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October 2021 Two-level parallel flats designs
Chunyan Wang, Robert W. Mee
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Ann. Statist. 49(5): 3015-3042 (October 2021). DOI: 10.1214/21-AOS2071


Regular 2np designs are also known as single flat designs. Parallel flats designs (PFDs) consisting of three parallel flats (3-PFDs) are the most frequently utilized PFDs, due to their simple structure. Generalizing to f-PFD with f>3 is more challenging. This paper aims to study the general theory for the f-PFD for any f3. We propose a method for obtaining the confounding frequency vectors for all nonequivalent f-PFDs, and to find the least G-aberration (or highest D-efficiency) f-PFD constructed from any single flat. PFDs are particularly useful for constructing nonregular fraction, split-plot or randomized block designs. We also characterize the quaternary code design series as PFDs. Finally, we show how designs constructed by concatenating regular fractions from different families may also have a parallel flats structure. Examples are given throughout to illustrate the results.


This research was accomplished during the first author’s 8-month research visit at the University of Tennessee, a visit made possible by Professor M.Q. Liu’s National Natural Science Foundation of China (Grant No. 11771220) and the National Ten Thousand Talents Program. We are also grateful to Boxin Tang and David Edwards for their helpful suggestions, and to a referee for suggesting that we explore concatenation of designs from different families.


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Chunyan Wang. Robert W. Mee. "Two-level parallel flats designs." Ann. Statist. 49 (5) 3015 - 3042, October 2021.


Received: 1 June 2020; Revised: 1 January 2021; Published: October 2021
First available in Project Euclid: 12 November 2021

Digital Object Identifier: 10.1214/21-AOS2071

Primary: 62K15
Secondary: 62K05

Keywords: Confounding frequency vector , equivalent design , G-aberration , nonregular fractional factorial design , quaternary code , Sylvester Hadamard matrix , two-factor interaction model

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 5 • October 2021
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