First and second order rotatability of experimental designs, moment matrices, and information surfaces

SummaryWe place the well-known notion of rotatable experimental designs into the more general context of invariant design problems. Rotatability is studied as it pertains to the experimental designs themselves, as well as to moment matrices, or to information surfaces. The distinct aspects become visible even in the case of first order rotatability. The case of second order rotatability then is conceptually similar, but technically more involved. Our main result is that second order rotatability may be characterized through a finite subset of the orthogonal group, generated by sign changes, permutations, and a single reflection. This is a great reduction compared to the usual definition of rotatability which refers to the full orthogonal group. Our analysis is based on representing the second order terms in the regression function by a Kronecker power. We show that it is essentially the same as using the Schläflian powers, or the usual minimal set of second order monomials, but it allows a more transparent calculus.

[1]  T. Muir.  The Theory of Determinants in the Historical Order of Development. Vol. II , 1912 .

[2]  H. W. Turnbull,et al.  Lectures on Matrices , 1934 .

[3]  A. C. Aitken.  Determinants and matrices , 1940 .

[4]  A. Ac.  On the Wishart distribution in statistics. , 1949 .

[5]  J. S. Hunter,et al.  Multi-Factor Experimental Designs for Exploring Response Surfaces , 1957 .

[6]  R. C. Bose,et al.  Complex Representation in the Construction of Rotatable Designs , 1959 .

[7]  R. C. Bose,et al.  Second Order Rotatable Designs in Three Dimensions , 1959 .

[8]  G. Box,et al.  Some New Three Level Designs for the Study of Quantitative Variables , 1960 .

[9]  George E. P. Box,et al.  Simplex-Sum Designs: A Class of Second Order Rotatable Designs Derivable From Those of First Order , 1960 .

[10]  Norman R. Draper,et al.  Second Order Rotatable Designs in Four or More Dimensions , 1960 .

[11]  R. Bellman,et al.  A Survey of Matrix Theory and Matrix Inequalities , 1965 .

[12]  On a Method of Construction of Rotatable Designs with Smaller Number of Points Controlling the Number of Levels , 1966 .

[13]  A Method for the Construction of Second Order Rotatable Designs in $k$ Dimensions , 1967 .

[14]  N. Draper,et al.  Further Second Order Rotatable Designs , 1968 .

[15]  Four and Six Level Second Order Rotatable Designs , 1970 .

[16]  Justus Seely,et al.  Quadratic Subspaces and Completeness , 1971 .

[17]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[18]  Friedrich Pukelsheim,et al.  On hsu's model in regression analysis , 1977 .

[19]  M. Singh.  Group Divisible Second Order Rotatable Designs , 1979 .

[20]  Henry P. Wynn,et al.  Optimum continuous block designs , 1981, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[21]  S. R. Searle,et al.  The Vec-Permutation Matrix, the Vec Operator and Kronecker Products: A Review , 1981 .

[22]  A Method for Constructing Second-Order Rotatable Designs , 1981 .

[23]  S. R. Searle,et al.  Matrix Algebra Useful for Statistics , 1982 .

[24]  S. R. Searle,et al.  On the history of the kronecker product , 1983 .

[25]  James V. Bondar.  Universal optimality of experimental designs: definitions and a criterion , 1983 .

[26]  P. Seymour,et al.  Averaging sets: A generalization of mean values and spherical designs , 1984 .

[27]  H. Wynn,et al.  G-majorization with applications to matrix orderings , 1985 .

[28]  I. Olkin,et al.  Collected Papers III , 1985 .

[29]  L. Brown,et al.  Jack Carl Kiefer collected papers , 1985 .

[30]  F. Pukelsheim.  Information increasing orderings in experimental design theory , 1987 .

[31]  F. Pukelsheim,et al.  GROUP INVARIANT ORDERINGS AND EXPERIMENTAL DESIGNS , 1987 .

[32]  George E. P. Box,et al.  Empirical Model‐Building and Response Surfaces , 1988 .

[33]  Norbert Gaffke,et al.  Further Characterizations of Design Optimality and Admissibility for Partial Parameter Estimation in Linear Regression , 1987 .

[34]  M. Marcus,et al.  A Survey of Matrix Theory and Matrix Inequalities , 1965 .

[35]  Norman R. Draper,et al.  Another look at rotatability , 1990 .

[36]  J. J. Seidel,et al.  Measures of strength $2e$ and optimal designs of degree $e$ , 1992 .