459 research outputs found
Some results on designs of resolution IV with (weak) minimum aberration
It is known that all resolution IV regular designs of run size
where must be projections of the maximal even design
with factors and, therefore, are even designs. This paper derives a
general and explicit relationship between the wordlength pattern of any even
design and that of its complement in the maximal even design. Using
these identities, we identify some (weak) minimum aberration designs
of resolution IV and the structures of their complementary designs. Based on
these results, several families of minimum aberration designs of
resolution IV are constructed.Comment: Published in at http://dx.doi.org/10.1214/08-AOS670 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Uniform fractional factorial designs
The minimum aberration criterion has been frequently used in the selection of
fractional factorial designs with nominal factors. For designs with
quantitative factors, however, level permutation of factors could alter their
geometrical structures and statistical properties. In this paper uniformity is
used to further distinguish fractional factorial designs, besides the minimum
aberration criterion. We show that minimum aberration designs have low
discrepancies on average. An efficient method for constructing uniform minimum
aberration designs is proposed and optimal designs with 27 and 81 runs are
obtained for practical use. These designs have good uniformity and are
effective for studying quantitative factors.Comment: Published in at http://dx.doi.org/10.1214/12-AOS987 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Blocked regular fractional factorial designs with minimum aberration
This paper considers the construction of minimum aberration (MA) blocked
factorial designs. Based on coding theory, the concept of minimum moment
aberration due to Xu [Statist. Sinica 13 (2003) 691--708] for unblocked designs
is extended to blocked designs. The coding theory approach studies designs in a
row-wise fashion and therefore links blocked designs with nonregular and
supersaturated designs. A lower bound on blocked wordlength pattern is
established. It is shown that a blocked design has MA if it originates from an
unblocked MA design and achieves the lower bound. It is also shown that a
regular design can be partitioned into maximal blocks if and only if it
contains a row without zeros. Sufficient conditions are given for constructing
MA blocked designs from unblocked MA designs. The theory is then applied to
construct MA blocked designs for all 32 runs, 64 runs up to 32 factors, and all
81 runs with respect to four combined wordlength patterns.Comment: Published at http://dx.doi.org/10.1214/009053606000000777 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Theory and Construction Methods for Large Regular Resolution IV Designs
We define 2k-p fractional factorial designs which use all of their degrees of freedom to estimate main effects and two-factor interactions as second order saturated (sos) designs. We prove that resolution IV sos designs project to every other resolution IV design, and show the details of these projections for every n = 32 and n = 64 run fraction. For k \u3e (5/16)n, all resolution IV designs are a projection from the even sos design at k = n/2. For k ≤ (5/16)n the minimum aberration design resolution IV designs are projections of sos designs with both even and odd words in the defining relation. While even resolution IV designs are limited to estimating fewer than n/2 two-factor interactions (in addition to the k main effects), resolution IV designs with odd-length words in the defining relation may devote more than half of their degrees of freedom to two-factor interactions. We propose a method to search for good resolution IV designs using naïve projections from even/odd sos designs. We introduce the alias length pattern as a tool to help characterize designs. We describe how the matrix T = DD\u27 for a design D is useful in searching for designs. We list the resolution IV even/odd minimum aberration designs for n = 128 and provide a catalog of the best resolution IV even/odd designs for n = 128. These results are based on an isomorphic check using a convenient function of T, as well as the set of projections of a design. Finally, we suggest a new method for finding good regular resolution IV designs for large n (\u3e 128) and provide a preliminary table of good resolution IV even/odd designs for n = 256
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Minimum aberration designs for discrete choice experiments
A discrete choice experiment (DCE) is a survey method that givesinsight into individual preferences for particular attributes.Traditionally, methods for constructing DCEs focus on identifyingthe individual effect of each attribute (a main effect). However, aninteraction effect between two attributes (a two-factor interaction)better represents real-life trade-offs, and provides us a better understandingof subjects’ competing preferences. In practice it is oftenunknown which two-factor interactions are significant. To address theuncertainty, we propose the use of minimum aberration blockeddesigns to construct DCEs. Such designs maximize the number ofmodels with estimable two-factor interactions in a DCE with two-levelattributes. We further extend the minimum aberration criteria toDCEs with mixed-level attributes and develop some general theoreticalresults
Quarter-fraction factorial designs constructed via quaternary codes
The research of developing a general methodology for the construction of good
nonregular designs has been very active in the last decade. Recent research by
Xu and Wong [Statist. Sinica 17 (2007) 1191--1213] suggested a new class of
nonregular designs constructed from quaternary codes. This paper explores the
properties and uses of quaternary codes toward the construction of
quarter-fraction nonregular designs. Some theoretical results are obtained
regarding the aliasing structure of such designs. Optimal designs are
constructed under the maximum resolution, minimum aberration and maximum
projectivity criteria. These designs often have larger generalized resolution
and larger projectivity than regular designs of the same size. It is further
shown that some of these designs have generalized minimum aberration and
maximum projectivity among all possible designs.Comment: Published in at http://dx.doi.org/10.1214/08-AOS656 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Construction of optimal multi-level supersaturated designs
A supersaturated design is a design whose run size is not large enough for
estimating all the main effects. The goodness of multi-level supersaturated
designs can be judged by the generalized minimum aberration criterion proposed
by Xu and Wu [Ann. Statist. 29 (2001) 1066--1077]. A new lower bound is derived
and general construction methods are proposed for multi-level supersaturated
designs. Inspired by the Addelman--Kempthorne construction of orthogonal
arrays, several classes of optimal multi-level supersaturated designs are given
in explicit form: Columns are labeled with linear or quadratic polynomials and
rows are points over a finite field. Additive characters are used to study the
properties of resulting designs. Some small optimal supersaturated designs of
3, 4 and 5 levels are listed with their properties.Comment: Published at http://dx.doi.org/10.1214/009053605000000688 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Generalized resolution for orthogonal arrays
The generalized word length pattern of an orthogonal array allows a ranking
of orthogonal arrays in terms of the generalized minimum aberration criterion
(Xu and Wu [Ann. Statist. 29 (2001) 1066-1077]). We provide a statistical
interpretation for the number of shortest words of an orthogonal array in terms
of sums of values (based on orthogonal coding) or sums of squared
canonical correlations (based on arbitrary coding). Directly related to these
results, we derive two versions of generalized resolution for qualitative
factors, both of which are generalizations of the generalized resolution by
Deng and Tang [Statist. Sinica 9 (1999) 1071-1082] and Tang and Deng [Ann.
Statist. 27 (1999) 1914-1926]. We provide a sufficient condition for one of
these to attain its upper bound, and we provide explicit upper bounds for two
classes of symmetric designs. Factor-wise generalized resolution values provide
useful additional detail.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1205 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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