regression) and binary data (quasi-binomial regression)

Model Fitting Techniques

• Iterative weighted least squares (IWLS) is a common method for fitting quasi-likelihood models
  • IWLS iteratively updates the parameter estimates by solving a weighted least squares problem
• The weights in IWLS are determined by the variance function and the current parameter estimates
• IWLS continues until the parameter estimates converge or a maximum number of iterations is reached
• Generalized estimating equations (GEE) is another approach for fitting quasi-likelihood models
  • GEE accounts for the correlation structure in clustered or longitudinal data
• GEE uses a working correlation matrix to model the dependence among observations within clusters
• The choice of the working correlation structure (exchangeable, autoregressive, unstructured) depends on the nature of the data
• Sandwich variance estimators are used to obtain robust standard errors in GEE models
  • These estimators are consistent even if the working correlation structure is misspecified

Diagnostic Tools

• Residual analysis is crucial for assessing the adequacy of quasi-likelihood models
• Pearson residuals and deviance residuals are commonly used in quasi-likelihood models
  • Pearson residuals are standardized residuals based on the Pearson chi-square statistic
  • Deviance residuals are based on the contribution of each observation to the deviance
• Residual plots (residuals vs. fitted values, residuals vs. covariates) can reveal patterns or anomalies in the data
• Outliers and influential observations can be identified using leverage values and Cook's distance
• Quasi-likelihood ratio tests can be used for hypothesis testing and model comparison
  • These tests compare the quasi-likelihood of nested models
• Quasi-AIC (QAIC) is an information criterion for model selection in quasi-likelihood models
  • QAIC balances the goodness-of-fit and the complexity of the model
• Overdispersion tests (dispersion test, score test) can formally assess the presence of overdispersion in the data

Applications and Examples

• Quasi-likelihood methods have been widely applied in various fields, including ecology, epidemiology, and social sciences
• Overdispersed count data examples:
  • Modeling the number of species in different habitats (biodiversity studies)
  • Analyzing the number of accidents in different road segments (traffic safety analysis)
• Overdispersed binary data examples:
  • Modeling the presence or absence of a disease in patients (medical research)
  • Analyzing the success or failure of a product in different markets (marketing research)
• Quasi-Poisson regression has been used to model overdispersed count data in ecological studies
  • Example: Modeling the abundance of a species across different environmental gradients
• Quasi-binomial regression has been applied to analyze overdispersed binary data in social sciences
  • Example: Modeling the voting behavior of individuals based on demographic and socioeconomic factors

Limitations and Considerations

• Quasi-likelihood methods rely on the correct specification of the mean and variance functions
  • Misspecification of these functions can lead to biased parameter estimates and incorrect inference
• The choice of the variance function is crucial in quasi-likelihood models
  • Different variance functions (constant, proportional to the mean, quadratic) may lead to different results
• Quasi-likelihood methods do not provide a full probabilistic model for the data
  • They do not allow for the calculation of exact likelihood ratios or posterior probabilities
• The interpretation of the dispersion parameter in quasi-likelihood models can be challenging
  • It is not always clear how to compare dispersion parameters across different models or datasets
• Quasi-likelihood methods may not be as efficient as full likelihood methods when the probability distribution is correctly specified
• The choice of the link function (log, logit, probit) in quasi-likelihood models can affect the interpretation of the regression coefficients
• Quasi-likelihood methods assume that the observations are independent
  • Extensions like GEE are needed to handle correlated or clustered data
• Model diagnostics and goodness-of-fit assessment in quasi-likelihood models may not be as straightforward as in likelihood-based models