  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.
