Quantifying Heteroscedasticity in Linear Models Using Quantile LOWESS Intervals

2024/06/28 by Martina Sladekova, Andy P. Field · 1 voice
Mathematics · Neuroscience · Psychology · #Categorical variable #Cognitive Abilities and Testing #Econometrics #Heteroscedasticity #Homoscedasticity #Linear model #Mathematics #Mental Health Research Topics #Neural and Behavioral Psychology Studies #Ordinary least squares #Quantile #Statistics #Studentized residual

paper · pdf · doi:10.31234/osf.io/gn4mr

openalex publication_date 2024/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/14

Abstract

Ordinary Least Squares (OLS) estimation, which is frequently applied in psychology, assumes constant variance of errors across predictor levels. This assumption is known as homoscedasticity, while its violation is referred to as heteroscedasticity. In categorical predictors, heteroscedasticity can be quantified by calculating the ratio of variances across groups. For continuous predictors, diagnostic residual plots are often used to assess whether the assumption had been met, but there is currently no measure that can quantify the amount of heteroscedasticity in an interpretable way. In this study, we have developed and evaluated a measure that constructs a quantile LOWESS interval (QLI) around the residuals and estimates the linear, quadratic, cubic and quartic change in the width of this interval as a function of the predictor or the fitted values. Further, we evaluated simple linear models under different patterns of heteroscedasticity in a simulation in order to provide benchmark values of QLI estimates associated with inadequate control over false-positive results, loss of power, and loss of coverage probability of confidence intervals. The QLI method provided consistent estimates of trends for models with 60 or more cases, and this was true across variance patterns. We discuss QLI-generated estimates in relation to performance of OLS linear models. Finally, we present an example of how to apply the QLI method to quantify heteroscedasticity and how to interpret the estimates it provides, focusing on the implications for the OLS analysis.

Citations

• ggplot2: Elegant Graphics for Data Analysis
• Robust Locally Weighted Regression and Smoothing Scatterplots
• A note on preliminary tests of equality of variances
• Robustness?
• Properties of sufficiency and statistical tests
• In Search of Unicorns: Assessing Statistical Assumptions in Real Psychology Datasets
• Reporting and Interpreting Research in PSPB: Practices, Principles, and Pragmatics
• A Simple Test for Heteroscedasticity and Random Coefficient Variation
• Reliability, Validity, and Factor Structure of the Creative Achievement Questionnaire

Discussions

• Chapter 4 was a several months long detour where we developed a measure for quantifying heteroscedasticity in models with continuous predictors: doi.org/10.31234/osf... (version with R package and cod [bsky, 0 points, 1 comments]

Related

• Using Heteroscedasticity Consistent Standard Errors in the Linear Regression Model
• In Search of Unicorns: Assessing Statistical Assumptions in Real Psychology Datasets
• Commonly Used Statistical Models in Psychology are Not Equipped to Deal with Real-World Conditions: A Simulation
• Monte Carlo power comparison of seven most commonly used heteroscedasticity tests
• Robust Statistical Methods and the Credibility Movement of Psychological Science