lm

A linear model, often abbreviated as lm in statistical literature, is a mathematical framework for relating a response variable to one or more predictor variables through a linear relationship plus an error term. In its most common form, the model for n observations is written as y = Xβ + ε, where y is an n-dimensional vector of responses, X is an n by p matrix of predictor values, β is a p-dimensional vector of coefficients, and ε is a vector of random errors with mean zero and constant variance. The errors are typically assumed to be independently and identically distributed, often with a normal distribution.

Estimation and inference are usually based on the least squares principle, which minimizes the sum of squared residuals. When X has full column rank, the coefficient estimates are β̂ = (X′X)⁻¹X′y. Predicted values are ŷ = Xβ̂, and model fit is commonly summarized with statistics such as R² and adjusted R². Inference about coefficients uses standard errors, t-tests, and F-tests, under the modeling assumptions.

Extensions and variants include simple linear regression (one predictor) and multiple linear regression (several predictors), as well as models that incorporate nonlinear terms or interactions. Generalized linear models extend linear models to responses that follow distributions other than normal and handle different link functions.

Software implementation is widespread; in R, the lm function fits linear models using a formula interface (for example, y ~ x1 + x2) and returns an object containing coefficients, residuals, fitted values, and diagnostic information. Diagnostics typically involve residual analysis, checks for linearity, homoscedasticity, normality, and multicollinearity. Limitations include sensitivity to outliers and model mis-specification.