🥖Linear Modeling Theory Unit 13 – Intro to Generalized Linear Models (GLMs)

Key Concepts and Definitions

• Generalized Linear Models (GLMs) extend ordinary linear regression to accommodate response variables with non-normal distributions
• GLMs consist of three components: a random component, a systematic component, and a link function
  • The random component specifies the probability distribution of the response variable (e.g., Gaussian, Binomial, Poisson)
  • The systematic component relates the linear predictor to the explanatory variables through a linear combination
  • The link function connects the mean of the response variable to the systematic component
• Exponential family distributions play a central role in GLMs, providing a unified framework for various types of response variables
• Maximum likelihood estimation is commonly used to estimate the parameters of GLMs, maximizing the likelihood function of the observed data
• Deviance is a measure of goodness of fit for GLMs, comparing the fitted model to the saturated model
• Overdispersion occurs when the variability in the data exceeds what is expected under the assumed probability distribution

Foundations of Linear Models

• Linear models assume a linear relationship between the response variable and the explanatory variables
• Ordinary least squares (OLS) is used to estimate the parameters of linear models, minimizing the sum of squared residuals
• Assumptions of linear models include linearity, independence, homoscedasticity, and normality of errors
  • Linearity assumes a straight-line relationship between the response and explanatory variables
  • Independence assumes that the observations are independent of each other
  • Homoscedasticity assumes constant variance of the errors across all levels of the explanatory variables
  • Normality assumes that the errors follow a normal distribution
• Residuals are the differences between the observed and predicted values, used to assess model assumptions and fit
• Hypothesis testing and confidence intervals can be used to make inferences about the model parameters
• Limitations of linear models include the inability to handle non-linear relationships, non-normal responses, and categorical predictors

Introduction to GLMs

• GLMs extend linear models to accommodate response variables with various distributions, such as binary, count, or continuous data
• The main idea behind GLMs is to model the relationship between the response variable and the explanatory variables through a link function
• GLMs allow for the modeling of non-linear relationships between the response and explanatory variables
• The choice of the appropriate GLM depends on the nature of the response variable and the research question
• GLMs provide a unified framework for regression analysis, encompassing linear regression, logistic regression, Poisson regression, and more
• GLMs are widely used in various fields, including biology, economics, social sciences, and engineering

Components of GLMs

• The random component of a GLM specifies the probability distribution of the response variable
  • The distribution must belong to the exponential family (e.g., Gaussian, Binomial, Poisson, Gamma)
  • The distribution determines the mean-variance relationship and the appropriate link function
• The systematic component of a GLM relates the linear predictor to the explanatory variables
  • The linear predictor is a linear combination of the explanatory variables and their coefficients
  • The coefficients represent the change in the response variable for a unit change in the corresponding explanatory variable
• The link function connects the mean of the response variable to the systematic component
  • The link function is chosen based on the distribution of the response variable
  • Common link functions include identity (linear regression), logit (logistic regression), and log (Poisson regression)
• The canonical link function is the natural choice for a given exponential family distribution, resulting in desirable statistical properties

Types of GLMs

• Linear regression is used when the response variable is continuous and normally distributed
  • The identity link function is used, assuming a direct linear relationship between the response and explanatory variables
• Logistic regression is used when the response variable is binary or categorical
  • The logit link function is used, modeling the log-odds of the response as a linear combination of the explanatory variables
• Poisson regression is used when the response variable represents count data
  • The log link function is used, modeling the log of the expected count as a linear combination of the explanatory variables
• Gamma regression is used when the response variable is continuous, positive, and right-skewed
  • The inverse link function is commonly used, modeling the reciprocal of the mean response as a linear combination of the explanatory variables
• Quasi-likelihood models extend GLMs to situations where the full probability distribution is not specified, using only the mean-variance relationship

Model Fitting and Estimation

• Maximum likelihood estimation (MLE) is the most common method for estimating the parameters of GLMs
  • MLE finds the parameter values that maximize the likelihood function of the observed data
  • The likelihood function measures the probability of observing the data given the model parameters
• Iteratively reweighted least squares (IRLS) is an algorithm used to solve the MLE equations for GLMs
  • IRLS iteratively updates the parameter estimates by solving a weighted least squares problem
  • The weights are determined by the current estimates and the link function
• Goodness of fit measures, such as deviance and Akaike information criterion (AIC), assess the model's fit to the data
  • Deviance compares the fitted model to the saturated model, with lower values indicating better fit
  • AIC balances the model's fit and complexity, favoring models with lower AIC values
• Residual analysis is used to assess the model assumptions and identify potential outliers or influential observations
• Hypothesis tests and confidence intervals can be constructed for the model parameters using the asymptotic normality of the MLE

Interpreting GLM Results

• The coefficients in a GLM represent the change in the linear predictor for a unit change in the corresponding explanatory variable
• The interpretation of the coefficients depends on the link function and the scale of the response variable
  • For the identity link (linear regression), the coefficients directly represent the change in the mean response
  • For the logit link (logistic regression), the coefficients represent the change in the log-odds of the response
  • For the log link (Poisson regression), the coefficients represent the change in the log of the expected count
• Exponentiated coefficients (e.g., odds ratios, rate ratios) provide a more intuitive interpretation for some GLMs
• Confidence intervals and p-values can be used to assess the significance and precision of the estimated coefficients
• Model predictions can be made for new observations by plugging in their values for the explanatory variables and inverting the link function

Applications and Examples

• GLMs are widely used in epidemiology to study the relationship between risk factors and disease outcomes (e.g., logistic regression for case-control studies)
• In ecology, GLMs are used to model species distribution, abundance, and habitat preferences (e.g., Poisson regression for count data)
• GLMs are applied in finance to model the probability of default, claim severity, and insurance pricing (e.g., gamma regression for loss amounts)
• In marketing, GLMs are used to analyze customer behavior, preferences, and response to promotions (e.g., logistic regression for purchase decisions)
• GLMs are employed in social sciences to study the factors influencing voting behavior, educational attainment, and social mobility (e.g., ordinal logistic regression for ordered categories)

Common Challenges and Solutions

• Model selection involves choosing the appropriate GLM and selecting the relevant explanatory variables
  • Stepwise procedures (forward, backward, or mixed) can be used to iteratively add or remove variables based on a selection criterion (e.g., AIC)
  • Regularization techniques (e.g., lasso, ridge) can be employed to shrink the coefficients and handle high-dimensional data
• Multicollinearity occurs when the explanatory variables are highly correlated, leading to unstable and unreliable estimates
  • Variance inflation factors (VIF) can be used to detect multicollinearity
  • Remedies include removing redundant variables, combining related variables, or using dimensionality reduction techniques (e.g., principal component analysis)
• Overdispersion arises when the variability in the data exceeds what is expected under the assumed probability distribution
  • Quasi-likelihood models or negative binomial regression can be used to account for overdispersion
  • Generalized estimating equations (GEE) can be employed for clustered or correlated data
• Zero-inflation occurs when there are excessive zeros in the response variable compared to the assumed distribution
  • Zero-inflated models (e.g., zero-inflated Poisson, zero-inflated negative binomial) can be used to handle zero-inflation
  • Hurdle models separately model the zero-generating process and the positive counts
• Model diagnostics and validation techniques should be used to assess the model's assumptions, fit, and predictive performance
  • Residual plots, QQ-plots, and goodness-of-fit tests can be used to check the model assumptions
  • Cross-validation or bootstrap resampling can be employed to evaluate the model's predictive accuracy and robustness