A theory on constructing blocked two-level designs with general minimum lower order confounding

Abstract

Completely random allocation of the treatment combinations to the experimental units is appropriate only if the experimental units are homogeneous. Such homogeneity may not always be guaranteed when the size of the experiment is relatively large. Suitably partitioning inhomogeneous units into homogeneous groups, known as blocks, is a practical design strategy. How to partition the experimental units for a given design is an important issue. The blocked general minimum lower order confounding is a new criterion for selecting blocked designs. With the help of doubling theory and second order saturated design, we present a theory on constructing optimal blocked designs under the blocked general minimum lower order confounding criterion.

Keywords

Aliased effect-number pattern / general minimum lower order confounding / second order saturated design / Yates order

References

[1] Ai M Y, He S Y. An efficient method for identifying clear effects in blocked fractional factorial designs. Statist Probab Lett, 2006, 76: 1889–1894

[2] Block R M, Mee R W. Second order saturated resolution IV designs. J Stat Theory Appl, 2003, 2: 96–112

[3] Chen B J, Li P F, Liu M Q, Zhang R C. Some results on blocked regular 2-level fractional factorial designs with clear effects. J Statist Plann Inference, 2006, 136:4436–4449

[4] Chen H, Cheng C-S. Theory of optimal blocking of 2^n–m designs. Ann Statist, 1999, 27: 1948–1973

[5] Chen H, Cheng C-S. Doubling and projection: a method of constructing two-level designs of resolution IV. Ann Statist, 2006, 34: 546–558

[6] Chen J, Liu M Q. Some theory for constructing general minimum lower order confounding designs. Statist Sinica, 2011, 21: 1541–1555

[7] Cheng C-S, Mukerjee R. Blocked regular fractional factorial designs with maximum estimation capacity. Ann Statist, 2001, 29: 530–548

[8] Cheng Y, Zhang R C. On construction of general minimum lower order confounding 2^n−m designs with N4 +1≤n≤9N/32. J Statist Plann Inference, 2010, 140: 2384–2394

[9] Davydov A A, Tombak L M. Quasi-perfect linear binary codes with distance 4 and complete caps in projective geometry. Probl Inf Transm, 1990, 25: 265–275

[10] Fries A, Hunter W G. Minimum aberration 2k−p designs. Technometrics, 1980, 22: 601–608

[11] Li P F, Zhao S L, Zhang R C. A theory on constructing 2^n−m designs with general minimum lower-order confounding. Statist Sinica, 2011, 21: 1571–1589

[12] Sitter R R, Chen J, Feder M. Fractional resolution and minimum aberration in blocked 2^n−k designs. Technometrics, 1997, 39: 382–390

[13] Sun D X. Estimation capacity and related topics in experimental designs. Ph D Dissertation. Waterloo: University of Waterloo, 1993

[14] Wei J L, Li P, Zhang R C. Blocked two-level regular designs with general minimum lower order confounding. J Stat Theory Pract, 2014, 8: 46–65

[15] Wu C F J, Chen Y. A graph-aided method for planning two-level experiments when certain interactions are important. Technometrics, 1992, 34: 162–175

[16] Xu H Q. Blocked regular fractional factorial designs with minimum aberration. Ann Statist, 2006, 34: 2534–2553

[17] Xu H Q, Mee R W. Minimum aberration blocking schemes for 128-run designs. J Statist Plann Inference, 2010, 140: 3213–3229

[18] Zhang R C, Cheng Y. General minimum lower order confounding designs: an overview and a construction theory. J Statist Plann Inference, 2010, 140: 1719–1730

[19] Zhang R C, Li P, Zhao S L, Ai M Y. A general minimum lower-order confounding criterion for two-level regular designs. Statist Sinica, 2008, 18: 1689–1705

[20] Zhang R C, Mukerjee R. Characterization of general minimum lower order confounding via complementary sets. Statist Sinica, 2009, 19: 363–375

[21] Zhang R C, Mukerjee R. General minimum lower order confounding in block designs using complementary sets. Statist Sinica, 2009, 19: 1787–1802

[22] Zhang R C, Park D K. Optimal blocking of two-level fractional factorial designs. J Statist Plann Inference, 2000, 91: 107–121

[23] Zhao S L, Li P F, Karunamuni R. Blocked two-level regular factorial designs with weak minimum aberration. Biometrika, 2013, 100: 249–253

[24] Zhao S L, Li P F, Liu M Q. On blocked resolution IV designs containing clear twofactor interactions. J Complexity, 2013, 29: 389–395

[25] Zhao S L, Li P F, Zhang R C, Karunamuni R. Construction of blocked two-level regular designs with general minimum lower order confounding. J Statist Plann Inference, 2013,143: 1082–1090

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg