Browsing by Author "Jensen, D. R."
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- Determinant Efficiencies in Ill-Conditioned ModelsJensen, D. R. (Hindawi, 2011-10-05)The canonical correlations between subsets of OLS estimators are identifiedwith 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 concernin 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 estimators—by duality, the Alienation Coefficients betweensubsets 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-orderdesigns. 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.
- Revision: Variance Inflation in RegressionJensen, D. R.; Ramirez, D. E. (Hindawi, 2013-02-25)Variance Inflation Factors (VIFs) are reexamined as conditioning diagnostics for models with intercept, with and without centering regressors to their means as oft debated. Conventional VIFs, both centered and uncentered, are flawed. To rectify matters, two types of orthogonality are noted: vector-space orthogonality and uncorrelated centered regressors. The key to our approach lies in feasible Reference models encoding orthogonalities of these types. For models with intercept it is found that (i)uncentered VIFs are not ratios of variances as claimed, owing to infeasible Reference models; (ii) instead they supply informative angles between subspaces of regressors; (iii) centered VIFs are incomplete if not misleading, masking collinearity of regressors withthe intercept; and (iv) variance deflation may occur, where ill-conditioned data yield smaller variances than their orthogonal surrogates. Conventional VIFs have all regressors linked, or none, often untenable in practice. Beyond these, our models enable the unlinking of regressors that can be unlinked, while preserving dependence among those intrinsically linked. Moreover, known collinearity indices are extended to encompass angles between subspaces of regressors. To reaccess ill-conditioned data, we consider case studies ranging from elementary examples to data from the literature.