Linear Regression Assumptions and Residual Analysis

Linear regression models a response as a linear function of predictors plus an error term. The assumptions are often summarized too broadly, so separate the ones used for estimating coefficients from those needed for familiar standard errors and tests. The conditional mean should be correctly represented by the chosen predictors and functional form. Observations or errors should be independent under the sampling design. Constant conditional error variance, called homoscedasticity, supports the usual OLS standard-error formula. Normally distributed errors are chiefly relevant to exact small-sample inference under the classical model, rather than a requirement for calculating the least-squares fit itself. A residual is the observed outcome minus the fitted value. Plot residuals against fitted values and important predictors. A cloud centered around zero without a systematic curve supports the chosen mean shape; a wave or U-shaped band points toward missing structure. A fan-shaped spread suggests variance changes with the fitted level. A normal quantile plot can reveal heavy tails or skew, but it does not diagnose independence. For data ordered over time, plot residuals in time order and consider design-aware checks for serial correlation. For a hypothetical model of repair time, residuals may be small for routine repairs and much larger for complex ones. That could indicate nonconstant variance, omitted complexity, or both. First inspect whether predictors and measurements are correct; then consider a transformed outcome, a better-specified mean, robust standard errors, or a model that represents changing variance. These choices answer different problems and should be justified by the data-generating context. An unusual residual is not automatically an error. A point can have a large residual, high leverage in predictor space, or substantial influence on fitted coefficients; these are distinct properties. Investigate records and report sensitivity to influential cases. Diagnostics are evidence for refining a model, not a mechanical checklist that certifies it. Assess predictive performance on data that reflect the intended use, and communicate remaining limitations.