Facilitating heterogeneous effect estimation via statistically efficient categorical modifiers

TL;DR

This paper introduces an alternative parametrization called abundance-based constraints (ABCs) for regression models with categorical modifiers, enabling unbiased, interpretable, and more powerful estimation of heterogeneous effects without affecting main effect estimates.

Contribution

It proposes ABCs as a new approach to include categorical modifiers in linear models, preserving main effect estimates and increasing statistical power, with demonstrated benefits through simulations and real data.

Findings

ABCs leave main effect estimates unchanged when adding categorical modifiers.

Using ABCs increases the statistical power of detecting heterogeneous effects.

Application to educational data reveals demographic heterogeneities in social and environmental impacts.

Abstract

Categorical covariates such as race, sex, or group are ubiquitous in regression analysis. While main-only (or ANCOVA) linear models are predominant, cat-modified linear models that include categorical-continuous or categorical-categorical interactions are increasingly important and allow heterogeneous, group-specific effects. However, with standard approaches, the addition of cat-modifiers fundamentally alters the estimates and interpretations of the main effects, often inflates their standard errors, and introduces significant concerns about group (e.g., racial) biases. We advocate an alternative parametrization and estimation scheme using abundance-based constraints (ABCs). ABCs induce a model parametrization that is both interpretable and equitable. Crucially, we show that with ABCs, the addition of cat-modifiers 1) leaves main effect estimates unchanged and 2) enhances their statistical power, under reasonable conditions. Thus, analysts can, and arguably should include cat-modifiers in linear regression models to discover potential heterogeneous effects--without compromising estimation, inference, and interpretability for the main effects. Using simulated data, we verify these invariance properties for estimation and inference and showcase the capabilities of ABCs to increase statistical power. We apply these tools to study demographic heterogeneities among the effects of social and environmental factors on STEM educational outcomes for children in North Carolina. An R package lmabc is available.