The canonical correlations between subsets of OLS estimators are identified
with design linkage parameters between their regressors. Known collinearity indices are extended to encompass angles between each regressor vector and remaining vectors. One such angle quantifies the collinearity of regressors with the intercept, of concern
in the corruption of all estimates due to ill-conditioning. Matrix identities factorize a determinant in terms of principal subdeterminants and the canonical Vector Alienation Coefficients between subset estimatorsâ€”by duality, the Alienation Coefficients between
subsets of regressors. These identities figure in the study of D and Ds as determinant efficiencies for estimators and their subsets, specifically, Ds-efficiencies for the constant, linear, pure quadratic, and interactive coefficients in eight known small second-order
designs. Studies on D- and Ds-efficiencies confirm that designs are seldom efficient for both. Determinant identities demonstrate the propensity for Ds-inefficient subsets to be masked through near collinearities in overall D-efficient designs.