Supersaturated design including an orthogonal base

Recently, many supersaturated designs have been proposed. A supersaturated design is a fractional factorial design in which the number of factors is greater than the number of experimental runs. The main thrust of the previous studies has been to generate more columns while avoiding large values of squared inner products among all design columns. These designs would be appropriate if the probability for each factor being active is uniformly distributed. When factors can be partitioned into two groups, namely, with high and low probabilities of each factor being active, it is desirable to maintain orthogonality among columns to be assigned to the factors in the high-probability group. We discuss a supersaturated design including an orthogonal base which is suitable for this common situation. Mathematical results on the existence of the supersaturated designs are shown, and the construction of supersaturated designs is presented. We next discuss some properties of the proposed supersaturated designs based on the squared inner products.     

Analysis of a supersaturated design using Entropy Prior Complexity for binary responses via generalized linear models

A supersaturated design is a factorial design in which the number of effects to be estimated is greater than the available number of experimental runs. It is used in many experiments for screening purposes, i.e., for studying a large number of factors and then identifying the active ones. The goal with such a design is to identify just a few of the factors under consideration, that have dominant effects and to do this at minimum cost. While most of the literature on supersaturated designs has focused on the construction of designs and their optimality, the data analysis of such designs remains still at an early stage. In this paper, we incorporate the parameter model complexity into the supersaturated design analysis process, by assuming generalized linear models for a Bernoulli response, for analyzing main effects designs and discovering simultaneously the effects that are significant.     

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.     

Construction of three-level supersaturated design

Supersaturated design is one type of fractional factorial design where the number of columns is greater than the number of rows. This type of design would be useful when costs of experiments are expensive and the number of factors is large, and there is a limitation on the number of runs. This paper presents some theorems on three-level supersaturated design and their application to construction. The construction methods proposed in this paper can be regarded as an extension of the methods developed for two-level supersaturated designs.     

Construction of Efficient Multi-Level k-Circulant Supersaturated Designs

This article proposes an algorithm to construct efficient balanced multi-level k-circulant supersaturated designs with m factors and n runs. The algorithm generates efficient balanced multi-level k-circulant supersaturated designs very fast. Using the proposed algorithm many balanced multi-level supersaturated designs are constructed and cataloged. A list of many optimal and near optimal, multi-level supersaturated designs is also provided for m ≤ 60 and number of levels (q) ≤10. The algorithm can be used to generate two-level k-circulant supersaturated designs also and some large optimal two-level supersaturated designs are presented. An upper bound to the number of factors in a balanced multi-level supersaturated design such that no two columns are fully aliased is also provided.     

An optimality criterion for supersaturated designs with quantitative factors

A supersaturated design (SSD) is a factorial design in which the degrees of freedom for all its main effects exceed the total number of distinct factorial level-combinations (runs) of the design. Designs with quantitative factors, in which level permutation within one or more factors could result in different geometrical structures, are very different from designs with nominal ones which have been treated as traditional designs. In this paper, a new criterion is proposed for SSDs with quantitative factors. Comparison and analysis for this new criterion are made. It is shown that the proposed criterion has a high efficiency in discriminating geometrically nonisomorphic designs and an advantage in computation.     

Lower bounds of various criteria in experimental designs

Criterion is essential for measuring the goodness of an experimental design. In this paper, lower bounds of various criteria in experimental designs will be reviewed according to methodology of their construction. The criteria include most well-known ones which are frequently used as benchmarks for orthogonal array, uniform design, supersaturated design and other types of designs. To derive the lower bounds of these criteria, five different approaches are explored. Some new results are given. Throughout the paper, some relationships among different types of lower bounds are also discussed.     

A Method for Analyzing Supersaturated Designs

ABSTRACT

Analyzing supersaturated designs is challenging because the number of experiments is less than the number of factors. In this article we propose a new contrasts based method to analyze supersaturated designs. The method is discussed and explained through some simulation examples. The performance of the method is evaluated using several known designs from the literature.     

Adjusted Wilcoxon signed rank test tables for ratio of percentiles

The ordinary Wilcoxon signed rank test table provides confidence intervals for the median of one population. Adjusted Wilcoxon signed rank test tables which can provide confidence intervals for the median and the 10th percentile of one population are created in this paper. Base-(n + 1) number system and theorems about property of symmetry of the adjusted Wilcoxon signed rank test statistic are derived for programming. Theorem 1 states that the adjusted Wilcoxon signed rank test statistic are symmetric around n(n + 1)/4. Theorem 2 states that the adjusted Wilcoxon signed rank test statistic with the same number of negative ranks m are symmetric around m(n+1)/2. 87.5% and 85% confidence intervals of the median are given in the table for n = 12, 13,…, 29 to create approximated 95% confidence intervals of the ratio of medians for two independent populations. 95% and 92.5% confidence intervals of the 10th percentile are given in the table for n = 26, 27, 28, 29 to create approximated 95% confidence regions of the ratio of the 10th percentiles for two independent populations. Finally two large datasets from wood industry will be partitioned to verify the correctness of adjusted Wilcoxon signed rank test tables for small samples.     

Model-robust supersaturated and partially supersaturated designs

Supersaturated designs are an increasingly popular tool for screening factors in the presence of effect sparsity. The advantage of this class of designs over resolution III factorial designs or Plackett–Burman designs is that n, the number of runs, can be substantially smaller than the number of factors, m. A limitation associated with most supersaturated designs produced thus far is that the capability of these designs for estimating g active effects has not been discussed. In addition to exploring this capability, we develop a new class of model-robust supersaturated designs that, for a given n and m, maximizes the number g   of active effects that can be estimated simultaneously. The capabilities of model-robust supersaturated designs for model discrimination are assessed using a model-discrimination criterion, the subspace angle. Finally, we introduce the class of partially supersaturated designs, intended for use when we require a specific subset of _m1$m_1$ core factors to be estimable, and the sparsity of effects principle applies to the remaining (m-_m1$m-m_1$) factors.     

Optimality of mixed-level supersaturated designs

This paper presents generalized theorems on the optimality of supersaturated designs in terms of low dependency over all pairs of column vectors. Some mixed-level supersaturated designs are constructed using a method based on these theorems. An index is proposed for measuring the efficiency of supersaturated design and applied to evaluate the constructed mixed-level supersaturated designs.     

On the number of 2n-p fractional factorials of resolution III

An algorithm is specified and demonstrated which will compute the total number of ways a 2ⁿ factorial design may be partitioned into 2^p mutually exclusive 2^n-p fractional factorial designs, each having resolution III. The results of its application to all designs possessing resolution III fractions for n=5,…,20 are also given.     

Minimum Cost Linear Trend Free 2n-(n-k) Designs of Resolution IV

This article extends the resolution of time trend free designs for sequential 2^n-p experiments from III into IV and minimizes the number of factor level changes between runs (i.e., cost) by constructing a catalog of (2^k?2 ?1) minimum cost linear trend free resolution IV 2^n?(n?k) designs (2^k?2 ≤ n ≤ 2^k?1?2) from the full 2^k factorial experiment using the interactions-main effects assignment technique. Each systematic 2^n?(n?k) design in the catalog is economic in minimum number of factor level changes and allows for the estimation of all n main effects unbiased by either the linear time trend (which may be present in the 2^n?(n?k) sequentially generated responses) or the non negligible two-factor interactions. This article provides for each 2^n?(n?k) design: (1) the defining relation or the alias structure; (2) the k independent generators for sequencing the 2^n?(n?k) runs by the generalized foldover scheme; and (3) the minimum cost represented by the total number of factor level changes between the 2^n?(n?k) runs. All k main effects of the 2^k experiment are excluded from the selection assignment process due to their nonlinear time trend resistance as well as excluding a total of (2^k?1 –k +1) interactions violating the resolution IV requirement.     

A general method of constructing E(s2)-optimal supersaturated designs

There has been much recent interest in supersaturated designs and their application in factor screening experiments. Supersaturated designs have mainly been constructed by using the E ( s ²)-optimality criterion originally proposed by Booth and Cox in 1962. However, until now E ( s ²)-optimal designs have only been established with certainty for n experimental runs when the number of factors m is a multiple of n-1 , and in adjacent cases where m = q ( n -1) + r (| r | 2, q an integer). A method of constructing E ( s ²)-optimal designs is presented which allows a reasonably complete solution to be found for various numbers of runs n including n ,=8 12, 16, 20, 24, 32, 40, 48, 64.     

C334. On the neutron and proton masses: a numerological case study

The intercomponent rank test suggested by Thompson (1991a) for the bivariate two sample problem is compared with the intracomponent rank test discussed by Puri and Sen (1971) and Hettmansperger (1984) and with the Hotelling T ² test. Asymptotic relative efficiencies are discussed and the results of a simulation study are presented. Power studies show that for small sample sizes and small Type 1 error rates, say n = 5 and α = .01, the intercomponent rank test of Thompson (1991a) is somewhat liberal and the intracomponent test is quite conservative. For larger sample sizes and larger Type 1 error rates, both rank tests have improved properties under the null hypothesis. In almost all simulated cases, the intercomponent test is more powerful. In light of these studies it is suggested that the intercomponent rank test of Thompson (1991a), which has the added advantage of being easily computed with standard statistical software, is a strong competitor to the intracomponent rank test.     

Balanced 2m factorial designs of resolution v which allow search and estimation of one extra unknown effect, 4 ≤ m ≤ 8

In this paper, we obtain balanced resolution V plans for 2^m factorial experiments (4 ≤ m ≤ 8), which have an additional feature. Instead of assuming that the three factor and higher order effects are all zero, we assume that there is at most one nonnegligible effect among them; however, we do not know which particular effect is nonnegligible. The problem is to search which effect is non-negligible and to estimate it, along with estimating the main effects and two factor interactions etc., as in an ordinary resolution V design. For every value of N (the number of treatments) within a certain practical range, we present a design using which the search and estimation can be carried out. (Of course, as in all statistical problems, the probability of correct search will depend upon the size of “error” or “noise” present in the observations. However, the designs obtained are such that, at least in the noiseless case, this probability equals 1.) It is found that many of these designs are identical with optimal balanced resolution V designs obtained earlier in the work of Srivastava and Chopra.     

Analyzing supersaturated designs with entropic measures

A supersaturated design is a design for which there are fewer runs than effects to be estimated. In this paper, we propose a method for screening out the important factors from a large set of potentially active variables, based on an information theoretical approach. Three entropy measures: Rényi entropy, Tsallis entropy and Havrda–Charvát entropy, have been associated with the measure of information gain, in order to identify the significant factors using data and assuming generalized linear models. The investigation of the proposed method performance and the comparison of each entropic measure application have been accomplished through simulation experiments. A noteworthy advantage of this paper is the use of generalized linear models for analyzing data from supersaturated designs, a fact that, to the best of our knowledge, has not yet been studied.     

Aligned rank statistics for repeated measurement models with orthonormal design, employing a Chernoff–Savage approach

In this paper, aligned rank statistics are considered for testing hypotheses regarding the location in repeated measurement designs, where the design matrix for each set of measurements is orthonormal. Such a design may, for instance, be used when testing for linearity. It turns out that the centered design matrix is not of full rank, and therefore it does not quite satisfy the usual conditions in the literature. The number of degrees of freedom of the limiting chi-square distribution of the test statistic under the null hypothesis, however, is not affected, unless rather special hypotheses are tested. An independent derivation of this limiting distribution is given, using the Chernoff–Savage approach. In passing, it is observed that independence of the choice of aligner, which in the location problem is well-known to be due to cancellation, may in scale problems occur as a result of the type of score function suitable for scale tests. A possible extension to multivariate data is briefly indicated.     

The Effectiveness of Risk Scores: the Logit Rank Plot

A risk score s for event E is a function of covariates with the property that P ( E | s ) is an increasing function of s . Motivated by applications in medicine and in criminology, we suggest the logit rank plot as a good way of summarizing the effectiveness of such a score. Explicitly, plot logit{ P ( E | s )} against logit( r ), where r is the proportional rank of s in a sample or population. The slope of this plot gives an overall measure of effectiveness, and the logit rank transformation provides a common basis on which different risk scores can be compared. Some practical and theoretical aspects are discussed.     

A method for screening active effects in supersaturated designs

A supersaturated design (SSD) is a design whose run size is not enough for estimating all the main effects. The goal in conducting such a design is to identify, presumably only a few, relatively dominant active effects with a cost as low as possible. However, data analysis of such designs remains primitive: traditional approaches are not appropriate in such a situation and several methods which were proposed in the literature in recent years are effective when used to analyze two-level SSDs. In this paper, we introduce a variable selection procedure, called the PLSVS method, to screen active effects in mixed-level SSDs based on the variable importance in projection which is an important concept in the partial least-squares regression. Simulation studies show that this procedure is effective.