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

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 $R^2$ 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

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

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

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

A trigonometric approach to quaternary code designs with application to one-eighth and one-sixteenth fractions

The study of good nonregular fractional factorial designs has received significant attention over the last two decades. Recent research indicates that designs constructed from quaternary codes (QC) are very promising in this regard. The present paper shows how a trigonometric approach can facilitate a systematic understanding of such QC designs and lead to new theoretical results covering hitherto unexplored situations. We focus attention on one-eighth and one-sixteenth fractions of two-level factorials and show that optimal QC designs often have larger generalized resolution and projectivity than comparable regular designs. Moreover, some of these designs are found to have maximum projectivity among all designs.Comment: Published in at http://dx.doi.org/10.1214/10-AOS815 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A complementary design theory for doubling

Chen and Cheng [Ann. Statist. 34 (2006) 546--558] discussed the method of doubling for constructing two-level fractional factorial designs. They showed that for $9N/32\le n\le 5N/16$, all minimum aberration designs with $N$ runs and $n$ factors are projections of the maximal design with $5N/16$ factors which is constructed by repeatedly doubling the $2^{5-1}$ design defined by $I=ABCDE$. This paper develops a general complementary design theory for doubling. For any design obtained by repeated doubling, general identities are established to link the wordlength patterns of each pair of complementary projection designs. A rule is developed for choosing minimum aberration projection designs from the maximal design with $5N/16$ factors. It is further shown that for $17N/64\le n\le 5N/16$, all minimum aberration designs with $N$ runs and $n$ factors are projections of the maximal design with $N$ runs and $5N/16$ factors.Comment: Published in at http://dx.doi.org/10.1214/009005360700000712 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org