Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. The traditional analysis focuses on main e�ffects only. Hamada and Wu (1992) went beyond the traditional approach and proposed an analysis strategy to demonstrate that some interactions could be entertained and estimated beyond a few significant main effects. Their groundbreaking work stimulated much of the recent developments in design criterion creation, construction and analysis of nonregular designs. This paper reviews important developments in optimality criteria and comparison, including projection properties, generalized resolution, various generalized minimum aberration criteria, optimality results, construction methods and analysis strategies for nonregular designs.

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Recent developments in nonregular fractional factorial designs

Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. The traditional analysis focuses on main effects only. Hamada and Wu (1992) went beyond the traditional approach and proposed an analysis strategy to demonstrate that some interactions could be entertained and estimated beyond a few significant main effects. Their groundbreaking work stimulated much of the recent developments in design criterion creation, construction and analysis of nonregular designs. This paper reviews important developments in optimality criteria and comparison, including projection properties, generalized resolution, various generalized minimum aberration criteria, optimality results, construction methods and analysis strategies for nonregular designs.

Recent Developments in Nonregular Fractional Factorial Designs

(k, l) [1885]. (n− k + 1) [1753, 1623]. 0 [772]. 0 < ρ < ρ0 [2026]. 1 [772, 548]. $109.95 [2218]. $125.00 [2161]. $129.95 [2259]. 1 ≤ p <∞ [562]. 2 [616, 407]. $229.00 [2019]. 24 [1398]. 24 [1476]. 2s−k [2039]. 3 [852, 2125]. $49.95 [2161]. $59.95 [2219]. $89.95 [2217]. (R) [2002, 1962]. 1 [1276]. 2 [1417]. k [798]. A [1577, 2312, 2346]. α [858, 1840, 2242, 1465, 1271, 1945, 2327, 1926]. AR(1) [1650, 790]. AR(p) [1346]. R̄ [652]. β [668, 856, 497]. c [2332]. χ [919]. D [2135, 2268, 836, 948, 1324, 2309, 2346, 2098, 2176]. δ [1959]. e [596]. E(x⊗ xx′) [1805]. E(xx′ ⊗ xx′) [1805]. F [1901, 915, 1361, 407, 681]. G [548]. Γ [722]. H [1664]. I(1) [845]. K [2308, 1551, 1223, 1837, 368, 479, 2177, 1322, 1198, 2224, 2309, 407, 1932]. k − 2 [1781]. L [439, 1974, 1584, 1399, 1322]. L1 [895]. L2 [2195, 2363, 2039]. LP [562]. LM [1343]. LR [1343]. M [1054, 1107, 2094, 2180]. Lp [1552]. N [816, 752, 48, 1753, 1623, 548]. p [1245, 1461, 2378, 1794, 1868, 1604, 1791, 1589, 2074, 2227]. P (X < Y ) [1867, 1492, 1955]. P (Y < X) [2222, 1380]. Φ [1345]. Pr(a < x < b) [263]. q [1651]. R [1238, 1848, 616, 1932]. r − k [2082, 1480]. R/S [1996]. R = Pr(X > Y ) [2007]. r [1446]. ρ∗ [1884]. S [2151, 1105]. S [848, 790, 909]. σ [1141, 684]. SUNn,2 [2046]. t [1054, 1777, 327, 1533, 1722, 982, 856, 1031, 1780, 1559, 1094, 1391, 1342, 1680]. T 2

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A Complete Bibliography of Publications in Statistical Papers

Many construction methods for (nearly) uniform designs have been proposed under the centered $$L_2$$
 -discrepancy, and most of them are only suitable for constructing designs with small size. This paper proposes a new method, called mixture method (MM), to construct nearly symmetrical/asymmetrical uniform designs with large number of runs and/or large number of factors. The new method has the “better than given” property, i.e., the resulting design is better than existing designs in the sense of the pre-decided criterion. Moreover, the computational speed of MM is faster than most existing methods.

An efficient method for constructing uniform designs with large size

Designing their experiments is the significant problem that experimenters face. Maximin distance designs, supersaturated designs, minimum aberration designs, uniform designs, minimum moment designs and orthogonal arrays are arguably the most exceedingly used designs for many real-life experiments. From different perspectives, several criteria have been proposed for constructing these designs for investigating quantitative or qualitative factors. Each of those criteria has its pros and cons and thus an optimal criterion does not exist, which may confuse investigators searching for a suitable criterion for their experiment. Some logical questions are now arising, such as are these designs consistent, can an optimal design via a specific criterion perform well based on another criterion and can an optimal design for screening quantitative factors be optimal for qualitative factors? Through theoretical justifications, this paper tries to answer these interesting questions by building some bridges among these designs. Some conditions under which these designs agree with each other are discussed. These bridges can be used to select a suitable criterion for studying some hard problems effectively, such as detection of (combinatorial/geometrical) non-isomorphism among designs and construction of optimal designs. Benchmarks for reducing the computational complexity are given.

Building some bridges among various experimental designs

Discrete discrepancy has been utilized as a uniformity measure for comparing and evaluating factorial designs. In this paper, for asymmetrical factorials, we give some linkages between uniformity measured by the discrete discrepancy and other criteria, such as generalized minimum aberration (Xu and Wu, 2001) and minimum projection variance (Ai and Zhang, 2004). These close linkages show a significant justification for the discrete discrepancy used to measure uniformity of factorial designs, and provide an additional rationale for using uniform designs.

Hong Qin

Papers

A note on the connection between uniformity and generalized minimum aberration