Reversals of Least-Squares Estimates and Model-Independent Estimation for Directions of Unique Effects

TL;DR

This paper investigates when and why regression coefficients can reverse signs upon model adjustment, introducing model-independent estimation methods to identify unique effects despite model uncertainty, with applications to health data.

Contribution

It develops a geometric and theoretical framework for understanding coefficient reversals and introduces model-independent estimation techniques applicable to large datasets.

Findings

Reversal regions form elliptical cones with geometric properties.

Model-independent estimation can be guided by subject matter knowledge.

Conditions for Simpson's paradox are derived.

Abstract

When a linear model is adjusted to control for additional explanatory variables the sign of a fitted coefficient may reverse. Here these reversals are studied using coefficients of determination. The resulting theory can be used to determine directions of unique effects in the presence of substantial model uncertainty. This process is called model-independent estimation when the estimates are invariant across changes to the model structure. When a single covariate is added, the reversal region can be understood geometrically as an elliptical cone of two nappes with an axis of symmetry relating to a best-possible condition for a reversal using a single coefficient of determination. When a set of covariates are added to a model with a single explanatory variable, model-independent estimation can be implemented using subject matter knowledge. More general theory with partial coefficients is applicable to analysis of large data sets. Applications are demonstrated with dietary health data from the United Nations. Necessary conditions for Simpson's paradox are derived.