Generalized Linear Models

Overview

Generalized Linear Models (GLM) represent a powerful extension of classical linear regression that accommodates response variables with non-normal distributions. While ordinary least squares (OLS) regression assumes responses follow a normal distribution Y \sim N(\mu, \sigma^2), GLMs generalize this framework to handle any distribution from the Exponential Family. This flexibility makes GLMs essential for analyzing diverse data types: binary outcomes (logistic regression), count data (Poisson and negative binomial regression), positive continuous measurements (Gamma and Inverse Gaussian regression), and zero-inflated or composite data (Tweedie regression).

The power of GLMs lies in their unified mathematical framework consisting of three core components. The Random Component specifies the probability distribution of the response variable Y, chosen based on the nature of your data and domain knowledge. The Systematic Component represents the linear predictor \eta = X\beta, combining predictor variables and regression coefficients in a linear fashion. The Link Function g(\cdot) bridges these two components by transforming the expected value of the response: g(\mu) = \eta = X\beta, where \mu = E[Y]. This relationship is crucial because it allows the linear model to flexibly accommodate different scales and ranges of response variables while maintaining interpretability.

Model Estimation and Inference in GLMs relies on Maximum Likelihood Estimation (MLE) rather than least squares, providing a principled approach to parameter estimation. The likelihood function depends entirely on the chosen exponential family distribution. Python’s statsmodels library and SciPy provide comprehensive implementations of GLM families with efficient MLE algorithms. These libraries handle the computational complexity of iteratively reweighted least squares (IRLS), the standard algorithm for GLM fitting, and provide diagnostics for model assessment including residual plots, influence measures, and goodness-of-fit statistics.

Choosing the Appropriate Family is a critical decision when building a GLM. The Binomial Family handles binary outcomes (yes/no, success/failure) and proportions, using the logit link by default to map probabilities to the real line. The Poisson Family models count data with the constraint that mean equals variance; it’s ideal for events occurring at a constant rate but can be restrictive when data is overdispersed. The Negative Binomial Family extends Poisson regression by allowing mean and variance to differ, making it superior for count data with extra variability. The Gamma Family suits continuous positive responses with right-skewed distributions, common in survival times, income, and spending data. The Inverse Gaussian Family handles positive data with even more pronounced right skew and is frequently used for reliability and lifetime data. The Tweedie Family provides remarkable flexibility by encompassing Poisson, Gamma, and inverse Gaussian as special cases, enabling simultaneous modeling of zero-inflation and heavy tails.

Link Functions transform the linear predictor scale to the appropriate scale for the response variable. The Logit Link \log\left(\frac{\mu}{1-\mu}\right) is standard for binomial GLMs, converting linear predictors to probabilities. The Log Link \log(\mu) ensures positive predictions for Poisson and other families working with positive responses. The Identity Link \mu maintains the linear scale and is sometimes used despite violating domain constraints. The Probit Link, based on the normal cumulative distribution function, provides an alternative for binary responses with different asymptotic behavior than logit. Selecting the appropriate link function should balance theoretical justification with empirical model fit.

Practical Applications span numerous fields. Logistic regression (binomial GLM with logit link) predicts binary outcomes in medical diagnosis, credit approval, and marketing response. Poisson and negative binomial regression model count outcomes like customer complaints, disease incidence, or accident frequencies. Gamma regression handles continuous positive responses such as healthcare costs, insurance claims, and reliability data. These tools form the foundation of modern statistical modeling, bridging the gap between the assumptions of classical regression and the diversity of real-world data.

Tools

Tool	Description
GLM_BINOMIAL	Fits a Generalized Linear Model with binomial family for binary or proportion data.
GLM_GAMMA	Fit a Generalized Linear Model with Gamma family for positive continuous data.
GLM_INV_GAUSS	Fits a Generalized Linear Model with Inverse Gaussian family for right-skewed positive data.
GLM_NEG_BINOM	Fits a Generalized Linear Model with Negative Binomial family for overdispersed count data.
GLM_POISSON	Fits a Generalized Linear Model with Poisson family for count data.
GLM_TWEEDIE	Fits a Generalized Linear Model with Tweedie family for flexible distribution modeling.