Generalized Linear Models (GLMs) extend the capabilities of traditional ordinary least squares architectures to accommodate non-normal, skewed, or discrete target variables. A foundational component of the GLM framework is the link function, which establishes a strict mathematical connection between the un-bounded linear predictor space and the bounded expected value of the target distribution.
Purpose: Maps the expected value of the target response variable directly to the un-bounded linear combination of predictors without transformation.
Primary Application: Continuous response variables tracking a normal Gaussian error distribution, making the GLM equivalent to standard linear regression.
Purpose: Models the natural logarithm of the expected target value. This transformation ensures that inverse projected model values always remain strictly positive numbers.
Primary Application: Open-ended count data pipelines (Poisson regression) or skewed continuous positive variables (Gamma distributions).
Purpose: Models the log-odds (logit transformation) of an event probability. This mathematical boundary constrains target probability predictions to map cleanly between 0 and 1.
Primary Application: The baseline standard link function used to optimize binary classification tasks (logistic regression models).
Purpose: Models the inverse cumulative distribution function (CDF) of a standard normal distribution to map target probability limits.
Primary Application: Binary response classification models, frequently deployed in structural contexts where an unobserved latent variable follows a normal distribution shape.
Purpose: Models the natural logarithm of the negative log tracking the complementary probability of an event happening.
Primary Application: Asymmetrical binary datasets where target event probabilities track as extremely rare (near 0) or heavily ubiquitous (near 1), alongside specialized survival analysis models.
Purpose: Models the strict mathematical inverse of the expected value parameter of the response target.
Primary Application: Continuous positive variables displaying exponential skews, standardly acting as the canonical link function for Gamma distributions.
Purpose: Models the precise square root parameter of the expected value of the target distribution.
Primary Application: Over-dispersed count data environments where local variance tracks and escalates faster than the baseline distribution mean.
Isolating the appropriate link function parameter configuration depends on three architecture rules:
Selecting an inappropriate link function model specification introduces structural estimation bias and renders downstream hypothesis inferences invalid.