Statistical Diagnostics and Reliability Evaluation of Linear Regression Models

Abstract

Linear regression is often used to explain, infer statistically and predict, but a good fit within the training data does not guarantee the reliability of the model. The main diagnostic and validation tools for assessing linear regression models are introduced in this paper, including residual analysis, multicollinearity, influential observations and out-of-sample evaluation. First, it presents the assumptions of ordinary least squares and explains how residual plots, heteroscedasticity tests, autocorrelation diagnostics, variance inflation factors, leverage measures, Cook's distance and other statistics can reveal different sources of model instability. Then it will present recent studies on improved multicollinearity diagnosis, robust estimation, influence measures and cross-validation. Based on the above comparisons, the diagnostic conclusions may differ depending on the sample size, model selection, data quality and decision-making threshold. Other reasons are that multiple assumptions have been violated simultaneously, there is sensitivity to influential observations, instability in high-dimensional spaces, and discrepancies among validation methods. In the future, good estimation, uncertainty-aware diagnosis, high-dimensional methods and transparent out-of-sample validation will be integrated to enhance the reliability and reproducibility of linear regression analysis.