Statistical properties of Rechtschaffner designs

Rechtschaffner designs are saturated designs of resolution V   in which main effects and two-factor interactions are estimable if three-factor and higher order interactions are negligible. Statistical properties of Rechtschaffner designs are studied in this paper. Best linear unbiased estimators of main effects and two-factor interactions are given explicitly and asymptotic properties of correlations between these estimators are studied as well. It is shown that designs recommended by Rechtschaffner [1967. Saturated fractions of 2ⁿ$2^n$ and 3ⁿ$3^n$ factorial designs, Technometrics 9, 569–576] are not only A-optimal but also D-optimal. Comparisons of Rechtschaffner designs with other A- and D-optimal designs of resolution V are also discussed.     

Optimal block sequences for blocked fractional factorial split-plot designs

It is known that for blocked 2^n-k$2^{n-k}$ designs a judicious sequencing of blocks may allow one to obtain early and insightful results regarding influential parameters in the experiment. Such findings may justify the early termination of the experiment thereby producing cost and time savings. This paper introduces an approach for selecting the optimal sequence of blocks for regular two-level blocked fractional factorial split-plot screening experiments. An optimality criterion is developed so as to give priority to the early estimation of low-order factorial effects. This criterion is then applied to the minimum aberration blocked fractional factorial split-plot designs tabled in McLeod and Brewster [2004. The design of blocked fractional factorial split-plot experiments. Technometrics 46, 135–146]. We provide a catalog of optimal block sequences for 16 and 32-run minimum aberration blocked fractional factorial split-plot designs run in either 4 or 8 blocks.     

Invariance of generalized wordlength patterns

The generalized wordlength pattern (GWLP) introduced by Xu and Wu [2001. Generalized minimum aberration for asymmetrical fractional factorial designs. Ann. Statist. 29, 1066–1077] for an arbitrary fractional factorial design allows one to extend the use of the minimum aberration criterion to such designs. Ai and Zhang [2004. Projection justification of generalized minimum aberration for asymmetrical fractional factorial designs. Metrika 60, 279–285] defined the J$J$-characteristics of a design and showed that they uniquely determine the design. While both the GWLP and the J$J$-characteristics require indexing the levels of each factor by a cyclic group, we see that the definitions carry over with appropriate changes if instead one uses an arbitrary abelian group. This means that the original definitions rest on an arbitrary choice of group structure. We show that the GWLP of a design is independent of this choice, but that the J$J$-characteristics are not. We briefly discuss some implications of these results.     

An isomorphism check for two-level fractional factorial designs

Two fractional factorial designs are isomorphic if one can be obtained from the other by reordering the treatment combinations, relabelling the factor levels and relabelling the factors. By defining a word-pattern matrix, we are able to create a new isomorphism check which is much faster than existing checks for certain situations. We combine this with a new, extremely fast, sufficient condition for non-isomorphism to avoid checking certain cases. We then create a faster search algorithm by combining the Bingham and Sitter [1999. Minimum aberration fractional factorial split-plot designs. Technometrics 41, 62–70] search algorithm, the isomorphism check algorithm of Clark and Dean [2001. Equivalence of fractional factorial designs. Statist. Sinica 11, 537–547] with our proposed isomorphism check. The algorithm is used to extend the known set of existing non-isomorphic 128-run two-level regular designs with resolution ?4$?4$ to situations with 12, 13, 14, 15 and 16 factors, 256- and 512-run designs with resolution ?5$?5$ and ?17$?17$ factors and 1024-run even designs with resolution ?6$?6$ and ?18$?18$ factors.     

Utilization of higher order moments of scrambling variables in randomized response sampling

In this paper, a new estimator for estimating the proportion of a potentially sensitive attribute in survey sampling has been introduced. The proposed estimator makes use of higher order moments of the scrambling variable at the estimation stage. The proposed estimator has been found to be more efficient than the estimator due to Kuk [1990. Asking sensitive questions indirectly. Biomerika 77(2), 436–438] and Franklin [1989. A comparison of estimators for randomized response sampling with continuous distributions from a dichotomous population. Comm. Statist. Theory Methods 18, 489–505] type estimators in randomized response sampling. Recently, Guerriero and Sandri [2007. A note on the comparison of some randomized response procedures. J. Statist. Plann. Inference 137, 2184–2190] have shown that the family of randomized response models proposed by Kuk [1990. Asking sensitive questions indirectly. Biomerika 77(2), 436–438] is better than the Simmons’ family in terms of efficiency and protection.     

On E(s)-optimal supersaturated designs

A popular measure to assess 2-level supersaturated designs is the E(s²)$E(s^2)$ criterion. In this paper, improved lower bounds on E(s²)$E(s^2)$ are obtained. The same improvement has recently been established by Ryan and Bulutoglu [2007. E(s²)$E(s^2)$-optimal supersaturated designs with good minimax properties. J. Statist. Plann. Inference 137, 2250–2262]. However, our analysis provides more details on precisely when an improvement is possible, which is lacking in Ryan and Bulutoglu [2007. E(s²)$E(s^2)$-optimal supersaturated designs with good minimax properties. J. Statist. Plann. Inference 137, 2250–2262]. The equivalence of the bounds obtained by Butler et al. [2001. A general method of constructing E(s²)$E(s^2)$-optimal supersaturated designs. J. Roy. Statist. Soc. B 63, 621–632] (in the cases where their result applies) and those obtained by Bulutoglu and Cheng [2004. Construction of E(s²)$E(s^2)$-optimal supersaturated designs. Ann. Statist. 32, 1662–1678] is established. We also give two simple methods of constructing E(s²)$E(s^2)$-optimal designs.     

Identification of factors influencing dispersion in split-plot experiments

As split-plot designs are commonly used in robust design it is important to identify factors in these designs that influence the dispersion of the response variable. In this article, the Bergman-Hynén method, developed for identification of dispersion effects in unreplicated experiments, is modified to be used in the context of split-plot experiments. The modification of the Bergman-Hynén method enables identification of factors that influence specific variance components in unreplicated two-level fractional factorial splitplot experiments. An industrial example is used to illustrate the proposed method.     

On the equivalence of definitions for regular fractions of mixed-level factorial designs

The notion of regularity for fractional factorial designs was originally defined only for two-level factorial designs. Recently, rather different definitions for regular fractions of mixed-level factorial designs have been proposed by Collombier [1996. Plans d’Expérience Factoriels. Springer, Berlin], Wu and Hamada [2000. Experiments. Wiley, New York] and Pistone and Rogantin [2008. Indicator function and complex coding for mixed fractional factorial designs. J. Statist. Plann. Inference 138, 787–802]. In this paper we prove that, surprisingly, these definitions are equivalent. The proof of equivalence relies heavily on the character theory of finite Abelian groups. The group-theoretic framework provides a unified approach to deal with mixed-level factorial designs and treat symmetric factorial designs as a special case. We show how within this framework each regular fraction is uniquely characterized by a defining relation as for two-level factorial designs. The framework also allows us to extend the result that every regular fraction is an orthogonal array of a strength that is related to its resolution, as stated in Dey and Mukerjee [1999. Fractional Factorial Plans. Wiley, New York] to mixed-level factorial designs.     

Optimal mixed-level k-circulant supersaturated designs

Supersaturated designs (SSDs) offer a potentially useful way to investigate many factors with only few experiments in the preliminary stages of experimentation. This paper explores how to construct E(_fNOD)$E(f_{\mathrm{NOD}})$-optimal mixed-level SSDs using k-cyclic generators. The necessary and sufficient conditions for the existence of mixed-level k-circulant SSDs with the equal occurrence property are provided. Properties of the mixed-level k  -circulant SSDs are investigated, in particular, the sufficient condition under which the generator vector produces an E(_fNOD)$E(f_{\mathrm{NOD}})$-optimal SSD is obtained. Moreover, many new E(_fNOD)$E(f_{\mathrm{NOD}})$-optimal mixed-level SSDs are constructed and listed. The method here generalizes the one proposed by Liu and Dean [2004. k$k$-circulant supersaturated designs. Technometrics 46, 32–43] for two-level SSDs and the one due to Georgiou and Koukouvinos [2006. Multi-level k-circulant supersaturated designs. Metrika 64, 209–220] for the multi-level case.     

New 3-level response surface designs constructed from incomplete block designs

Box and Behnken [1958. Some new three level second-order designs for surface fitting. Statistical Technical Research Group Technical Report No. 26. Princeton University, Princeton, NJ; 1960. Some new three level designs for the study of quantitative variables. Technometrics 2, 455–475.] introduced a class of 3-level second-order designs for fitting the second-order response surface model. These 17 Box–Behnken designs (BB designs) are available for 3–12 and 16 factors. Although BB designs were developed nearly 50 years ago, they and the central-composite designs of Box and Wilson [1951. On the experimental attainment of optimum conditions. J. Royal Statist. Soc., Ser. B 13, 1–45.] are still the most often recommended response surface designs. Of the 17 aforementioned BB designs, 10 were constructed from balanced incomplete block designs (BIBDs) and seven were constructed from partially BIBDs (PBIBDs). In this paper we show that these seven BB designs constructed from PBIBDs can be improved in terms of rotatability as well as average prediction variance, D$D$- and G$G$-efficiency. In addition, we also report new orthogonally blocked solutions for 5, 8, 9, 11 and 13 factors. Note that an 11-factor BB design is available but cannot be orthogonally blocked. All new designs can be found at http://www.math.montana.edu/∼$\sim$jobo/bbd/.     

Consistency and asymptotic normality of the estimated effective doses in bioassay

In order to estimate the effective dose such as the 0.5 quantile _ED50${\mathrm{ED}}_{50}$ in a bioassay problem various parametric and semiparametric models have been used in the literature. If the true dose–response curve deviates significantly from the model, the estimates will generally be inconsistent. One strategy is to analyze the data making only a minimal assumption on the model, namely, that the dose–response curve is non-decreasing. In the present paper we first define an empirical dose–response curve based on the estimated response probabilities by using the “pool-adjacent-violators” (PAV) algorithm, then estimate effective doses _ED100p${\mathrm{ED}}_{100p}$ for a large range of p by taking inverse of this empirical dose–response curve. The consistency and asymptotic distribution of these estimated effective doses are obtained. The asymptotic results can be extended to the estimated effective doses proposed by Glasbey [1987. Tolerance-distribution-free analyses of quantal dose–response data. Appl. Statist. 36 (3), 251–259] and Schmoyer [1984. Sigmoidally constrained maximum likelihood estimation in quantal bioassay. J. Amer. Statist. Assoc. 79, 448–453] under the additional assumption that the dose–response curve is symmetric or sigmoidal. We give some simulations on constructing confidence intervals using different methods.     

Pseudo-Bayes and pseudo-empirical Bayes estimators in randomized response sampling

In this article, new pseudo-Bayes and pseudo-empirical Bayes estimators for estimating the proportion of a potentially sensitive attribute in a survey sampling have been introduced. The proposed estimators are compared with the recent estimator proposed by Odumade and Singh [Efficient use of two decks of cards in randomized response sampling, Comm. Statist. Theory Methods 38 (2009), pp. 439–446] and Warner [Randomized response: A survey technique for eliminating evasive answer bias, J. Amer. Statist. Assoc. 60 (1965), pp. 63–69].     

Alternatives to maximum likelihood estimation based on spacings and the Kullback–Leibler divergence

An alternative to the maximum likelihood (ML) method, the maximum spacing (MSP) method, is introduced in Cheng and Amin [1983. Estimating parameters in continuous univariate distributions with a shifted origin. J. Roy. Statist. Soc. Ser. B 45, 394–403], and independently in Ranneby [1984. The maximum spacing method. An estimation method related to the maximum likelihood method. Scand. J. Statist. 11, 93–112]. The method, as described by Ranneby [1984. The maximum spacing method. An estimation method related to the maximum likelihood method. Scand. J. Statist. 11, 93–112], is derived from an approximation of the Kullback–Leibler divergence. Since the introduction of the MSP method, several closely related methods have been suggested. This article is a survey of such methods based on spacings and the Kullback–Leibler divergence. These estimation methods possess good properties and they work in situations where the ML method does not. Important issues such as the handling of ties and incomplete data are discussed, and it is argued that by using Moran's [1951. The random division of an interval—Part II. J. Roy. Statist. Soc. Ser. B 13, 147–150] statistic, on which the MSP method is based, we can effectively combine: (a) a test on whether an assigned model of distribution functions is correct or not, (b) an asymptotically efficient estimation of an unknown parameter _θ0${\theta }_0$, and (c) a computation of a confidence region for _θ0${\theta }_0$.     

Designs for cropping systems research

The present article establishes equivalence between extended group divisible (EGD) designs and designs for crop sequence experiments. This equivalence has encouraged the agricultural experimenters to use EGD designs for their experimentation. Some real life applications of EGD designs have been given. It has also been shown that several existing association schemes are special cases of EGD association scheme. Some methods of construction of EGD designs are also given. A catalogue of EGD designs obtainable through methods of construction along with efficiency factors of various factorial effects is also presented. In some crop sequence experiments that are conducted to develop suitable integrated nutrient supply system of a crop sequence, the treatments do not comprise of a complete factorial structure. The experimenter is interested in estimating the residual and direct effect of the treatments along with their cumulative effects. For such experimental settings block designs with two sets of treatments applied in succession are the appropriate designs. The correspondence established between row–column designs and block designs for two stage experiments by Parsad et al. [2003. Structurally incomplete row–column designs. Comm. Statist. Theory Methods 32(1), 239–261] has been exploited in obtaining designs for such experimental situations. Some open problems related to designing of crop sequence experiments are also given.     

Supersaturated designs with high searching probability

A supersaturated design is essentially a fractional factorial design whose number of experimental variables is greater than or equal to its number of experimental runs. Under the effect sparsity assumption, a supersaturated design can be very cost-effective. In this paper, our prime objective is to compare the existing two-level supersaturated designs for the noisy case through the probability of correct searching—a powerful criterion proposed by Shirakura et al. [1996. Searching probabilities for nonzeroeffects in search designs for the noisy case. Ann. Statist. 24, 2560–2568]. An algorithm is proposed to construct supersaturated designs with high probability of correct searching. Examples are given for illustration.     

Minimum aberration blocking schemes for 128-run designs

Several criteria have been proposed for ranking blocked fractional factorial designs. For large fractional factorial designs, the most appropriate minimum aberration criterion was one proposed by Cheng and Wu (2002). We justify this assertion and propose a novel construction method to overcome the computational challenge encountered in large fractional factorial designs. Tables of minimum aberration blocked designs are presented for N=128 runs and n=8–64 factors.     

Experimental design for product improvement

Fractional factorial designs have been used successfullyin industry and elsewhere todetect and estimate sparse factor effects , The effectsusually evisioned measure changes in location associated with the experimental factors . Here we consider the possibility of detecting and estimating sparse dispersion effects measuring changes in variance associated with the factors .

( 2 ) In industrial experimentation it is frequently true thata large proportion of process v ariation is associated with a smallproportion of the process variables . In such circum stancs of“effect sparsity”unreplicated fractional designs have frequently been effectivein islolating preponderant factors.A very useful graphical analysis for such experiments

due to Cuthbert Daniel(1959)employs normal probability plotting.A more formal analysis is presented here which might be used to supplement such plots.     

D-Optimal Two-Level Fractional Factorial Designs of Resolution III with Two Dispersion Factors

The problem of finding D-optimal designs, with two dispersion factors, for the estimation of all location main effects is investigated in the class of regular unreplicated two-level fractional factorial designs of resolution III. Designs having length three words involving both of the dispersion factors in the defining relation are shown to be inferior in terms of D-optimality. Tables of factors that are named as the two dispersion factors so that the resulting design is either D-optimal or has the largest determinant of the information matrix are provided. Rank-order of designs is studied when the number of length three words involving either one of the dispersion factors and the number of length four words involving both of the dispersion factors are fixed. Rank-order of designs when the numbers of aforementioned words are less than or equal to ten is given.     

A note on the selection of optimal foldover plans for 16- and 32-run fractional factorial designs

The minimum aberration criterion has been advocated for ranking foldovers of 2^k−p$2^{k-p}$ fractional factorial designs (Li and Lin, 2003); however, a minimum aberration design may not maximize the number of clear low-order effects. We propose using foldover plans that sequentially maximize the number of clear low-order effects in the combined (initial plus foldover) design and investigate the extent to which these foldover plans differ from those that are optimal under the minimum aberration criterion. A small catalog is provided to summarize the results.