Summary
We 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.
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References
Aitken AC (1949) On the Wishart distribution in statistics. Biometrika 36:59–62
Aitken AC (1951) Determinants and matrices. Oliver and Boyd, Edinburgh and London
BenIsrael A, Greville T (1974) Generalized inverses: theory and applications. Wiley, New York
Bondar JV (1983) Universal optimality of experimental designs: definitions and a criterion. Canad J Statist 11:325–331
Bose RC, Carter RL (1959) Complex representation in the construction of rotatable designs. Ann Math Statist 30:771–780
Bose RC, Draper NR (1959) Second order rotatable designs in three dimensions. Ann Math Statist 30:1097–1112
Box GEP, Behnken DW (1960a) Some new three level designs for the study of quantitative variables. Technometrics 2:455–475
Box GEP, Behnken DW (1960b) Simplex-sum designs: a class of second order rotatable designs derivable from those of first order. Ann Math Statist 31:838–864
Box GEP, Draper NR (1987) Empirical model-building and response surfaces. Wiley, New York
Box GEP, Hunter JS (1957) Multi-factor experimental designs for exploring response surfaces. Ann Math Statist 28:195–241
Coxeter HSM (1963) Regular Polytopes. Dover, New York
Draper NR (1960) Second order rotatable designs in four or more dimensions. Ann Math Statist 31:23–33
Draper NR, Herzberg AM (1968) Further second order rotatable designs. Ann Math Statist 39:1995–2001
Draper NR, Pukelsheim F (1990) Another look at rotatability. Technometrics 32:195–202
Gaffke N (1987) Further characterizations of design optimality and admissibility for partial parameter estimation in linear regression. Ann Statist 15:942–957
Giovagnoli A, Wynn HP (1981) Optimum continuous block designs. Proc Roy Soc London Ser A 377:405–416
Giovagnoli A, Wynn HP (1985a) Schur-optimal continuous block designs for treatments with a control. In: Le Cam LM, Olshen RA (eds) Proceedings of the Berkeley Conference in honor of Jerzy Neyman and Jack Kiefer, vol 1. Wadsworth, Belmont CA, pp 418–433
Giovagnoli A, Wynn HP (1985b) G-majorization with applications to matrix orderings. Linear Algebra Appl 67:111–135
Giovagnoli A, Pukelsheim F, Wynn HP (1987) Group invariant orderings and experimental designs. J Statist Plann Inference 17:159–171
Henderson HV, Pukelsheim F, Searle SR (1983) On the history of the Kronecker product. Linear and Multilinear Algebra 14:113–120
Henderson HV, Searle SR (1981) The vec-permutation matrix, the vec operator and Kronecker products: A review. Linear and Multilinear Algebra 9:271–288
Herzberg AM (1967) A method for the construction of second order rotatable designs ink dimensions. Ann Math Statist 38:177–180
Huda S (1981) A method for constructing second order rotatable designs. Calcutta Statist Assoc Bull 30:139–144
Kiefer JC (1985) Jack Carl Kiefer collected papers, vol III. Brown RD, Olkin I, Sacks J, Wynn HP (eds). Springer, New York
Koll K (1980) Drehbare Versuchspläne erster und zweiter Ordnung. Diplomarbeit, RWTH Aachen
Marcus M, Minc H (1964) A survey of matrix theory and matrix inequalities. Prindle, Weber and Schmidt, Boston MA, London, Syndney
Minc H (1978) Permanents. Encyclopedia of mathematics and its applications, vol 6. Addison-Wesley, Reading MA
Muir T (1911) The theory of determinants in the historical order of development, vol II. The Period 1841–1860. Macmillan, London
Neumaier A, Seidel JJ (1990) Measures of strength 2e, and optimal designs of degreee. Sankhya, forthcoming
Nigam AK (1977) A note on four and six level second order rotatable designs. J Indian Soc Agric Statist 29:89–91
Nigam AK, Das MN (1986) On a method of construction of rotatable designs with smaller number of points controlling the number of levels. Calcutta Statist Assoc Bull 15:153–174
Nigam AK, Dey A (1970) Four and six level second order rotatable designs. Calcutta Statist Assoc Bull 19:155–157
Pukelsheim F (1977) On Hsu’s model in regression analysis. Math Operationsforsch Statist Ser Statist 8:323–331
Pukelsheim F (1987a) Majorization orderings for linear regression designs. In: Pukkila T, Puntanen S (eds) Proceedings of the second international Tampere Conference in statistics. Department of Mathematical Sci, Tampere, pp 261–274
Pukelsheim F (1987b) Information increasing orderings in experimental design theory. Internat Statist Rev 55:203–219
Pukelsheim F (1987c) Ordering experimental designs. In: Prohorov Yu A, Sazonov VV (eds) Proceedings of the 1st World Congress of the Bernoulli Society, vol 2. Tashkent, USSR, 8–14 Sept 1986. VNU Science Press, Utrecht, pp 157–165
Raghavarao D (1963) Construction of second order rotatable designs through incomplete block designs. J Ind Statist Assoc 1:221–225
Schläfli L (1851) Über die Resultante eines Systems mehrerer algebraischer Gleichungen. Ein Beitrag zur Theorie der Elimination. Denkschriften der Kaiserlichen Akademie der Wissenschaften, mathematisch-naturwissenschaftliche Klasse, 4. Band (1852) Wien. Reprinted in: Ludwig Schläfli (1814–1895) Gesammelte Mathematische Abhandlungen, Band II, herausgegeben vom Steiner-Schläfli-Komitee der Schweizerischen Naturforschenden Gesellschaft, Birkhäuser, Basel 1953
Searle SR (1982) Matrix algebra useful for statistics. Wiley, New York
Seely J (1971) Quadratic subspaces and completeness. Ann Math Statist 42:710–721
Seymour PD, Zaslavski T (1984) Averaging sets: a generalization of mean values and spherical designs. Adv in Math 52:213–240
Singh M (1979) Group divisible second order rotatable designs. Biometrical J 21:579–589
Wedderburn JHM (1934) Lectures on Matrices. Colloquium Publ vol XVII. American Math Society, Providence RI
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Draper, N.R., Gaffke, N. & Pukelsheim, F. First and second order rotatability of experimental designs, moment matrices, and information surfaces. Metrika 38, 129–161 (1991). https://doi.org/10.1007/BF02613607
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DOI: https://doi.org/10.1007/BF02613607