Projection justification of generalized minimum aberration for asymmetrical fractional factorial designs

Recently, Xu and Wu (2001) presented generalized minimum aberration criterion for comparing and selecting general fractional factorial designs. This criterion is defined using a set of _u(D) values, called J-characteristics by us. In this paper, we find a set of linear equations that relate the set of design points to that of J-characteristics, which implies that a factorial design is uniquely determined by its J-characteristics once the orthonormal contrasts are designated. Thereto, a projection justification of generalized minimum aberration is established. All of these conclusions generalize the results for two-level symmetrical factorial designs in Tang (2001).Acknowledgements The authors are grateful to the editor, the associate editor and the referees for their valuable comments. This paper is supported by NNSF of P.R.China grant No. 10171051. and RFDP grant No. 1999005512.     

Discrete discrepancy in factorial designs

Discrepancy measure can be utilized as a uniformity measure for comparing factorial designs. A so-called discrete discrepancy has been used to evaluate the uniformity of factorials. In this paper we give linkages among uniformity measured by the discrete discrepancy, generalized minimum aberration, minimum moment aberration and uniformity measured by the centered L₂-discrepancy/the wrap-around L₂-discrepancy. These close linkages provide a significant justification for the discrete discrepancy used to measure uniformity of factorial designs.     

Choice of optimal initial designs in sequential experiments

Combined-optimal designs (Li and Lin, 2003) are obviously the best choices for the initial designs if we partition the experiment into two parts with equal size to obtain some information about the process, especially for the case not considering the blocking factor. In this paper, the definition of combined-optimal design is extended to the case when blocking factor is significant, and this new class of designs is called blocked combined-optimal designs. Some general results are obtained which relate 2^k−p_III initial designs with their complementary designs when , where n=2^k−p. By applying these results, we are able to characterize 2^k−p_III combined-optimal designs or blocked combined-optimal designs in terms of their complementary designs. It is also proved that both 2^k−p_III combined-optimal and blocked combined-optimal designs are not minimum aberration designs when and n−1−k > 2. And some combined-optimal and blocked combined-optimal designs with 16 and 32 runs are constructed for illustration. 2000 Mathematics Subject Classifications: 62K15, 62K05     

Optimal criteria and equivalence for nonregular fractional factorial designs

By considering the so-called algebraic orthogonality among experimental runs, this paper introduces a new criterion, minimum inner-product moment (MIPM), for general asymmetrical designs, and shows that MIPM is equivalent to the minimum moment aberration (MMA) criterion for the natural weights. Furthermore, the relationship between the generalized minimum aberration (GMA) and some model-dependent efficiency criteria is investigated by using the complex contrasts. Thus, two new justifications of GMA criterion is given from the points of view of orthogonality among experimental runs and design efficiency. They are generalizations of the related results of Butler (2003) and Cheng, Deng, and Tang (2002) for two-level factorial designs.     

A note on the robustness of Box-Behnken designs to the unavailability of data

Box-Behnken designs and central composite designs are efficient designs for fitting second order polynomials to response surfaces, because they use relatively small numbers of observations to estimate the parameters. In this paper we investigate the robustness of Box-Behnken designs to the unavailability of observations, in the sense of finding t _max , the maximum number of arbitrary rows in the design matrix that can be removed and still leave all of the parameters of interest estimable. The results are compared to the known results for the central composite designs found in MacEachern, Notz, Whittinghill & Zhu (1995). The blocked Box-Behnken designs are equally as robust as those that are not blocked. Received December 1997     

A comment on Mendeş and Yiğit (2013), ‘Type I error and test power of different tests for testing interaction effects in factorial experiments’, Statistica Neerlandica, 67:1–26

In their advocacy of the rank‐transformation (RT) technique for analysis of data from factorial designs, Mende? and Yi?it (Statistica Neerlandica, 67, 2013, 1–26) missed important analytical studies identifying the statistical shortcomings of the RT technique, the recommendation that the RT technique not be used, and important advances that have been made for properly analyzing data in a non‐parametric setting. Applied data analysts are at risk of being misled by Mende? and Yi?it, when statistically sound techniques are available for the proper non‐parametric analysis of data from factorial designs. The appropriate methods express hypotheses in terms of normalized distribution functions, and the test statistics account for variance heterogeneity.