The Same Problem by Different Names: Unifying Regression Dilution and Regression to the Mean

TL;DR

This paper unifies the concepts of Regression to the Mean and Regression Dilution, demonstrating they are essentially the same issue caused by measurement error, and compares various estimators to guide proper method selection.

Contribution

It provides an analytical framework and practical maps to identify the most accurate estimator for different noise levels and sample sizes, bridging clinical and ecological approaches.

Findings

Berry correction is effective for clinical 1:1 relationships.

Standard estimators perform variably depending on noise and slope.

Guidelines are provided for choosing the optimal estimator based on data conditions.

Abstract

Regression to the Mean and Regression Dilution are often viewed as unrelated issues in the clinical and ecological literatures. In reality, they are different names for the same problem: measurement error in an independent variable that biases the perceived relationship between two factors. This study unifies these traditions by comparing specialized clinical tools, like the Berry correction, with standard structural estimators such as Major Axis and Reduced Major Axis regression. Using an analytical framework, we evaluate how these methods perform across various noise levels and sample sizes. Our results show that the Berry method is a specialized tool designed for clinical scenarios where a 1:1 relationship is expected. However, applying it to ecological trade-offs with negative slopes can lead to severe errors. We provide maps of optimality to identify which estimator most accurately recovers the true biological signal under different conditions. By reconciling these disparate methods, we offer a principled guide for researchers to choose the correct tool based on their data's noise profile rather than their disciplinary tradition.