The effect of collinearity and sample size on linear regression results: a simulation study

TL;DR

This simulation study investigates how collinearity and sample size jointly influence linear regression accuracy, emphasizing that VIF thresholds should be context-dependent and highlighting the risks of ignoring sample size and bias.

Contribution

It provides a comprehensive simulation-based analysis of collinearity effects across various sample sizes, offering practical guidance for interpreting VIFs and understanding bias amplification.

Findings

Collinearity reduces precision in small samples but has limited impact on coverage in large samples.

Bias amplification due to collinearity is significant under model misspecification.

VIF thresholds should be adapted based on sample size and study context.

Abstract

Background: Multicollinearity inflates the variance of OLS coefficients, widening confidence intervals and reducing inferential reliability. Yet fixed variance inflation factor (VIF) cut-offs are often applied uniformly across studies with very different sample sizes, even though collinearity is a finite-sample problem. We quantify how collinearity and sample size jointly affect linear regression performance and provide practical guidance for interpreting VIFs. Methods: We simulated data across sample sizes N=100-100,000 and collinearity levels VIF=1-50. For each scenario we generated 1,000 datasets, fitted OLS models, and assessed coverage, mean absolute error (MAE), bias, traditional power (CI excludes 0), and precision assurance (probability the 95% CI lies within a prespecified margin around the true effect). We also evaluated a biased, misspecified setting by omitting a relevant predictor to study bias amplification. Results: Under correct specification, collinearity did not materially affect nominal coverage and did not introduce systematic bias, but it reduced precision in small samples: at N=100, even mild collinearity (VIF<2) inflated MAE and markedly reduced both power metrics, whereas at N>=50,000 estimates were robust even at VIF=50. Under misspecification, collinearity strongly amplified bias, increasing errors, reducing coverage, and sharply degrading both precision assurance and traditional power even at low VIF. Conclusion: VIF thresholds should not be applied mechanically. Collinearity must be interpreted in relation to sample size and potential sources of bias; removing predictors solely to reduce VIF can worsen inference via omitted-variable bias. The accompanying heatmaps provide a practical reference across study sizes and modelling assumptions.