Singular majorants and minorants: enhanced design conditioning

Abstract

ABSTRACTLower and upper spectral bounds are known for matrices under Loewner [Uber monotone matrixfunktionen. Math Z. 1934;38:177–216] order, as are corresponding bounds for the factor under an induced order. Least upper bounds for the latter give designs with dominating Fisher Information, with consequent gains in linear inference; see Jensen DR, Ramirez DE [Enhanced design efficiency through least upper bounds. J Stat Comput Simul. 2016;86:1798–1817]. The present study examines properties on ordering the singular values of a design matrix using majorization as in Marshall and Olkin [Inequalities: theory of majorization and its applications. New York: Academic Press; 1979]. The principal focus includes conditioning through condition numbers, variance inflation factors, and lengths and efficiencies of OLS solutions. Functions monotone under the induced order are identified; equivalence classes of designs are displayed preserving a dispersion matrix or its eigenvalues; a minimal element is characterized; as are equivalence classes of -optimal designs showing the latter not to be unique. Algorithms to achieve enhanced designs are given on modifying a single design, or on amalgamating two designs, with essential consequences in linear inference. A collateral procedure, based on mixtures of Fisher information matrices, serves effectively to ameliorate the ill effects of near collinearity. Case studies illustrate gains to be made in practice, to include a substantial improvement in the analysis of classically ill-conditioned data from the literature.