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 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.     

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.     

Marginally oversaturated designs

A special class of supersaturated design, called marginally over saturated design (MOSD), in which the number of variables under investigation (k) is only slightly larger than the number of experimental runs (n), is presented. Several optimality criteria for supersaturated designs are discussed. It is shown that the resolution rank criterion is most appropriate for screening situations. The construction method builds on two major theorems which provide an efficient way to evaluate resolution rank. Examples are given for the cases n=8, 12, 16, and 20. Potential extensions for future work are discussed.     

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.     

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.     

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.     

Optimal k-circulant supersaturated designs

Optimal k-circulant supersaturated designs have been constructed in literature using computer intensive methods. A systematic method of construction for multi-level experiments based on balanced incomplete block designs is presented in this paper. The method is also applicable to two-level experiments. Illustrative examples are also given.     

A general construction of E(fNOD)-optimal multi-level supersaturated designs

The present paper deals with E(f_NOD)-optimal multi-level supersaturated designs. We present a new technique for the construction of supplementary difference sets. Based on the new supplementary difference sets, we also provide E(f_NOD)-optimal multi-level supersaturated designs with a large number of columns when compared with other designs. Moreover, these designs retain the equal occurrence property.     

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.     

A three-stage variable selection method for supersaturated designs

A supersaturated design (SSD) is a design whose run size is not enough for estimating all main effects. Such a design is commonly used in screening experiments to screen active effects based on the effect sparsity principle. Traditional approaches, such as the ordinary stepwise regression and the best subset variable selection, may not be appropriate in this situation. In this article, a new variable selection method is proposed based on the idea of staged dimensionality reduction. Simulations and several real data studies indicate that the newly proposed method is more effective than the existing data analysis methods.     

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.     

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.     

A method for analyzing supersaturated designs inspired by control charts

The identification of active effects in supersaturated designs (SSDs) constitutes a problem of considerable interest to both scientists and engineers. The complicated structure of the design matrix renders the analysis of such designs a complicated issue. Although several methods have been proposed so far, a solution to the problem beyond one or two active factors seems to be inadequate. This article presents a heuristic approach for analyzing SSDs using the cumulative sum control chart (CUSUM) under a sure independence screening approach. Simulations are used to investigate the performance of the method comparing the proposed method with other well-known methods from the literature. The results establish the powerfulness of the proposed methodology.     

A comparison of test statistics for the recovery of rapid growth‐based enumeration tests

This paper considers five test statistics for comparing the recovery of a rapid growth‐based enumeration test with respect to the compendial microbiological method using a specific nonserial dilution experiment. The finite sample distributions of these test statistics are unknown, because they are functions of correlated count data. A simulation study is conducted to investigate the type I and type II error rates. For a balanced experimental design, the likelihood ratio test and the main effects analysis of variance (ANOVA) test for microbiological methods demonstrated nominal values for the type I error rate and provided the highest power compared with a test on weighted averages and two other ANOVA tests. The likelihood ratio test is preferred because it can also be used for unbalanced designs. It is demonstrated that an increase in power can only be achieved by an increase in the spiked number of organisms used in the experiment. The power is surprisingly not affected by the number of dilutions or the number of test samples. A real case study is provided to illustrate the theory. Copyright © 2013 John Wiley & Sons, Ltd.     

A method for constructing supersaturated designs and its Es2 optimality

A lower bound for the Es² value of an arbitrary supersaturated design is derived. A general method for constructing supersaturated designs is proposed and shown to produce designs with n runs and m = k(n — 1) factors that achieve the lower bound for Es² and are thus optimal with respect to the Es² criterion. Within the class of designs given by the construction method, further discrimination can be made by minimizing the pairwise correlations and using the generalized D and A criteria proposed by Wu (1993). Efficient designs of 12, 16, 20 and 24 runs are constructed by following this approach.     

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.     

Repeated measurements designs - a review

Any experiment in which one or more of the experimental units is used more than once is called a repeated measurements experiment. The associated design of a repeated measurements experiment is referred to as a repeated measurements design. This review covers some known results on repeated measurements designs. Emphasis is placed on the impact of optimal design theory. Some construction methods for these designs are presented. Hedayat and Afsarinejad (1975) has an extensive bibliography of earlier literature. A bibliography of these designs published after 1974 is provided at the end of this paper.     

Determining the minimum duration of treatment in tuberculosis: An order restricted non-inferiority trial design

Tuberculosis (TB) is one of the biggest killers among infectious diseases worldwide. Together with the identification of drugs that can provide benefits to patients, the challenge in TB is also the optimisation of the duration of these treatments. While conventional duration of treatment in TB is 6 months, there is evidence that shorter durations might be as effective but could be associated with fewer side effects and may be associated with better adherence. Based on a recent proposal of an adaptive order-restricted superiority design that employs the ordering assumptions within various duration of the same drug, we propose a non-inferiority (typically used in TB trials) adaptive design that effectively uses the order assumption. Together with the general construction of the hypothesis testing and expression for type I and type II errors, we focus on how the novel design was proposed for a TB trial concept. We consider a number of practical aspects such as choice of the design parameters, randomisation ratios, and timings of the interim analyses, and how these were discussed with the clinical team.     

Bayesian methods for mark-resighting surveys

We consider the problem of estimating the size of a closed population based on the results of a certain type of mark-resighting sampling design. The design is similar to the commonly used multiple capture-recapture design, yet in some cases economically more feasible and easy to use. Sampling is done by first tagging a number of randomly selected animals with visible markers and later randomly sighting them (for instance, for large animals by visually sampling from a helicopter) and counting the number of tagged animals. In this paper, we look at Bayesian methods for point and interval estimation of population size for this design. An example involving estimation of mountain sheep, a couple of simulated examples and simulation studies are given to demonstrate the advantages of the proposed procedure over the other available approximate procedures.