Reversals of Least-Square Estimates and Model-Invariant Estimation for Directions of Unique Effects

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 model uncertainty. This process is called model-invariant 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-invariant estimation can be implemented using subject matter knowledge. More general theory with partial coefficients is applicable to analysis of large datasets. Applications are demonstrated with dietary health data from the United Nations.